3.3.90 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^7 (d+e x)} \, dx\)
Optimal. Leaf size=498 \[ \frac {\left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}+\frac {\left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac {\left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 x^5} \]
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Rubi [A] time = 0.72, antiderivative size = 498, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used =
{849, 834, 806, 720, 724, 206} \begin {gather*} -\frac {\left (21 a^2 c d^2 e^4-105 a^3 e^6+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}+\frac {\left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}+\frac {\left (6 a^2 c^2 d^4 e^4-21 a^4 e^8+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (21 a^2 c d^2 e^4+21 a^3 e^6+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 x^5}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]
[Out]
((7*c^4*d^8 + 8*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 21*a^4*e^8)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2])/(512*a^4*d^5*e^4*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(6*d*x^6) -
((c/(a*e) - (3*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*x^5) + ((7*c^2*d^4 + 6*a*c*d^2*e^2
- 21*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*a^2*d^3*e^2*x^4) - ((35*c^3*d^6 + 33*a*c^2*d
^4*e^2 + 21*a^2*c*d^2*e^4 - 105*a^3*e^6)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*a^3*d^4*e^3*x^3)
- ((c*d^2 - a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*ArcTanh[(2*a*d*e + (c*d^2
+ a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(1024*a^(9/2)*d^(11/2)*e
^(9/2))
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 720
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]
Rule 724
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 806
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]
Rule 834
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])
Rule 849
Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] && !IntegerQ[p] && ( !IntegerQ[n] || !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx &=\int \frac {(a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^7} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\int \frac {\left (-\frac {3}{2} a e \left (c d^2-3 a e^2\right )+3 a c d e^2 x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx}{6 a d e}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\int \frac {\left (-\frac {3}{4} a e \left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right )-3 a c d e^2 \left (c d^2-3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx}{30 a^2 d^2 e^2}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac {\int \frac {\left (-\frac {3}{8} a e \left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right )-\frac {3}{4} a c d e^2 \left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4} \, dx}{120 a^3 d^3 e^3}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac {\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac {\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{128 a^3 d^4 e^3}\\ &=\frac {\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac {\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 a^4 d^5 e^4}\\ &=\frac {\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac {\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 a^4 d^5 e^4}\\ &=\frac {\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac {\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac {\left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 380, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {16 x^2 (d+e x) \left (63 a^2 e^4+54 a c d^2 e^2+35 c^2 d^4\right ) (a e+c d x)^2}{a^2 d^2 e^2}+\frac {5 x^3 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {d+e x} \sqrt {a e+c d x} \left (a^2 e^2 \left (-8 d^2-2 d e x+3 e^2 x^2\right )-2 a c d^2 e x (7 d+4 e x)-3 c^2 d^4 x^2\right )+3 x^3 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )\right )}{a^{7/2} d^{9/2} e^{7/2} \sqrt {d+e x} \sqrt {a e+c d x}}-\frac {128 x (d+e x) \left (9 a e^2+7 c d^2\right ) (a e+c d x)^2}{a d e}+1280 (d+e x) (a e+c d x)^2\right )}{7680 a d e x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]
[Out]
-1/7680*(Sqrt[(a*e + c*d*x)*(d + e*x)]*(1280*(a*e + c*d*x)^2*(d + e*x) - (128*(7*c*d^2 + 9*a*e^2)*x*(a*e + c*d
*x)^2*(d + e*x))/(a*d*e) + (16*(35*c^2*d^4 + 54*a*c*d^2*e^2 + 63*a^2*e^4)*x^2*(a*e + c*d*x)^2*(d + e*x))/(a^2*
d^2*e^2) + (5*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*x^3*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt
[a*e + c*d*x]*Sqrt[d + e*x]*(-3*c^2*d^4*x^2 - 2*a*c*d^2*e*x*(7*d + 4*e*x) + a^2*e^2*(-8*d^2 - 2*d*e*x + 3*e^2*
x^2)) + 3*(c*d^2 - a*e^2)^3*x^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(7/2
)*d^(9/2)*e^(7/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(a*d*e*x^6)
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IntegrateAlgebraic [F] time = 180.88, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x]
[Out]
$Aborted
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fricas [A] time = 55.73, size = 1072, normalized size = 2.15 \begin {gather*} \left [-\frac {15 \, {\left (7 \, c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} - 3 \, a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} e^{8} + 42 \, a^{5} c d^{2} e^{10} - 21 \, a^{6} e^{12}\right )} \sqrt {a d e} x^{6} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (1280 \, a^{6} d^{6} e^{6} - {\left (105 \, a c^{5} d^{11} e - 55 \, a^{2} c^{4} d^{9} e^{3} - 54 \, a^{3} c^{3} d^{7} e^{5} - 78 \, a^{4} c^{2} d^{5} e^{7} + 525 \, a^{5} c d^{3} e^{9} - 315 \, a^{6} d e^{11}\right )} x^{5} + 2 \, {\left (35 \, a^{2} c^{4} d^{10} e^{2} - 16 \, a^{3} c^{3} d^{8} e^{4} - 18 \, a^{4} c^{2} d^{6} e^{6} + 168 \, a^{5} c d^{4} e^{8} - 105 \, a^{6} d^{2} e^{10}\right )} x^{4} - 8 \, {\left (7 \, a^{3} c^{3} d^{9} e^{3} - 3 \, a^{4} c^{2} d^{7} e^{5} + 33 \, a^{5} c d^{5} e^{7} - 21 \, a^{6} d^{3} e^{9}\right )} x^{3} + 16 \, {\left (3 \, a^{4} c^{2} d^{8} e^{4} + 14 \, a^{5} c d^{6} e^{6} - 9 \, a^{6} d^{4} e^{8}\right )} x^{2} + 128 \, {\left (13 \, a^{5} c d^{7} e^{5} + a^{6} d^{5} e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{30720 \, a^{5} d^{6} e^{5} x^{6}}, \frac {15 \, {\left (7 \, c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} - 3 \, a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} e^{8} + 42 \, a^{5} c d^{2} e^{10} - 21 \, a^{6} e^{12}\right )} \sqrt {-a d e} x^{6} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (1280 \, a^{6} d^{6} e^{6} - {\left (105 \, a c^{5} d^{11} e - 55 \, a^{2} c^{4} d^{9} e^{3} - 54 \, a^{3} c^{3} d^{7} e^{5} - 78 \, a^{4} c^{2} d^{5} e^{7} + 525 \, a^{5} c d^{3} e^{9} - 315 \, a^{6} d e^{11}\right )} x^{5} + 2 \, {\left (35 \, a^{2} c^{4} d^{10} e^{2} - 16 \, a^{3} c^{3} d^{8} e^{4} - 18 \, a^{4} c^{2} d^{6} e^{6} + 168 \, a^{5} c d^{4} e^{8} - 105 \, a^{6} d^{2} e^{10}\right )} x^{4} - 8 \, {\left (7 \, a^{3} c^{3} d^{9} e^{3} - 3 \, a^{4} c^{2} d^{7} e^{5} + 33 \, a^{5} c d^{5} e^{7} - 21 \, a^{6} d^{3} e^{9}\right )} x^{3} + 16 \, {\left (3 \, a^{4} c^{2} d^{8} e^{4} + 14 \, a^{5} c d^{6} e^{6} - 9 \, a^{6} d^{4} e^{8}\right )} x^{2} + 128 \, {\left (13 \, a^{5} c d^{7} e^{5} + a^{6} d^{5} e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15360 \, a^{5} d^{6} e^{5} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm="fricas")
[Out]
[-1/30720*(15*(7*c^6*d^12 - 6*a*c^5*d^10*e^2 - 3*a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 42
*a^5*c*d^2*e^10 - 21*a^6*e^12)*sqrt(a*d*e)*x^6*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 +
4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d
*e^3)*x)/x^2) + 4*(1280*a^6*d^6*e^6 - (105*a*c^5*d^11*e - 55*a^2*c^4*d^9*e^3 - 54*a^3*c^3*d^7*e^5 - 78*a^4*c^2
*d^5*e^7 + 525*a^5*c*d^3*e^9 - 315*a^6*d*e^11)*x^5 + 2*(35*a^2*c^4*d^10*e^2 - 16*a^3*c^3*d^8*e^4 - 18*a^4*c^2*
d^6*e^6 + 168*a^5*c*d^4*e^8 - 105*a^6*d^2*e^10)*x^4 - 8*(7*a^3*c^3*d^9*e^3 - 3*a^4*c^2*d^7*e^5 + 33*a^5*c*d^5*
e^7 - 21*a^6*d^3*e^9)*x^3 + 16*(3*a^4*c^2*d^8*e^4 + 14*a^5*c*d^6*e^6 - 9*a^6*d^4*e^8)*x^2 + 128*(13*a^5*c*d^7*
e^5 + a^6*d^5*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^5*d^6*e^5*x^6), 1/15360*(15*(7*c^6*d^12
- 6*a*c^5*d^10*e^2 - 3*a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 42*a^5*c*d^2*e^10 - 21*a^6*e
^12)*sqrt(-a*d*e)*x^6*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqr
t(-a*d*e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)) - 2*(1280*a^6*d^6*e^6 - (105*a*c^5*d^11
*e - 55*a^2*c^4*d^9*e^3 - 54*a^3*c^3*d^7*e^5 - 78*a^4*c^2*d^5*e^7 + 525*a^5*c*d^3*e^9 - 315*a^6*d*e^11)*x^5 +
2*(35*a^2*c^4*d^10*e^2 - 16*a^3*c^3*d^8*e^4 - 18*a^4*c^2*d^6*e^6 + 168*a^5*c*d^4*e^8 - 105*a^6*d^2*e^10)*x^4 -
8*(7*a^3*c^3*d^9*e^3 - 3*a^4*c^2*d^7*e^5 + 33*a^5*c*d^5*e^7 - 21*a^6*d^3*e^9)*x^3 + 16*(3*a^4*c^2*d^8*e^4 + 1
4*a^5*c*d^6*e^6 - 9*a^6*d^4*e^8)*x^2 + 128*(13*a^5*c*d^7*e^5 + a^6*d^5*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x))/(a^5*d^6*e^5*x^6)]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-2*exp(1)^5*a^2*exp(2)^2+4*exp(1)^7*
a^2*exp(2)-2*exp(1)^9*a^2)/2/d^5/sqrt(-a*d*exp(1)^3+a*d*exp(1)*exp(2))*atan((-d*sqrt(c*d*exp(1))+(sqrt(a*d*exp
(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*exp(1))/sqrt(-a*d*exp(1)^3+a*d*exp(1)*exp(2)))+(7*a
^6*exp(2)^6+12*exp(1)^2*a^6*exp(2)^5+24*exp(1)^4*a^6*exp(2)^4+64*exp(1)^6*a^6*exp(2)^3+384*exp(1)^8*a^6*exp(2)
^2-1536*exp(1)^10*a^6*exp(2)+1024*exp(1)^12*a^6+42*c*d^2*a^5*exp(2)^5+105*c^2*d^4*a^4*exp(2)^4-120*c^2*d^4*exp
(1)^2*a^4*exp(2)^3+140*c^3*d^6*a^3*exp(2)^3-240*c^3*d^6*exp(1)^2*a^3*exp(2)^2+96*c^3*d^6*exp(1)^4*a^3*exp(2)+1
05*c^4*d^8*a^2*exp(2)^2-180*c^4*d^8*exp(1)^2*a^2*exp(2)+72*c^4*d^8*exp(1)^4*a^2+42*c^5*d^10*a*exp(2)-48*c^5*d^
10*exp(1)^2*a+7*c^6*d^12)/512/d^5/exp(1)^4/a^4/2/sqrt(-a*d*exp(1))*atan((sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*
d*x^2*exp(1))-sqrt(c*d*exp(1))*x)/sqrt(-a*d*exp(1)))-(-105*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))
-sqrt(c*d*exp(1))*x)^11*a^6*exp(2)^6-180*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d
*exp(1))*x)^11*a^6*exp(2)^5-360*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*
x)^11*a^6*exp(2)^4-960*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^6
*exp(2)^3+9600*exp(1)^8*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^6*exp(2)^
2-7680*exp(1)^10*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^6*exp(2)-630*c*d
^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^5*exp(2)^5-1575*c^2*d^4*(sqrt(
a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^4*exp(2)^4+1800*c^2*d^4*exp(1)^2*(sqrt(
a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^4*exp(2)^3-2100*c^3*d^6*(sqrt(a*d*exp(1
)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^3*exp(2)^3+3600*c^3*d^6*exp(1)^2*(sqrt(a*d*exp(1
)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^3*exp(2)^2-1440*c^3*d^6*exp(1)^4*(sqrt(a*d*exp(1
)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^3*exp(2)-1575*c^4*d^8*(sqrt(a*d*exp(1)+a*x*exp(2
)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^2*exp(2)^2+2700*c^4*d^8*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2
)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^2*exp(2)-1080*c^4*d^8*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+
c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^11*a^2-630*c^5*d^10*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*ex
p(1))-sqrt(c*d*exp(1))*x)^11*a*exp(2)+720*c^5*d^10*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1)
)-sqrt(c*d*exp(1))*x)^11*a-105*c^6*d^12*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x
)^11+15360*d*exp(1)^7*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)
^10*a^6*exp(2)^2-30720*d*exp(1)^9*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*
d*exp(1))*x)^10*a^6*exp(2)+15360*d*exp(1)^11*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(
1))-sqrt(c*d*exp(1))*x)^10*a^6+595*d*exp(1)*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1
))*x)^9*a^7*exp(2)^6+1020*d*exp(1)^3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9
*a^7*exp(2)^5+2040*d*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^7*ex
p(2)^4+320*d*exp(1)^7*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^7*exp(2)^3-4
4160*d*exp(1)^9*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^7*exp(2)^2+38400*d
*exp(1)^11*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^7*exp(2)+3570*c*d^3*exp
(1)*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^6*exp(2)^5+15360*c*d^3*exp(1)^
7*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^6*exp(2)^2-30720*c*d^3*exp(1)^9*
(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^6*exp(2)+15360*c*d^3*exp(1)^11*(sq
rt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^6+8925*c^2*d^5*exp(1)*(sqrt(a*d*exp(1
)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^5*exp(2)^4-10200*c^2*d^5*exp(1)^3*(sqrt(a*d*exp(1
)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^5*exp(2)^3+11900*c^3*d^7*exp(1)*(sqrt(a*d*exp(1)+
a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^4*exp(2)^3-20400*c^3*d^7*exp(1)^3*(sqrt(a*d*exp(1)+
a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^4*exp(2)^2+8160*c^3*d^7*exp(1)^5*(sqrt(a*d*exp(1)+a
*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^4*exp(2)+8925*c^4*d^9*exp(1)*(sqrt(a*d*exp(1)+a*x*ex
p(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^3*exp(2)^2-15300*c^4*d^9*exp(1)^3*(sqrt(a*d*exp(1)+a*x*ex
p(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^3*exp(2)+6120*c^4*d^9*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2
)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^3+3570*c^5*d^11*exp(1)*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c
*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^2*exp(2)-4080*c^5*d^11*exp(1)^3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d
*x^2*exp(1))-sqrt(c*d*exp(1))*x)^9*a^2+595*c^6*d^13*exp(1)*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))
-sqrt(c*d*exp(1))*x)^9*a-15360*d^2*exp(1)^6*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1
))-sqrt(c*d*exp(1))*x)^8*a^7*exp(2)^3-46080*d^2*exp(1)^8*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+
c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^8*a^7*exp(2)^2+138240*d^2*exp(1)^10*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*
exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^8*a^7*exp(2)-76800*d^2*exp(1)^12*sqrt(c*d*exp(1))*(sqrt(a*d
*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^8*a^7+15360*c*d^4*exp(1)^6*sqrt(c*d*exp(1))*(sq
rt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^8*a^6*exp(2)^2-30720*c*d^4*exp(1)^8*sqrt(
c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^8*a^6*exp(2)+15360*c*d^4*e
xp(1)^10*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^8*a^6-30720*
c^3*d^8*exp(1)^6*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^8*a^
4-1386*d^2*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^8*exp(2)^6-237
6*d^2*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^8*exp(2)^5-1680*d^2
*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^8*exp(2)^4+5760*d^2*exp(
1)^8*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^8*exp(2)^3+80640*d^2*exp(1)^1
0*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^8*exp(2)^2-76800*d^2*exp(1)^12*(
sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^8*exp(2)-8316*c*d^4*exp(1)^2*(sqrt(
a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^7*exp(2)^5-30720*c*d^4*exp(1)^6*(sqrt(a*
d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^7*exp(2)^3+92160*c*d^4*exp(1)^10*(sqrt(a*d
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exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^6*exp(2)^3+15360*c^2*d^6*exp(1)^6*(sqrt(a*d*exp(1)+a*x*
exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^6*exp(2)^2-30720*c^2*d^6*exp(1)^8*(sqrt(a*d*exp(1)+a*x*
exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^6*exp(2)+15360*c^2*d^6*exp(1)^10*(sqrt(a*d*exp(1)+a*x*e
xp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^6-27720*c^3*d^8*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d
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d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^5*exp(2)-20790*c^4*d^10*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d
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x^2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^3*exp(2)+9504*c^5*d^12*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^
2*exp(1))-sqrt(c*d*exp(1))*x)^7*a^3-1386*c^6*d^14*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))
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(1))-sqrt(c*d*exp(1))*x)^6*a^8*exp(2)^4+30720*d^3*exp(1)^7*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*
x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^8*exp(2)^3+46080*d^3*exp(1)^9*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*
exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^8*exp(2)^2-245760*d^3*exp(1)^11*sqrt(c*d*exp(1))*(sqrt(
a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^8*exp(2)+153600*d^3*exp(1)^13*sqrt(c*d*e
xp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^8-51200*c*d^5*exp(1)^5*sqrt
(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^7*exp(2)^3+61440*c*d^
5*exp(1)^7*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^7*exp(
2)^2+30720*c*d^5*exp(1)^9*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1)
)*x)^6*a^7*exp(2)-40960*c*d^5*exp(1)^11*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-s
qrt(c*d*exp(1))*x)^6*a^7-245760*c^2*d^7*exp(1)^5*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*
exp(1))-sqrt(c*d*exp(1))*x)^6*a^6*exp(2)^2-30720*c^2*d^7*exp(1)^7*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)
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p(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^6-276480*c^3*d^9*exp(1)^5*sqrt(c*d*exp(1))*(sq
rt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^5*exp(2)-61440*c^3*d^9*exp(1)^7*sqrt(
c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^5-97280*c^4*d^11*exp(1
)^5*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^6*a^4+1686*d^3*ex
p(1)^3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^9*exp(2)^6+696*d^3*exp(1)^5
*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^9*exp(2)^5-1680*d^3*exp(1)^7*(sqr
t(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^9*exp(2)^4-9600*d^3*exp(1)^9*(sqrt(a*d
*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^9*exp(2)^3-72960*d^3*exp(1)^11*(sqrt(a*d*ex
p(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^9*exp(2)^2+76800*d^3*exp(1)^13*(sqrt(a*d*exp(1
)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^9*exp(2)+10116*c*d^5*exp(1)^3*(sqrt(a*d*exp(1)+a*
x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^8*exp(2)^5+46080*c*d^5*exp(1)^5*(sqrt(a*d*exp(1)+a*x*
exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^8*exp(2)^4-46080*c*d^5*exp(1)^9*(sqrt(a*d*exp(1)+a*x*ex
p(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^8*exp(2)^2-92160*c*d^5*exp(1)^11*(sqrt(a*d*exp(1)+a*x*exp
(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^8*exp(2)+92160*c*d^5*exp(1)^13*(sqrt(a*d*exp(1)+a*x*exp(2)
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c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^7*exp(2)^4+177360*c^2*d^7*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x
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x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^7*exp(2)^2-46080*c^2*d^7*exp(1)^9*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*
x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^7*exp(2)-15360*c^2*d^7*exp(1)^11*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x
+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^7+33720*c^3*d^9*exp(1)^3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*
exp(1))-sqrt(c*d*exp(1))*x)^5*a^6*exp(2)^3+262560*c^3*d^9*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2
*exp(1))-sqrt(c*d*exp(1))*x)^5*a^6*exp(2)^2+269760*c^3*d^9*exp(1)^7*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^
2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^6*exp(2)+15360*c^3*d^9*exp(1)^9*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*
exp(1))-sqrt(c*d*exp(1))*x)^5*a^6+25290*c^4*d^11*exp(1)^3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-
sqrt(c*d*exp(1))*x)^5*a^5*exp(2)^2+173880*c^4*d^11*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1)
)-sqrt(c*d*exp(1))*x)^5*a^5*exp(2)+133200*c^4*d^11*exp(1)^7*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1)
)-sqrt(c*d*exp(1))*x)^5*a^5+10116*c^5*d^13*exp(1)^3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c
*d*exp(1))*x)^5*a^4*exp(2)+43296*c^5*d^13*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*
d*exp(1))*x)^5*a^4+1686*c^6*d^15*exp(1)^3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))
*x)^5*a^3-15360*d^4*exp(1)^4*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp
(1))*x)^4*a^9*exp(2)^5-15360*d^4*exp(1)^6*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))
-sqrt(c*d*exp(1))*x)^4*a^9*exp(2)^4-15360*d^4*exp(1)^8*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*
d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^9*exp(2)^3-15360*d^4*exp(1)^10*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp
(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^9*exp(2)^2+215040*d^4*exp(1)^12*sqrt(c*d*exp(1))*(sqrt(a*d
*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^9*exp(2)-153600*d^4*exp(1)^14*sqrt(c*d*exp(
1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^9-76800*c*d^6*exp(1)^4*sqrt(c*
d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^8*exp(2)^4-122880*c*d^6*
exp(1)^6*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^8*exp(2)
^3-46080*c*d^6*exp(1)^8*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*
x)^4*a^8*exp(2)^2+30720*c*d^6*exp(1)^10*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-s
qrt(c*d*exp(1))*x)^4*a^8*exp(2)+30720*c*d^6*exp(1)^12*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d
*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^8-153600*c^2*d^8*exp(1)^4*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*
d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^7*exp(2)^3-276480*c^2*d^8*exp(1)^6*sqrt(c*d*exp(1))*(sqrt(a*d*ex
p(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^7*exp(2)^2-138240*c^2*d^8*exp(1)^8*sqrt(c*d*ex
p(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^7*exp(2)+15360*c^2*d^8*exp(1
)^10*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^7-153600*c^3
*d^10*exp(1)^4*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^6*
exp(2)^2-245760*c^3*d^10*exp(1)^6*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*
d*exp(1))*x)^4*a^6*exp(2)-107520*c^3*d^10*exp(1)^8*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^
2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^6-76800*c^4*d^12*exp(1)^4*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2
*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^5*exp(2)-76800*c^4*d^12*exp(1)^6*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+
a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^5-15360*c^5*d^14*exp(1)^4*sqrt(c*d*exp(1))*(sqrt(a*
d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^4+595*d^4*exp(1)^4*(sqrt(a*d*exp(1)+a*x*ex
p(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^10*exp(2)^6+1020*d^4*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)
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^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^10*exp(2)^4+5440*d^4*exp(1)^10*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*
x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^10*exp(2)^3+32640*d^4*exp(1)^12*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+
c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^10*exp(2)^2-38400*d^4*exp(1)^14*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*
d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^10*exp(2)+3570*c*d^6*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^
2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^9*exp(2)^5+30720*c*d^6*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*
exp(1))-sqrt(c*d*exp(1))*x)^3*a^9*exp(2)^4+30720*c*d^6*exp(1)^8*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*ex
p(1))-sqrt(c*d*exp(1))*x)^3*a^9*exp(2)^3+30720*c*d^6*exp(1)^10*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp
(1))-sqrt(c*d*exp(1))*x)^3*a^9*exp(2)^2+30720*c*d^6*exp(1)^12*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(
1))-sqrt(c*d*exp(1))*x)^3*a^9*exp(2)-61440*c*d^6*exp(1)^14*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))
-sqrt(c*d*exp(1))*x)^3*a^9+8925*c^2*d^8*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*
exp(1))*x)^3*a^8*exp(2)^4+112680*c^2*d^8*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d
*exp(1))*x)^3*a^8*exp(2)^3+138240*c^2*d^8*exp(1)^8*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*
d*exp(1))*x)^3*a^8*exp(2)^2+61440*c^2*d^8*exp(1)^10*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c
*d*exp(1))*x)^3*a^8*exp(2)-15360*c^2*d^8*exp(1)^12*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*
d*exp(1))*x)^3*a^8+11900*c^3*d^10*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1)
)*x)^3*a^7*exp(2)^3+163920*c^3*d^10*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(
1))*x)^3*a^7*exp(2)^2+192480*c^3*d^10*exp(1)^8*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*ex
p(1))*x)^3*a^7*exp(2)+51200*c^3*d^10*exp(1)^10*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*ex
p(1))*x)^3*a^7+8925*c^4*d^12*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^
3*a^6*exp(2)^2+107580*c^4*d^12*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x
)^3*a^6*exp(2)+82920*c^4*d^12*exp(1)^8*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)
^3*a^6+3570*c^5*d^14*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^5*ex
p(2)+26640*c^5*d^14*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^5+595
*c^6*d^16*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^4-92160*d^5*exp
(1)^13*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^10*exp(2)+
76800*d^5*exp(1)^15*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2
*a^10-30720*c*d^7*exp(1)^7*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1
))*x)^2*a^9*exp(2)^3-30720*c*d^7*exp(1)^9*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))
-sqrt(c*d*exp(1))*x)^2*a^9*exp(2)^2-30720*c*d^7*exp(1)^11*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x
+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^9*exp(2)-92160*c^2*d^9*exp(1)^7*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x
*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^8*exp(2)^2-73728*c^2*d^9*exp(1)^9*sqrt(c*d*exp(1))*(sq
rt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^8*exp(2)-27648*c^2*d^9*exp(1)^11*sqrt
(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^8-92160*c^3*d^11*exp(
1)^7*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^7*exp(2)-430
08*c^3*d^11*exp(1)^9*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^
2*a^7-30720*c^4*d^13*exp(1)^7*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*ex
p(1))*x)^2*a^6-105*d^5*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^11*e
xp(2)^6-180*d^5*exp(1)^7*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^11*exp(2)^5
-360*d^5*exp(1)^9*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^11*exp(2)^4-960*d^
5*exp(1)^11*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^11*exp(2)^3-5760*d^5*exp
(1)^13*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^11*exp(2)^2+7680*d^5*exp(1)^1
5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^11*exp(2)-630*c*d^7*exp(1)^5*(sqrt
(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^10*exp(2)^5+15360*c*d^7*exp(1)^15*(sqrt(a
*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^10-1575*c^2*d^9*exp(1)^5*(sqrt(a*d*exp(1)+a
*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^9*exp(2)^4+1800*c^2*d^9*exp(1)^7*(sqrt(a*d*exp(1)+a*x*
exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^9*exp(2)^3+15360*c^2*d^9*exp(1)^9*(sqrt(a*d*exp(1)+a*x*ex
p(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^9*exp(2)^2+15360*c^2*d^9*exp(1)^11*(sqrt(a*d*exp(1)+a*x*exp
(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^9*exp(2)+15360*c^2*d^9*exp(1)^13*(sqrt(a*d*exp(1)+a*x*exp(2)
+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^9-2100*c^3*d^11*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*
d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^8*exp(2)^3+3600*c^3*d^11*exp(1)^7*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*
x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^8*exp(2)^2+29280*c^3*d^11*exp(1)^9*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x
^2*exp(1))-sqrt(c*d*exp(1))*x)*a^8*exp(2)+15360*c^3*d^11*exp(1)^11*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2
*exp(1))-sqrt(c*d*exp(1))*x)*a^8-1575*c^4*d^13*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sq
rt(c*d*exp(1))*x)*a^7*exp(2)^2+2700*c^4*d^13*exp(1)^7*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt
(c*d*exp(1))*x)*a^7*exp(2)+14280*c^4*d^13*exp(1)^9*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*
d*exp(1))*x)*a^7-630*c^5*d^15*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)
*a^6*exp(2)+720*c^5*d^15*exp(1)^7*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^6-
105*c^6*d^17*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^5+15360*d^6*ex
p(1)^14*sqrt(c*d*exp(1))*a^11*exp(2)-15360*d^6*exp(1)^16*sqrt(c*d*exp(1))*a^11-5120*c*d^8*exp(1)^14*sqrt(c*d*e
xp(1))*a^10-3072*c^2*d^10*exp(1)^10*sqrt(c*d*exp(1))*a^9*exp(2)-3072*c^2*d^10*exp(1)^12*sqrt(c*d*exp(1))*a^9-3
072*c^3*d^12*exp(1)^10*sqrt(c*d*exp(1))*a^8)/15360/d^5/exp(1)^4/a^4/((sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x
^2*exp(1))-sqrt(c*d*exp(1))*x)^2-d*exp(1)*a)^6)
________________________________________________________________________________________
maple [B] time = 0.04, size = 3387, normalized size = 6.80 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)/x^7/(e*x+d),x)
[Out]
1/8/d^4*e^5*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+19/60/d^3/a/x^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^
(5/2)+1/64*e/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^3-23/96/d/a^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/
2)*c^3-21/512/d^6*e^7*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-1/8/d^4*e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^
(1/2)*c-15/512/d^2*e^3/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^2-703/1536/d^3*e^2/a^2*(c*d*e*x^2+a*d*e+(a*
e^2+c*d^2)*x)^(3/2)*c^2-491/768/d^6*e^3/a/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)+257/768/d^3/a^3/x*(c*d*e
*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2+107/192/d^5*e^2/a/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)-1/3/d^7*e^
6*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)-533/1536/d^7*e^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+1/4/d
^7*e^8*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+1/8/d^8*e^9*a^2/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-3
/16/d^6*e^9*a^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(
1/2))/(c*d*e)^(1/2)+1/4/d^5*e^6*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-1/16/d^2*e^5*c^2*ln((1/2*a*e
^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+7/1536*
d^5/a^6/e^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^6+7/512*d^6/a^5/e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1
/2)*c^6+7/256*d/a^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^4-1/6/d^2/a/e/x^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^
2)*x)^(5/2)-7/96/a^3/e^3/x^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2-1/4/d^7*e^8*a*((x+d/e)^2*c*d*e+(a*e^2
-c*d^2)*(x+d/e))^(1/2)*x-1/8/d^8*e^9*a^2/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/12/e^3/a^4/x^2*(c*d
*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^3-109/768/e/a^4*c^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+3/1024*d^3
/a^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^4
+65/192/d^3/a^2/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c-5/384/e^4*d^3/a^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)
*x)^(3/2)*c^5-43/96*e/d^4/a/x^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)+13/512/e*d^2/a^3*(c*d*e*x^2+a*d*e+(a*e
^2+c*d^2)*x)^(1/2)*c^4+1/64/e^3*d^4/a^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^5-131/1536/e^2*d/a^4*(c*d*e*
x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^4+3/16/d^6*e^9*a^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^
2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)+1/16/d^2*e^5*c^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+
(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)+1045/1536/d^7*e^4/a/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)
^(5/2)-235/384/d^5*e^4/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c-149/512/d^5*e^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d
^2)*x)^(1/2)*x*c-91/384/e/d^2/a^3/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2+29/192/e^2/d/a^3/x^3*(c*d*e*
x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2+109/768/e^2/d/a^4/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^3+41/1536/e
^4*d/a^5/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^4+3/512/e^2*d^5/a^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^
2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^5-41/1536/e^3*d^2/a^5*c^5*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(3/2)*x+15/512/e^2*d^3/a^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^5-11/48/e/d^2/a^2/x^4*
(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c-1/16/d^8*e^11*a^3/c*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(
c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)-43/96/d^4*e/a^2/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5
/2)*c+1/256*d*e^2/a/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x
)^(1/2))/x)*c^3+3/256/d*e^2/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^3-257/768/d^2*e/a^3*c^3*(c*d*e*x^2
+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-877/1536/d^4*e^3/a^2*c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+877/1536/d^
5*e^2/a^2/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c-21/512/d^3*e^4/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)
*x*c^2-1045/1536/d^6*e^5*c/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-21/512/d^3*e^6*a/(a*d*e)^(1/2)*ln((2*a*
d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c-3/16/d^4*e^7*a*ln((c*d*e*x+1
/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*c+7/192*d/a^4/e^4/x^3
*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^3-7/1536*d^3/a^6/e^6/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^5-
7/768*d^2/a^5/e^5/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^4-7/1024*d^7/a^4/e^4/(a*d*e)^(1/2)*ln((2*a*d*e
+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^6+7/1536*d^4/a^6/e^5*c^6*(c*d*e
*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+7/60/d/a^2/e^2/x^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c+7/512*d^5/a^5
/e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^6+1/16/d^8*e^11*a^3/c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/
(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+3/16/d^4*e^7*a*c*ln((1/2*a*e^2-1/2*
c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+21/1024/d^5*e^
8*a^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)+15
/1024/d*e^4/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))
/x)*c^2
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x, algorithm="maxima")
[Out]
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^7), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^7\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)),x)
[Out]
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**7/(e*x+d),x)
[Out]
Timed out
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