3.5.56 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^7 (d+e x)} \, dx\) [456]

Optimal. Leaf size=498 \[ \frac {\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac {\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac {\left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}} \]

[Out]

-1/6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x^6-1/20*(c/a/e-3*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2
)/x^5+1/160*(-21*a^2*e^4+6*a*c*d^2*e^2+7*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a^2/d^3/e^2/x^4-1/96
0*(-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)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a^3/d^4/
e^3/x^3-1/1024*(-a*e^2+c*d^2)^3*(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)*arctanh(1/2*(2*a*d*e+
(a*e^2+c*d^2)*x)/a^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/a^(9/2)/d^(11/2)/e^(9/2)+1/5
12*(-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)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)/a^4/d^5/e^4/x^2

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Rubi [A]
time = 0.45, 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 = {863, 848, 820, 734, 738, 212} \begin {gather*} \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} \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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

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] /; Free
Q[{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 738

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 820

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)/(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[S
implify[m + 2*p + 3], 0]

Rule 848

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 863

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 ) \text {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 = 1.17, size = 402, normalized size = 0.81 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-105 c^5 d^{10} x^5+5 a c^4 d^8 e x^4 (14 d+11 e x)-2 a^2 c^3 d^6 e^2 x^3 \left (28 d^2+16 d e x-27 e^2 x^2\right )+6 a^3 c^2 d^4 e^3 x^2 \left (8 d^3+4 d^2 e x-6 d e^2 x^2+13 e^3 x^3\right )+a^4 c d^2 e^4 x \left (1664 d^4+224 d^3 e x-264 d^2 e^2 x^2+336 d e^3 x^3-525 e^4 x^4\right )+a^5 e^5 \left (1280 d^5+128 d^4 e x-144 d^3 e^2 x^2+168 d^2 e^3 x^3-210 d e^4 x^4+315 e^5 x^5\right )\right )}{x^6}-\frac {15 \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 {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 a^{9/2} d^{11/2} e^{9/2}} \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]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-105*c^5*d^10*x^5 + 5*a*c^4*d^8*e*x^4*(14*d + 11*e
*x) - 2*a^2*c^3*d^6*e^2*x^3*(28*d^2 + 16*d*e*x - 27*e^2*x^2) + 6*a^3*c^2*d^4*e^3*x^2*(8*d^3 + 4*d^2*e*x - 6*d*
e^2*x^2 + 13*e^3*x^3) + a^4*c*d^2*e^4*x*(1664*d^4 + 224*d^3*e*x - 264*d^2*e^2*x^2 + 336*d*e^3*x^3 - 525*e^4*x^
4) + a^5*e^5*(1280*d^5 + 128*d^4*e*x - 144*d^3*e^2*x^2 + 168*d^2*e^3*x^3 - 210*d*e^4*x^4 + 315*e^5*x^5)))/x^6)
- (15*(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[(Sqrt[a]*Sqrt[
e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(7680*a^(9/2)*d^(11/2)*e^(
9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(16882\) vs. \(2(460)=920\).
time = 0.09, size = 16883, normalized size = 33.90

method	result	size
default	\(\text {Expression too large to display}\)	\(16883\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^7/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((x*e + d)*x^7), x)

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Fricas [A]
time = 90.62, size = 1117, normalized size = 2.24 \begin {gather*} \left [-\frac {{\left (15 \, {\left (7 \, c^{6} d^{12} x^{6} - 6 \, a c^{5} d^{10} x^{6} e^{2} - 3 \, a^{2} c^{4} d^{8} x^{6} e^{4} - 4 \, a^{3} c^{3} d^{6} x^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} x^{6} e^{8} + 42 \, a^{5} c d^{2} x^{6} e^{10} - 21 \, a^{6} x^{6} e^{12}\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} + 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) - 4 \, {\left (105 \, a c^{5} d^{11} x^{5} e - 70 \, a^{2} c^{4} d^{10} x^{4} e^{2} - 315 \, a^{6} d x^{5} e^{11} + 210 \, a^{6} d^{2} x^{4} e^{10} + 21 \, {\left (25 \, a^{5} c d^{3} x^{5} - 8 \, a^{6} d^{3} x^{3}\right )} e^{9} - 48 \, {\left (7 \, a^{5} c d^{4} x^{4} - 3 \, a^{6} d^{4} x^{2}\right )} e^{8} - 2 \, {\left (39 \, a^{4} c^{2} d^{5} x^{5} - 132 \, a^{5} c d^{5} x^{3} + 64 \, a^{6} d^{5} x\right )} e^{7} + 4 \, {\left (9 \, a^{4} c^{2} d^{6} x^{4} - 56 \, a^{5} c d^{6} x^{2} - 320 \, a^{6} d^{6}\right )} e^{6} - 2 \, {\left (27 \, a^{3} c^{3} d^{7} x^{5} + 12 \, a^{4} c^{2} d^{7} x^{3} + 832 \, a^{5} c d^{7} x\right )} e^{5} + 16 \, {\left (2 \, a^{3} c^{3} d^{8} x^{4} - 3 \, a^{4} c^{2} d^{8} x^{2}\right )} e^{4} - {\left (55 \, a^{2} c^{4} d^{9} x^{5} - 56 \, a^{3} c^{3} d^{9} x^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-5\right )}}{30720 \, a^{5} d^{6} x^{6}}, \frac {{\left (15 \, {\left (7 \, c^{6} d^{12} x^{6} - 6 \, a c^{5} d^{10} x^{6} e^{2} - 3 \, a^{2} c^{4} d^{8} x^{6} e^{4} - 4 \, a^{3} c^{3} d^{6} x^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} x^{6} e^{8} + 42 \, a^{5} c d^{2} x^{6} e^{10} - 21 \, a^{6} x^{6} e^{12}\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (105 \, a c^{5} d^{11} x^{5} e - 70 \, a^{2} c^{4} d^{10} x^{4} e^{2} - 315 \, a^{6} d x^{5} e^{11} + 210 \, a^{6} d^{2} x^{4} e^{10} + 21 \, {\left (25 \, a^{5} c d^{3} x^{5} - 8 \, a^{6} d^{3} x^{3}\right )} e^{9} - 48 \, {\left (7 \, a^{5} c d^{4} x^{4} - 3 \, a^{6} d^{4} x^{2}\right )} e^{8} - 2 \, {\left (39 \, a^{4} c^{2} d^{5} x^{5} - 132 \, a^{5} c d^{5} x^{3} + 64 \, a^{6} d^{5} x\right )} e^{7} + 4 \, {\left (9 \, a^{4} c^{2} d^{6} x^{4} - 56 \, a^{5} c d^{6} x^{2} - 320 \, a^{6} d^{6}\right )} e^{6} - 2 \, {\left (27 \, a^{3} c^{3} d^{7} x^{5} + 12 \, a^{4} c^{2} d^{7} x^{3} + 832 \, a^{5} c d^{7} x\right )} e^{5} + 16 \, {\left (2 \, a^{3} c^{3} d^{8} x^{4} - 3 \, a^{4} c^{2} d^{8} x^{2}\right )} e^{4} - {\left (55 \, a^{2} c^{4} d^{9} x^{5} - 56 \, a^{3} c^{3} d^{9} x^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-5\right )}}{15360 \, a^{5} d^{6} 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*x^6 - 6*a*c^5*d^10*x^6*e^2 - 3*a^2*c^4*d^8*x^6*e^4 - 4*a^3*c^3*d^6*x^6*e^6 - 15*a^4*
c^2*d^4*x^6*e^8 + 42*a^5*c*d^2*x^6*e^10 - 21*a^6*x^6*e^12)*sqrt(a*d)*e^(1/2)*log((c^2*d^4*x^2 + 8*a*c*d^3*x*e
+ a^2*x^2*e^4 + 8*a^2*d*x*e^3 + 4*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sq
rt(a*d)*e^(1/2) + 2*(3*a*c*d^2*x^2 + 4*a^2*d^2)*e^2)/x^2) - 4*(105*a*c^5*d^11*x^5*e - 70*a^2*c^4*d^10*x^4*e^2
- 315*a^6*d*x^5*e^11 + 210*a^6*d^2*x^4*e^10 + 21*(25*a^5*c*d^3*x^5 - 8*a^6*d^3*x^3)*e^9 - 48*(7*a^5*c*d^4*x^4
- 3*a^6*d^4*x^2)*e^8 - 2*(39*a^4*c^2*d^5*x^5 - 132*a^5*c*d^5*x^3 + 64*a^6*d^5*x)*e^7 + 4*(9*a^4*c^2*d^6*x^4 -
56*a^5*c*d^6*x^2 - 320*a^6*d^6)*e^6 - 2*(27*a^3*c^3*d^7*x^5 + 12*a^4*c^2*d^7*x^3 + 832*a^5*c*d^7*x)*e^5 + 16*(
2*a^3*c^3*d^8*x^4 - 3*a^4*c^2*d^8*x^2)*e^4 - (55*a^2*c^4*d^9*x^5 - 56*a^3*c^3*d^9*x^3)*e^3)*sqrt(c*d^2*x + a*x
*e^2 + (c*d*x^2 + a*d)*e))*e^(-5)/(a^5*d^6*x^6), 1/15360*(15*(7*c^6*d^12*x^6 - 6*a*c^5*d^10*x^6*e^2 - 3*a^2*c^
4*d^8*x^6*e^4 - 4*a^3*c^3*d^6*x^6*e^6 - 15*a^4*c^2*d^4*x^6*e^8 + 42*a^5*c*d^2*x^6*e^10 - 21*a^6*x^6*e^12)*sqrt
(-a*d*e)*arctan(1/2*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(-a*d*e)/(a*
c*d^3*x*e + a^2*d*x*e^3 + (a*c*d^2*x^2 + a^2*d^2)*e^2)) + 2*(105*a*c^5*d^11*x^5*e - 70*a^2*c^4*d^10*x^4*e^2 -
315*a^6*d*x^5*e^11 + 210*a^6*d^2*x^4*e^10 + 21*(25*a^5*c*d^3*x^5 - 8*a^6*d^3*x^3)*e^9 - 48*(7*a^5*c*d^4*x^4 -
3*a^6*d^4*x^2)*e^8 - 2*(39*a^4*c^2*d^5*x^5 - 132*a^5*c*d^5*x^3 + 64*a^6*d^5*x)*e^7 + 4*(9*a^4*c^2*d^6*x^4 - 56
*a^5*c*d^6*x^2 - 320*a^6*d^6)*e^6 - 2*(27*a^3*c^3*d^7*x^5 + 12*a^4*c^2*d^7*x^3 + 832*a^5*c*d^7*x)*e^5 + 16*(2*
a^3*c^3*d^8*x^4 - 3*a^4*c^2*d^8*x^2)*e^4 - (55*a^2*c^4*d^9*x^5 - 56*a^3*c^3*d^9*x^3)*e^3)*sqrt(c*d^2*x + a*x*e
^2 + (c*d*x^2 + a*d)*e))*e^(-5)/(a^5*d^6*x^6)]

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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3289 vs. \(2 (449) = 898\).
time = 3.62, size = 3289, normalized size = 6.60 \begin {gather*} \text {Too large to display} \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]

1/512*(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)*arctan(-(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))/sqrt(-a*d*e
))*e^(-4)/(sqrt(-a*d*e)*a^4*d^5) - 1/7680*(105*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d
*e))*a^5*c^6*d^17*e^5 - 595*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^4*c^6*d^16
*e^4 - 1686*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^3*c^6*d^15*e^3 + 1386*(sqr
t(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^2*c^6*d^14*e^2 - 595*(sqrt(c*d)*x*e^(1/2)
- sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a*c^6*d^13*e + 105*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*
d^2*x + a*x*e^2 + a*d*e))^11*c^6*d^12 - 15360*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*
e))^4*sqrt(c*d)*a^4*c^5*d^14*e^(9/2) - 90*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*
a^6*c^5*d^15*e^7 - 30210*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^5*c^5*d^14*e^
6 - 53412*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^4*c^5*d^13*e^5 - 1188*(sqrt(
c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^3*c^5*d^12*e^4 + 510*(sqrt(c*d)*x*e^(1/2) -
sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^2*c^5*d^11*e^3 - 90*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c
*d^2*x + a*x*e^2 + a*d*e))^11*a*c^5*d^10*e^2 - 30720*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e))^2*sqrt(c*d)*a^6*c^4*d^13*e^(15/2) - 153600*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^
2 + a*d*e))^4*sqrt(c*d)*a^5*c^4*d^12*e^(13/2) - 97280*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^
2 + a*d*e))^6*sqrt(c*d)*a^4*c^4*d^11*e^(11/2) - 15405*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^
2 + a*d*e))*a^7*c^4*d^13*e^9 - 199425*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^
6*c^4*d^12*e^8 - 332370*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^5*c^4*d^11*e^7
- 86610*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^4*c^4*d^10*e^6 + 255*(sqrt(c*
d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^3*c^4*d^9*e^5 - 45*(sqrt(c*d)*x*e^(1/2) - sqrt
(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^11*a^2*c^4*d^8*e^4 - 3072*sqrt(c*d)*a^8*c^3*d^12*e^(21/2) - 135168*(s
qrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a^7*c^3*d^11*e^(19/2) - 506880*(
sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4*sqrt(c*d)*a^6*c^3*d^10*e^(17/2) - 337920*
(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^6*sqrt(c*d)*a^5*c^3*d^9*e^(15/2) - 30720*(
sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^8*sqrt(c*d)*a^4*c^3*d^8*e^(13/2) - 46140*(s
qrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^8*c^3*d^11*e^11 - 419500*(sqrt(c*d)*x*e^(1
/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^7*c^3*d^10*e^10 - 581400*(sqrt(c*d)*x*e^(1/2) - sqrt(c*
d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^6*c^3*d^9*e^9 - 135960*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2
*x + a*x*e^2 + a*d*e))^7*a^5*c^3*d^8*e^8 + 340*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d
*e))^9*a^4*c^3*d^7*e^7 - 60*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^11*a^3*c^3*d^6
*e^6 - 6144*sqrt(c*d)*a^9*c^2*d^10*e^(25/2) - 193536*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e))^2*sqrt(c*d)*a^8*c^2*d^9*e^(23/2) - 552960*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e))^4*sqrt(c*d)*a^7*c^2*d^8*e^(21/2) - 261120*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e))^6*sqrt(c*d)*a^6*c^2*d^7*e^(19/2) - 46305*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e))*a^9*c^2*d^9*e^13 - 305925*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^8*
c^2*d^8*e^12 - 279450*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^7*c^2*d^7*e^11 -
2970*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^6*c^2*d^6*e^10 + 1275*(sqrt(c*d)
*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^5*c^2*d^5*e^9 - 225*(sqrt(c*d)*x*e^(1/2) - sqrt(
c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^11*a^4*c^2*d^4*e^8 - 5120*sqrt(c*d)*a^10*c*d^8*e^(29/2) - 92160*(sqrt(
c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a^9*c*d^7*e^(27/2) - 184320*(sqrt(c*
d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4*sqrt(c*d)*a^8*c*d^6*e^(25/2) - 14730*(sqrt(c*d)*
x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^10*c*d^7*e^15 - 65010*(sqrt(c*d)*x*e^(1/2) - sqrt(c
*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^9*c*d^6*e^14 - 10116*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*
x + a*x*e^2 + a*d*e))^5*a^8*c*d^5*e^13 + 8316*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*
e))^7*a^7*c*d^4*e^12 - 3570*(sqrt(c*d)*x*e^(1/2...

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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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