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

Optimal result

Integrand size = 40, antiderivative size = 498 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\frac {\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 x^5}+\frac {\left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}-\frac {\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac {\left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}} \]

[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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 848, 820, 734, 738, 212} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\frac {\left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 a^2 d^3 e^2 x^4}+\frac {\left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^4 d^5 e^4 x^2}-\frac {\left (-105 a^3 e^6+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 a^3 d^4 e^3 x^3}-\frac {\left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{9/2} d^{11/2} e^{9/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 d x^6}-\frac {\left (\frac {c}{a e}-\frac {3 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 x^5} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-105 c^5 d^{10} x^5+5 a c^4 d^8 e x^4 (14 d+11 e x)-2 a^2 c^3 d^6 e^2 x^3 \left (28 d^2+16 d e x-27 e^2 x^2\right )+6 a^3 c^2 d^4 e^3 x^2 \left (8 d^3+4 d^2 e x-6 d e^2 x^2+13 e^3 x^3\right )+a^4 c d^2 e^4 x \left (1664 d^4+224 d^3 e x-264 d^2 e^2 x^2+336 d e^3 x^3-525 e^4 x^4\right )+a^5 e^5 \left (1280 d^5+128 d^4 e x-144 d^3 e^2 x^2+168 d^2 e^3 x^3-210 d e^4 x^4+315 e^5 x^5\right )\right )}{x^6}-\frac {15 \left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 a^{9/2} d^{11/2} e^{9/2}} \]

[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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(16882\) vs. \(2(460)=920\).

Time = 1.93 (sec) , antiderivative size = 16883, normalized size of antiderivative = 33.90

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

[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

Fricas [A] (verification not implemented)

none

Time = 37.47 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.15 \[ \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=\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 ] \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\text {Timed out} \]

[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

Maxima [F]

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

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3251 vs. \(2 (460) = 920\).

Time = 0.48 (sec) , antiderivative size = 3251, normalized size of antiderivative = 6.53 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\text {Too large to display} \]

[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*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/(sq
rt(-a*d*e)*a^4*d^5*e^4) - 1/7680*(105*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^5*c^6*d^
17*e^5 - 90*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^6*c^5*d^15*e^7 - 15405*(sqrt(c*d*e
)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^7*c^4*d^13*e^9 - 46140*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 +
c*d^2*x + a*e^2*x + a*d*e))*a^8*c^3*d^11*e^11 - 46305*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*
d*e))*a^9*c^2*d^9*e^13 - 14730*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^10*c*d^7*e^15 -
315*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^11*d^5*e^17 - 3072*sqrt(c*d*e)*a^8*c^3*d^
12*e^10 - 6144*sqrt(c*d*e)*a^9*c^2*d^10*e^12 - 5120*sqrt(c*d*e)*a^10*c*d^8*e^14 - 595*(sqrt(c*d*e)*x - sqrt(c*
d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^4*c^6*d^16*e^4 - 30210*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a
*e^2*x + a*d*e))^3*a^5*c^5*d^14*e^6 - 199425*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a
^6*c^4*d^12*e^8 - 419500*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^7*c^3*d^10*e^10 - 3
05925*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^8*c^2*d^8*e^12 - 65010*(sqrt(c*d*e)*x
- sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^9*c*d^6*e^14 - 3335*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2
*x + a*e^2*x + a*d*e))^3*a^10*d^4*e^16 - 30720*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x
+ a*d*e))^2*a^6*c^4*d^13*e^7 - 135168*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e
))^2*a^7*c^3*d^11*e^9 - 193536*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^8
*c^2*d^9*e^11 - 92160*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^9*c*d^7*e^
13 - 15360*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^10*d^5*e^15 - 1686*(s
qrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^3*c^6*d^15*e^3 - 53412*(sqrt(c*d*e)*x - sqrt(c
*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^4*c^5*d^13*e^5 - 332370*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x +
a*e^2*x + a*d*e))^5*a^5*c^4*d^11*e^7 - 581400*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5
*a^6*c^3*d^9*e^9 - 279450*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^7*c^2*d^7*e^11 - 1
0116*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^8*c*d^5*e^13 + 5058*(sqrt(c*d*e)*x - sq
rt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^9*d^3*e^15 - 15360*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2
+ c*d^2*x + a*e^2*x + a*d*e))^4*a^4*c^5*d^14*e^4 - 153600*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*
x + a*e^2*x + a*d*e))^4*a^5*c^4*d^12*e^6 - 506880*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^
2*x + a*d*e))^4*a^6*c^3*d^10*e^8 - 552960*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*
d*e))^4*a^7*c^2*d^8*e^10 - 184320*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*
a^8*c*d^6*e^12 + 1386*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^2*c^6*d^14*e^2 - 1188*
(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^3*c^5*d^12*e^4 - 86610*(sqrt(c*d*e)*x - sqrt
(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^4*c^4*d^10*e^6 - 135960*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x
+ a*e^2*x + a*d*e))^7*a^5*c^3*d^8*e^8 - 2970*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*
a^6*c^2*d^6*e^10 + 8316*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^7*c*d^4*e^12 - 4158*
(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^8*d^2*e^14 - 97280*sqrt(c*d*e)*(sqrt(c*d*e)*
x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^4*c^4*d^11*e^5 - 337920*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt
(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^5*c^3*d^9*e^7 - 261120*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^
2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^6*c^2*d^7*e^9 - 595*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x +
a*d*e))^9*a*c^6*d^13*e + 510*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^2*c^5*d^11*e^3
+ 255*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^3*c^4*d^9*e^5 + 340*(sqrt(c*d*e)*x - s
qrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^4*c^3*d^7*e^7 + 1275*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x
+ a*e^2*x + a*d*e))^9*a^5*c^2*d^5*e^9 - 3570*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*
a^6*c*d^3*e^11 + 1785*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^7*d*e^13 - 30720*sqrt(
c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^8*a^4*c^3*d^8*e^6 + 105*(sqrt(c*d*e)*x -
sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*c^6*d^12 - 90*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^
2*x + a*d*e))^11*a*c^5*d^10*e^2 - 45*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^2*c^4*
d^8*e^4 - 60*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^3*c^3*d^6*e^6 - 225*(sqrt(c*d*
e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^4*c^2*d^4*e^8 + 630*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 +
c*d^2*x + a*e^2*x + a*d*e))^11*a^5*c*d^2*e^10 - 315*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d
*e))^11*a^6*e^12)/((a*d*e - (sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2)^6*a^4*d^5*e^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^7\,\left (d+e\,x\right )} \,d x \]

[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)