3.447 \(\int \frac{1}{x^4 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=143 \[ \frac{5 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{31104 c^{9/2}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{9/2}}-\frac{35 d}{2592 c^4 \sqrt{c+d x^3}}+\frac{5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

[Out]

(-35*d)/(2592*c^4*Sqrt[c + d*x^3]) + (5*d)/(864*c^3*(8*c - d*x^3)*Sqrt[c + d*x^3
]) - 1/(24*c^2*x^3*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (5*d*ArcTanh[Sqrt[c + d*x^3]
/(3*Sqrt[c])])/(31104*c^(9/2)) + (5*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(384*c^(
9/2))

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Rubi [A]  time = 0.502188, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ \frac{5 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{31104 c^{9/2}}+\frac{5 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{9/2}}-\frac{35 d}{2592 c^4 \sqrt{c+d x^3}}+\frac{5 d}{864 c^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]

[Out]

(-35*d)/(2592*c^4*Sqrt[c + d*x^3]) + (5*d)/(864*c^3*(8*c - d*x^3)*Sqrt[c + d*x^3
]) - 1/(24*c^2*x^3*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (5*d*ArcTanh[Sqrt[c + d*x^3]
/(3*Sqrt[c])])/(31104*c^(9/2)) + (5*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(384*c^(
9/2))

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Rubi in Sympy [A]  time = 75.1026, size = 129, normalized size = 0.9 \[ - \frac{1}{24 c^{2} x^{3} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} + \frac{5 d}{864 c^{3} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} - \frac{35 d}{2592 c^{4} \sqrt{c + d x^{3}}} + \frac{5 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{31104 c^{\frac{9}{2}}} + \frac{5 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{384 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)

[Out]

-1/(24*c**2*x**3*sqrt(c + d*x**3)*(8*c - d*x**3)) + 5*d/(864*c**3*sqrt(c + d*x**
3)*(8*c - d*x**3)) - 35*d/(2592*c**4*sqrt(c + d*x**3)) + 5*d*atanh(sqrt(c + d*x*
*3)/(3*sqrt(c)))/(31104*c**(9/2)) + 5*d*atanh(sqrt(c + d*x**3)/sqrt(c))/(384*c**
(9/2))

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Mathematica [C]  time = 0.385686, size = 350, normalized size = 2.45 \[ \frac{\frac{280 c d^2 x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{450 c d^2 x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (8 c-d x^3\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}+\frac{108 c^2+265 c d x^3-35 d^2 x^6}{d x^3-8 c}}{2592 c^4 x^3 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^4*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]

[Out]

((108*c^2 + 265*c*d*x^3 - 35*d^2*x^6)/(-8*c + d*x^3) + (280*c*d^2*x^6*AppellF1[1
, 1/2, 1, 2, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x^3)*(16*c*AppellF1[1, 1/2,
1, 2, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^3*(AppellF1[2, 1/2, 2, 3, -((d*x^3)/c)
, (d*x^3)/(8*c)] - 4*AppellF1[2, 3/2, 1, 3, -((d*x^3)/c), (d*x^3)/(8*c)]))) + (4
50*c*d^2*x^6*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), (8*c)/(d*x^3)])/((8*c - d*
x^3)*(5*d*x^3*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), (8*c)/(d*x^3)] + 16*c*App
ellF1[5/2, 1/2, 2, 7/2, -(c/(d*x^3)), (8*c)/(d*x^3)] - c*AppellF1[5/2, 3/2, 1, 7
/2, -(c/(d*x^3)), (8*c)/(d*x^3)])))/(2592*c^4*x^3*Sqrt[c + d*x^3])

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Maple [C]  time = 0.02, size = 1019, normalized size = 7.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)

[Out]

1/64/c^2*(-1/3*(d*x^3+c)^(1/2)/c^2/x^3-2/3*d/c^2/((x^3+c/d)*d)^(1/2)+d*arctanh((
d*x^3+c)^(1/2)/c^(1/2))/c^(5/2))+1/256/c^3*d*(2/3/c/((x^3+c/d)*d)^(1/2)-2/3*arct
anh((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))+1/64*d^2/c^2*(-1/243/d/c^2*(d*x^3+c)^(1/2)
/(d*x^3-8*c)-2/243/d/c^2/((x^3+c/d)*d)^(1/2)-1/1458*I/d^3/c^3*2^(1/2)*sum((-c*d^
2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^
(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/
3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2
)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*d^2
-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3
*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-
c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c*
d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/c,(I*3^(1/2)/d*(
-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_al
pha=RootOf(_Z^3*d-8*c)))-1/256*d^2/c^3*(2/27/d/c/((x^3+c/d)*d)^(1/2)+1/243*I/d^3
/c^2*2^(1/2)*sum((-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c
*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)
+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(
-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3
^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2
)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d
^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*
3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-
3*c*d)/c,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*
d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d-8*c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^4),x, algorithm="maxima")

[Out]

integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^4), x)

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Fricas [A]  time = 0.232454, size = 1, normalized size = 0.01 \[ \left [\frac{5 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 405 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 24 \,{\left (35 \, d^{2} x^{6} - 265 \, c d x^{3} - 108 \, c^{2}\right )} \sqrt{c}}{62208 \,{\left (c^{4} d x^{6} - 8 \, c^{5} x^{3}\right )} \sqrt{d x^{3} + c} \sqrt{c}}, -\frac{5 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt{d x^{3} + c} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 405 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt{d x^{3} + c} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 12 \,{\left (35 \, d^{2} x^{6} - 265 \, c d x^{3} - 108 \, c^{2}\right )} \sqrt{-c}}{31104 \,{\left (c^{4} d x^{6} - 8 \, c^{5} x^{3}\right )} \sqrt{d x^{3} + c} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^4),x, algorithm="fricas")

[Out]

[1/62208*(5*(d^2*x^6 - 8*c*d*x^3)*sqrt(d*x^3 + c)*log(((d*x^3 + 10*c)*sqrt(c) +
6*sqrt(d*x^3 + c)*c)/(d*x^3 - 8*c)) + 405*(d^2*x^6 - 8*c*d*x^3)*sqrt(d*x^3 + c)*
log(((d*x^3 + 2*c)*sqrt(c) + 2*sqrt(d*x^3 + c)*c)/x^3) - 24*(35*d^2*x^6 - 265*c*
d*x^3 - 108*c^2)*sqrt(c))/((c^4*d*x^6 - 8*c^5*x^3)*sqrt(d*x^3 + c)*sqrt(c)), -1/
31104*(5*(d^2*x^6 - 8*c*d*x^3)*sqrt(d*x^3 + c)*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(
-c))) + 405*(d^2*x^6 - 8*c*d*x^3)*sqrt(d*x^3 + c)*arctan(c/(sqrt(d*x^3 + c)*sqrt
(-c))) + 12*(35*d^2*x^6 - 265*c*d*x^3 - 108*c^2)*sqrt(-c))/((c^4*d*x^6 - 8*c^5*x
^3)*sqrt(d*x^3 + c)*sqrt(-c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221646, size = 171, normalized size = 1.2 \[ -\frac{1}{31104} \, d{\left (\frac{405 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{4}} + \frac{5 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{4}} + \frac{12 \,{\left (35 \,{\left (d x^{3} + c\right )}^{2} - 335 \,{\left (d x^{3} + c\right )} c + 192 \, c^{2}\right )}}{{\left ({\left (d x^{3} + c\right )}^{\frac{5}{2}} - 10 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} c + 9 \, \sqrt{d x^{3} + c} c^{2}\right )} c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^4),x, algorithm="giac")

[Out]

-1/31104*d*(405*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) + 5*arctan(1/3*s
qrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) + 12*(35*(d*x^3 + c)^2 - 335*(d*x^3 + c)
*c + 192*c^2)/(((d*x^3 + c)^(5/2) - 10*(d*x^3 + c)^(3/2)*c + 9*sqrt(d*x^3 + c)*c
^2)*c^4))