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

Optimal. Leaf size=100 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2592 c^{7/2}}+\frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 c^{7/2}}-\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}} \]

[Out]

(-25*d)/(216*c^3*Sqrt[c + d*x^3]) - 1/(24*c^2*x^3*Sqrt[c + d*x^3]) + (d*ArcTanh[
Sqrt[c + d*x^3]/(3*Sqrt[c])])/(2592*c^(7/2)) + (11*d*ArcTanh[Sqrt[c + d*x^3]/Sqr
t[c]])/(96*c^(7/2))

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Rubi [A]  time = 0.375128, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2592 c^{7/2}}+\frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 c^{7/2}}-\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(-25*d)/(216*c^3*Sqrt[c + d*x^3]) - 1/(24*c^2*x^3*Sqrt[c + d*x^3]) + (d*ArcTanh[
Sqrt[c + d*x^3]/(3*Sqrt[c])])/(2592*c^(7/2)) + (11*d*ArcTanh[Sqrt[c + d*x^3]/Sqr
t[c]])/(96*c^(7/2))

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Rubi in Sympy [A]  time = 51.3181, size = 92, normalized size = 0.92 \[ - \frac{1}{24 c^{2} x^{3} \sqrt{c + d x^{3}}} - \frac{25 d}{216 c^{3} \sqrt{c + d x^{3}}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{2592 c^{\frac{7}{2}}} + \frac{11 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{96 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(24*c**2*x**3*sqrt(c + d*x**3)) - 25*d/(216*c**3*sqrt(c + d*x**3)) + d*atanh(
sqrt(c + d*x**3)/(3*sqrt(c)))/(2592*c**(7/2)) + 11*d*atanh(sqrt(c + d*x**3)/sqrt
(c))/(96*c**(7/2))

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Mathematica [C]  time = 0.384107, size = 326, normalized size = 3.26 \[ \frac{\frac{200 c d^2 x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{330 c d^2 x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (8 c-d x^3\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-9 c-25 d x^3}{216 c^3 x^3 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-9*c - 25*d*x^3 + (200*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)]))) + (330*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*AppellF1[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)])))/
(216*c^3*x^3*Sqrt[c + d*x^3])

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Maple [C]  time = 0.04, size = 549, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/c*(-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))+1/64*d/c^2*(2/3/c/((x^3+c/d)*d)^(1/2)-2/3*arctanh(
(d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))-1/64*d^2/c^2*(2/27/d/c/((x^3+c/d)*d)^(1/2)+1/2
43*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. \[ -\int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.240818, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d x^{3} + c} d x^{3} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 297 \, \sqrt{d x^{3} + c} d x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 24 \,{\left (25 \, d x^{3} + 9 \, c\right )} \sqrt{c}}{5184 \, \sqrt{d x^{3} + c} c^{\frac{7}{2}} x^{3}}, -\frac{\sqrt{d x^{3} + c} d x^{3} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 297 \, \sqrt{d x^{3} + c} d x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 12 \,{\left (25 \, d x^{3} + 9 \, c\right )} \sqrt{-c}}{2592 \, \sqrt{d x^{3} + c} \sqrt{-c} c^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/5184*(sqrt(d*x^3 + c)*d*x^3*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c
)/(d*x^3 - 8*c)) + 297*sqrt(d*x^3 + c)*d*x^3*log(((d*x^3 + 2*c)*sqrt(c) + 2*sqrt
(d*x^3 + c)*c)/x^3) - 24*(25*d*x^3 + 9*c)*sqrt(c))/(sqrt(d*x^3 + c)*c^(7/2)*x^3)
, -1/2592*(sqrt(d*x^3 + c)*d*x^3*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) + 297*sq
rt(d*x^3 + c)*d*x^3*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) + 12*(25*d*x^3 + 9*c)*s
qrt(-c))/(sqrt(d*x^3 + c)*sqrt(-c)*c^3*x^3)]

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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)/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21834, size = 128, normalized size = 1.28 \[ -\frac{1}{2592} \, d{\left (\frac{297 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{12 \,{\left (25 \, d x^{3} + 9 \, c\right )}}{{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - \sqrt{d x^{3} + c} c\right )} c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2592*d*(297*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) + arctan(1/3*sqrt
(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) + 12*(25*d*x^3 + 9*c)/(((d*x^3 + c)^(3/2) -
sqrt(d*x^3 + c)*c)*c^3))