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

Optimal. Leaf size=106 \[ \frac{7 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{7776 c^{7/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 c^{7/2}}+\frac{5}{648 c^3 \sqrt{c+d x^3}}+\frac{1}{216 c^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

[Out]

5/(648*c^3*Sqrt[c + d*x^3]) + 1/(216*c^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (7*Arc
Tanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(7776*c^(7/2)) - ArcTanh[Sqrt[c + d*x^3]/Sqrt
[c]]/(96*c^(7/2))

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Rubi [A]  time = 0.380532, antiderivative size = 106, 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 \[ \frac{7 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{7776 c^{7/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 c^{7/2}}+\frac{5}{648 c^3 \sqrt{c+d x^3}}+\frac{1}{216 c^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

5/(648*c^3*Sqrt[c + d*x^3]) + 1/(216*c^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (7*Arc
Tanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(7776*c^(7/2)) - ArcTanh[Sqrt[c + d*x^3]/Sqrt
[c]]/(96*c^(7/2))

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Rubi in Sympy [A]  time = 55.6895, size = 92, normalized size = 0.87 \[ \frac{1}{216 c^{2} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} + \frac{5}{648 c^{3} \sqrt{c + d x^{3}}} + \frac{7 \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{7776 c^{\frac{7}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{96 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(216*c**2*sqrt(c + d*x**3)*(8*c - d*x**3)) + 5/(648*c**3*sqrt(c + d*x**3)) + 7
*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(7776*c**(7/2)) - atanh(sqrt(c + d*x**3)/sq
rt(c))/(96*c**(7/2))

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Mathematica [C]  time = 0.374883, size = 338, normalized size = 3.19 \[ \frac{-\frac{20 d x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{c^2 \left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{45 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{c^2 \left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}+\frac{43 c-5 d x^3}{16 c^4-2 c^3 d x^3}}{324 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((43*c - 5*d*x^3)/(16*c^4 - 2*c^3*d*x^3) - (20*d*x^3*AppellF1[1, 1/2, 1, 2, -((d
*x^3)/c), (d*x^3)/(8*c)])/(c^2*(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)]))) + (45*d*x^3*Appel
lF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), (8*c)/(d*x^3)])/(c^2*(-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)])))/(324*Sqrt[c + d*x^3])

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Maple [C]  time = 0.018, size = 953, normalized size = 9. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64/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)
)+1/8*d/c*(-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)),_alpha=RootOf(_Z^3*d-8*c)))-1/64*d/c^2*(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. \[ \int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )}^{2} x}\,{d x} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.229867, size = 1, normalized size = 0.01 \[ \left [\frac{7 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 81 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 24 \,{\left (5 \, d x^{3} - 43 \, c\right )} \sqrt{c}}{15552 \,{\left (c^{3} d x^{3} - 8 \, c^{4}\right )} \sqrt{d x^{3} + c} \sqrt{c}}, -\frac{7 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 81 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 12 \,{\left (5 \, d x^{3} - 43 \, c\right )} \sqrt{-c}}{7776 \,{\left (c^{3} d x^{3} - 8 \, c^{4}\right )} \sqrt{d x^{3} + c} \sqrt{-c}}\right ] \]

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),x, algorithm="fricas")

[Out]

[1/15552*(7*sqrt(d*x^3 + c)*(d*x^3 - 8*c)*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d
*x^3 + c)*c)/(d*x^3 - 8*c)) + 81*sqrt(d*x^3 + c)*(d*x^3 - 8*c)*log(((d*x^3 + 2*c
)*sqrt(c) - 2*sqrt(d*x^3 + c)*c)/x^3) + 24*(5*d*x^3 - 43*c)*sqrt(c))/((c^3*d*x^3
- 8*c^4)*sqrt(d*x^3 + c)*sqrt(c)), -1/7776*(7*sqrt(d*x^3 + c)*(d*x^3 - 8*c)*arc
tan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) - 81*sqrt(d*x^3 + c)*(d*x^3 - 8*c)*arctan(c/
(sqrt(d*x^3 + c)*sqrt(-c))) - 12*(5*d*x^3 - 43*c)*sqrt(-c))/((c^3*d*x^3 - 8*c^4)
*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/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223934, size = 126, normalized size = 1.19 \[ \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{96 \, \sqrt{-c} c^{3}} - \frac{7 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{7776 \, \sqrt{-c} c^{3}} + \frac{5 \, d x^{3} - 43 \, c}{648 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )} c^{3}} \]

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),x, algorithm="giac")

[Out]

1/96*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) - 7/7776*arctan(1/3*sqrt(d*
x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) + 1/648*(5*d*x^3 - 43*c)/(((d*x^3 + c)^(3/2) -
9*sqrt(d*x^3 + c)*c)*c^3)