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

Optimal. Leaf size=78 \[ -\frac{\sqrt{c+d x^3}}{24 x^3}+\frac{9 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{32 \sqrt{c}}-\frac{13 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 \sqrt{c}} \]

[Out]

-Sqrt[c + d*x^3]/(24*x^3) + (9*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(32*Sqrt[
c]) - (13*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(96*Sqrt[c])

_______________________________________________________________________________________

Rubi [A]  time = 0.289311, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\sqrt{c+d x^3}}{24 x^3}+\frac{9 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{32 \sqrt{c}}-\frac{13 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c + d*x^3]/(24*x^3) + (9*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(32*Sqrt[
c]) - (13*d*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(96*Sqrt[c])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.2769, size = 70, normalized size = 0.9 \[ - \frac{\sqrt{c + d x^{3}}}{24 x^{3}} + \frac{9 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{32 \sqrt{c}} - \frac{13 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{96 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)/(24*x**3) + 9*d*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(32*sqrt(c
)) - 13*d*atanh(sqrt(c + d*x**3)/sqrt(c))/(96*sqrt(c))

_______________________________________________________________________________________

Mathematica [C]  time = 0.278364, size = 322, normalized size = 4.13 \[ \frac{\frac{408 c d^2 x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{130 c d^2 x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-3 \left (c+d x^3\right )}{72 x^3 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*(c + d*x^3) + (408*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)]))) + (130*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)])))/
(72*x^3*Sqrt[c + d*x^3])

_______________________________________________________________________________________

Maple [C]  time = 0.037, size = 556, normalized size = 7.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/c*(-1/3*c*(d*x^3+c)^(1/2)/x^3+2/3*d*(d*x^3+c)^(1/2)-c^(1/2)*d*arctanh((d*x^3
+c)^(1/2)/c^(1/2)))+1/64*d/c^2*(2/9*d*x^3*(d*x^3+c)^(1/2)+8/9*c*(d*x^3+c)^(1/2)-
2/3*c^(3/2)*arctanh((d*x^3+c)^(1/2)/c^(1/2)))-1/64*d^2/c^2*(2/9*x^3*(d*x^3+c)^(1
/2)+56/9*c*(d*x^3+c)^(1/2)/d+3*I*c/d^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))
)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.254185, size = 1, normalized size = 0.01 \[ \left [\frac{27 \, d x^{3} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 13 \, d x^{3} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 8 \, \sqrt{d x^{3} + c} \sqrt{c}}{192 \, \sqrt{c} x^{3}}, -\frac{27 \, d x^{3} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 13 \, d x^{3} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 4 \, \sqrt{d x^{3} + c} \sqrt{-c}}{96 \, \sqrt{-c} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(27*d*x^3*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 - 8*c
)) + 13*d*x^3*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3 + c)*c)/x^3) - 8*sqrt(d*
x^3 + c)*sqrt(c))/(sqrt(c)*x^3), -1/96*(27*d*x^3*arctan(3*c/(sqrt(d*x^3 + c)*sqr
t(-c))) - 13*d*x^3*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) + 4*sqrt(d*x^3 + c)*sqrt
(-c))/(sqrt(-c)*x^3)]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221783, size = 92, normalized size = 1.18 \[ \frac{1}{96} \, d{\left (\frac{13 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{27 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c}} - \frac{4 \, \sqrt{d x^{3} + c}}{d x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*d*(13*arctan(sqrt(d*x^3 + c)/sqrt(-c))/sqrt(-c) - 27*arctan(1/3*sqrt(d*x^3
+ c)/sqrt(-c))/sqrt(-c) - 4*sqrt(d*x^3 + c)/(d*x^3))