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

Optimal. Leaf size=124 \[ \frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{10368 c^{7/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{7/2}}+\frac{5 d \sqrt{c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )} \]

[Out]

(5*d*Sqrt[c + d*x^3])/(864*c^3*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(24*c^2*x^3*(8*c
- d*x^3)) + (11*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(10368*c^(7/2)) + (d*Ar
cTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(384*c^(7/2))

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Rubi [A]  time = 0.385277, antiderivative size = 124, 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 \[ \frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{10368 c^{7/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{7/2}}+\frac{5 d \sqrt{c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5*d*Sqrt[c + d*x^3])/(864*c^3*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(24*c^2*x^3*(8*c
- d*x^3)) + (11*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(10368*c^(7/2)) + (d*Ar
cTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(384*c^(7/2))

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Rubi in Sympy [A]  time = 54.5882, size = 107, normalized size = 0.86 \[ - \frac{\sqrt{c + d x^{3}}}{24 c^{2} x^{3} \left (8 c - d x^{3}\right )} + \frac{5 d \sqrt{c + d x^{3}}}{864 c^{3} \left (8 c - d x^{3}\right )} + \frac{11 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{10368 c^{\frac{7}{2}}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{384 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)/(24*c**2*x**3*(8*c - d*x**3)) + 5*d*sqrt(c + d*x**3)/(864*c**3
*(8*c - d*x**3)) + 11*d*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(10368*c**(7/2)) + d
*atanh(sqrt(c + d*x**3)/sqrt(c))/(384*c**(7/2))

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Mathematica [C]  time = 0.407908, size = 347, normalized size = 2.8 \[ \frac{\frac{40 c d^2 x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{30 c d^2 x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (8 c-d x^3\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-\frac{\left (c+d x^3\right ) \left (5 d x^3-36 c\right )}{d x^3-8 c}}{864 c^3 x^3 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-(((c + d*x^3)*(-36*c + 5*d*x^3))/(-8*c + d*x^3)) + (40*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)]))) + (30*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*AppellF
1[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)])))/(864*c^3*x^3*Sqrt[c + d*x^3])

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Maple [C]  time = 0.019, size = 926, normalized size = 7.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64/c^2*(-1/3*(d*x^3+c)^(1/2)/c/x^3+1/3*d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(3
/2))-1/384*d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(7/2)+1/64*d^2/c^2*(-1/27/d/c*(d
*x^3+c)^(1/2)/(d*x^3-8*c)-1/486*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)))-1/6912*I/d/c^4*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. \[ \int \frac{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.24074, size = 1, normalized size = 0.01 \[ \left [-\frac{24 \,{\left (5 \, d x^{3} - 36 \, c\right )} \sqrt{d x^{3} + c} \sqrt{c} - 11 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 27 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{20736 \,{\left (c^{3} d x^{6} - 8 \, c^{4} x^{3}\right )} \sqrt{c}}, -\frac{12 \,{\left (5 \, d x^{3} - 36 \, c\right )} \sqrt{d x^{3} + c} \sqrt{-c} + 11 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 27 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{10368 \,{\left (c^{3} d x^{6} - 8 \, c^{4} x^{3}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/20736*(24*(5*d*x^3 - 36*c)*sqrt(d*x^3 + c)*sqrt(c) - 11*(d^2*x^6 - 8*c*d*x^3
)*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 - 8*c)) - 27*(d^2*x^
6 - 8*c*d*x^3)*log(((d*x^3 + 2*c)*sqrt(c) + 2*sqrt(d*x^3 + c)*c)/x^3))/((c^3*d*x
^6 - 8*c^4*x^3)*sqrt(c)), -1/10368*(12*(5*d*x^3 - 36*c)*sqrt(d*x^3 + c)*sqrt(-c)
+ 11*(d^2*x^6 - 8*c*d*x^3)*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) + 27*(d^2*x^6
- 8*c*d*x^3)*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))))/((c^3*d*x^6 - 8*c^4*x^3)*sqr
t(-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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216024, size = 153, normalized size = 1.23 \[ -\frac{1}{10368} \, d{\left (\frac{27 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{11 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{12 \,{\left (5 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 41 \, \sqrt{d x^{3} + c} c\right )}}{{\left ({\left (d x^{3} + c\right )}^{2} - 10 \,{\left (d x^{3} + c\right )} c + 9 \, c^{2}\right )} c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10368*d*(27*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) + 11*arctan(1/3*s
qrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) + 12*(5*(d*x^3 + c)^(3/2) - 41*sqrt(d*x^
3 + c)*c)/(((d*x^3 + c)^2 - 10*(d*x^3 + c)*c + 9*c^2)*c^3))