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

Optimal. Leaf size=124 \[ \frac{7 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{1152 c^{5/2}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{128 c^{5/2}}+\frac{d \sqrt{c+d x^3}}{96 c^2 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )} \]

[Out]

(d*Sqrt[c + d*x^3])/(96*c^2*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(24*c*x^3*(8*c - d*
x^3)) + (7*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(1152*c^(5/2)) - (d*ArcTanh[S
qrt[c + d*x^3]/Sqrt[c]])/(128*c^(5/2))

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Rubi [A]  time = 0.379592, 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 \[ \frac{7 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{1152 c^{5/2}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{128 c^{5/2}}+\frac{d \sqrt{c+d x^3}}{96 c^2 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c x^3 \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(d*Sqrt[c + d*x^3])/(96*c^2*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(24*c*x^3*(8*c - d*
x^3)) + (7*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(1152*c^(5/2)) - (d*ArcTanh[S
qrt[c + d*x^3]/Sqrt[c]])/(128*c^(5/2))

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Rubi in Sympy [A]  time = 46.1118, size = 97, normalized size = 0.78 \[ \frac{\sqrt{c + d x^{3}}}{24 c x^{3} \left (8 c - d x^{3}\right )} - \frac{\sqrt{c + d x^{3}}}{96 c^{2} x^{3}} + \frac{7 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{1152 c^{\frac{5}{2}}} - \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{128 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c + d*x**3)/(24*c*x**3*(8*c - d*x**3)) - sqrt(c + d*x**3)/(96*c**2*x**3) +
7*d*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(1152*c**(5/2)) - d*atanh(sqrt(c + d*x**
3)/sqrt(c))/(128*c**(5/2))

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Mathematica [C]  time = 0.431641, size = 338, normalized size = 2.73 \[ \frac{\frac{\frac{10 c d^2 x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}+4 c^2+3 c d x^3-d^2 x^6}{d x^3-8 c}+\frac{8 c d^2 x^6 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}}{96 c^2 x^3 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((8*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*c^2 + 3*c*d*x^3 - d^2*x^6 + (10*c*d^2*x^6*AppellF1[3/2,
1/2, 1, 5/2, -(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)]))/(-8*c
+ d*x^3))/(96*c^2*x^3*Sqrt[c + d*x^3])

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Maple [C]  time = 0.019, size = 957, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64/c^2*(-1/3*(d*x^3+c)^(1/2)/x^3-1/3*d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(1/2
))+1/256/c^3*d*(2/3*(d*x^3+c)^(1/2)-2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))*c^(1/2)
)+1/64*d^2/c^2*(-1/3/d*(d*x^3+c)^(1/2)/(d*x^3-8*c)+1/54*I/d^3/c*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)))-1/256*d^2/c^3*(2/3*(d*x^3+c)^(1/2)/d+1/3*I/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))*El
lipticPi(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. \[ \int \frac{\sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231498, size = 1, normalized size = 0.01 \[ \left [-\frac{24 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 4 \, c\right )} \sqrt{c} - 7 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 9 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{2304 \,{\left (c^{2} d x^{6} - 8 \, c^{3} x^{3}\right )} \sqrt{c}}, -\frac{12 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 4 \, c\right )} \sqrt{-c} + 7 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 9 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{1152 \,{\left (c^{2} d x^{6} - 8 \, c^{3} x^{3}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2304*(24*sqrt(d*x^3 + c)*(d*x^3 - 4*c)*sqrt(c) - 7*(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)) - 9*(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^2*d*x^6 - 8
*c^3*x^3)*sqrt(c)), -1/1152*(12*sqrt(d*x^3 + c)*(d*x^3 - 4*c)*sqrt(-c) + 7*(d^2*
x^6 - 8*c*d*x^3)*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) - 9*(d^2*x^6 - 8*c*d*x^3
)*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))))/((c^2*d*x^6 - 8*c^3*x^3)*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((d*x**3+c)**(1/2)/x**4/(-d*x**3+8*c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229742, size = 150, normalized size = 1.21 \[ \frac{1}{1152} \, d{\left (\frac{9 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{7 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{12 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 5 \, \sqrt{d x^{3} + c} c\right )}}{{\left ({\left (d x^{3} + c\right )}^{2} - 10 \,{\left (d x^{3} + c\right )} c + 9 \, c^{2}\right )} c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1152*d*(9*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) - 7*arctan(1/3*sqrt(
d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) - 12*((d*x^3 + c)^(3/2) - 5*sqrt(d*x^3 + c)*
c)/(((d*x^3 + c)^2 - 10*(d*x^3 + c)*c + 9*c^2)*c^2))