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

Optimal. Leaf size=121 \[ \frac{3 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{128 c^{3/2}}-\frac{7 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{3/2}}+\frac{5 d \sqrt{c+d x^3}}{96 c \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 x^3 \left (8 c-d x^3\right )} \]

[Out]

(5*d*Sqrt[c + d*x^3])/(96*c*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(24*x^3*(8*c - d*x^
3)) + (3*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(128*c^(3/2)) - (7*d*ArcTanh[Sq
rt[c + d*x^3]/Sqrt[c]])/(384*c^(3/2))

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Rubi [A]  time = 0.378754, antiderivative size = 121, 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 \[ \frac{3 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{128 c^{3/2}}-\frac{7 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{3/2}}+\frac{5 d \sqrt{c+d x^3}}{96 c \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 x^3 \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5*d*Sqrt[c + d*x^3])/(96*c*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(24*x^3*(8*c - d*x^
3)) + (3*d*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(128*c^(3/2)) - (7*d*ArcTanh[Sq
rt[c + d*x^3]/Sqrt[c]])/(384*c^(3/2))

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Rubi in Sympy [A]  time = 56.0711, size = 99, normalized size = 0.82 \[ \frac{3 \sqrt{c + d x^{3}}}{8 x^{3} \left (8 c - d x^{3}\right )} - \frac{5 \sqrt{c + d x^{3}}}{96 c x^{3}} + \frac{3 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{128 c^{\frac{3}{2}}} - \frac{7 d \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{384 c^{\frac{3}{2}}} \]

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)**2,x)

[Out]

3*sqrt(c + d*x**3)/(8*x**3*(8*c - d*x**3)) - 5*sqrt(c + d*x**3)/(96*c*x**3) + 3*
d*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(128*c**(3/2)) - 7*d*atanh(sqrt(c + d*x**3
)/sqrt(c))/(384*c**(3/2))

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Mathematica [C]  time = 0.374014, size = 333, normalized size = 2.75 \[ \frac{\frac{60 d^2 x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{\frac{70 d^2 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}-\frac{15 d^2 x^3}{c}+\frac{12 c}{x^3}-3 d}{2 \left (d x^3-8 c\right )}}{144 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((60*d^2*x^3*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)]))) + (-3*d + (12*c)/x^3 - (15*d^2*x^3)/c + (70*d^2*x^3*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)]))/(2
*(-8*c + d*x^3)))/(144*Sqrt[c + d*x^3])

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Maple [C]  time = 0.02, size = 1014, normalized size = 8.4 \[ \text{result too large to display} \]

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)^2,x)

[Out]

1/64/c^2*(-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)))+1/256/c^3*d*(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*(-3*c/d*(d*x^3+c)
^(1/2)/(d*x^3-8*c)+2/3*(d*x^3+c)^(1/2)/d+1/2*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))*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*(2/9*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)*_alph
a*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. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.240089, size = 1, normalized size = 0.01 \[ \left [-\frac{8 \,{\left (5 \, d x^{3} - 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{c} - 9 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 7 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{768 \,{\left (c d x^{6} - 8 \, c^{2} x^{3}\right )} \sqrt{c}}, -\frac{4 \,{\left (5 \, d x^{3} - 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{-c} + 9 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 7 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{384 \,{\left (c d x^{6} - 8 \, c^{2} x^{3}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/768*(8*(5*d*x^3 - 4*c)*sqrt(d*x^3 + c)*sqrt(c) - 9*(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)) - 7*(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*d*x^6 - 8*c
^2*x^3)*sqrt(c)), -1/384*(4*(5*d*x^3 - 4*c)*sqrt(d*x^3 + c)*sqrt(-c) + 9*(d^2*x^
6 - 8*c*d*x^3)*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) - 7*(d^2*x^6 - 8*c*d*x^3)*
arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))))/((c*d*x^6 - 8*c^2*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)**(3/2)/x**4/(-d*x**3+8*c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220367, size = 153, normalized size = 1.26 \[ \frac{1}{384} \, d{\left (\frac{7 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{9 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{4 \,{\left (5 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )}}{{\left ({\left (d x^{3} + c\right )}^{2} - 10 \,{\left (d x^{3} + c\right )} c + 9 \, c^{2}\right )} c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*d*(7*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c) - 9*arctan(1/3*sqrt(d*x
^3 + c)/sqrt(-c))/(sqrt(-c)*c) - 4*(5*(d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 + c)*c)/(
((d*x^3 + c)^2 - 10*(d*x^3 + c)*c + 9*c^2)*c))