3.301 \(\int \frac{\left (c+d x^3\right )^{3/2}}{x^7 \left (8 c-d x^3\right )} \, dx\)

Optimal. Leaf size=104 \[ \frac{9 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{3/2}}-\frac{37 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{3/2}}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}-\frac{\sqrt{c+d x^3}}{48 x^6} \]

[Out]

-Sqrt[c + d*x^3]/(48*x^6) - (11*d*Sqrt[c + d*x^3])/(192*c*x^3) + (9*d^2*ArcTanh[
Sqrt[c + d*x^3]/(3*Sqrt[c])])/(256*c^(3/2)) - (37*d^2*ArcTanh[Sqrt[c + d*x^3]/Sq
rt[c]])/(768*c^(3/2))

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Rubi [A]  time = 0.419337, antiderivative size = 104, 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 \[ \frac{9 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{3/2}}-\frac{37 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{3/2}}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}-\frac{\sqrt{c+d x^3}}{48 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3)^(3/2)/(x^7*(8*c - d*x^3)),x]

[Out]

-Sqrt[c + d*x^3]/(48*x^6) - (11*d*Sqrt[c + d*x^3])/(192*c*x^3) + (9*d^2*ArcTanh[
Sqrt[c + d*x^3]/(3*Sqrt[c])])/(256*c^(3/2)) - (37*d^2*ArcTanh[Sqrt[c + d*x^3]/Sq
rt[c]])/(768*c^(3/2))

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Rubi in Sympy [A]  time = 54.5681, size = 94, normalized size = 0.9 \[ - \frac{\sqrt{c + d x^{3}}}{48 x^{6}} - \frac{11 d \sqrt{c + d x^{3}}}{192 c x^{3}} + \frac{9 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{256 c^{\frac{3}{2}}} - \frac{37 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{768 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c)**(3/2)/x**7/(-d*x**3+8*c),x)

[Out]

-sqrt(c + d*x**3)/(48*x**6) - 11*d*sqrt(c + d*x**3)/(192*c*x**3) + 9*d**2*atanh(
sqrt(c + d*x**3)/(3*sqrt(c)))/(256*c**(3/2)) - 37*d**2*atanh(sqrt(c + d*x**3)/sq
rt(c))/(768*c**(3/2))

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Mathematica [C]  time = 0.327822, size = 332, normalized size = 3.19 \[ \frac{\frac{132 d^3 x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{185 d^3 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-\frac{33 d^2}{2 c}-\frac{6 c}{x^6}-\frac{45 d}{2 x^3}}{288 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(c + d*x^3)^(3/2)/(x^7*(8*c - d*x^3)),x]

[Out]

((-33*d^2)/(2*c) - (6*c)/x^6 - (45*d)/(2*x^3) + (132*d^3*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)]))) + (185*d^3*x^
3*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)])))/(288*Sqrt[c + d*x^3])

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Maple [C]  time = 0.038, size = 617, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^(3/2)/x^7/(-d*x^3+8*c),x)

[Out]

1/8/c*(-1/6*c*(d*x^3+c)^(1/2)/x^6-5/12*d*(d*x^3+c)^(1/2)/x^3-1/4*d^2*arctanh((d*
x^3+c)^(1/2)/c^(1/2))/c^(1/2))+1/64*d/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)))+1/512*d^2/c^3*(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)))-1/512*d^3/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)*_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. \[ -\int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^7),x, algorithm="maxima")

[Out]

-integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^7), x)

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Fricas [A]  time = 0.254815, size = 1, normalized size = 0.01 \[ \left [\frac{27 \, d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 37 \, d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) - 8 \,{\left (11 \, d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{c}}{1536 \, c^{\frac{3}{2}} x^{6}}, -\frac{27 \, d^{2} x^{6} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 37 \, d^{2} x^{6} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 4 \,{\left (11 \, d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{-c}}{768 \, \sqrt{-c} c x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^7),x, algorithm="fricas")

[Out]

[1/1536*(27*d^2*x^6*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 -
8*c)) + 37*d^2*x^6*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3 + c)*c)/x^3) - 8*(1
1*d*x^3 + 4*c)*sqrt(d*x^3 + c)*sqrt(c))/(c^(3/2)*x^6), -1/768*(27*d^2*x^6*arctan
(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) - 37*d^2*x^6*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c)
)) + 4*(11*d*x^3 + 4*c)*sqrt(d*x^3 + c)*sqrt(-c))/(sqrt(-c)*c*x^6)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221336, size = 127, normalized size = 1.22 \[ \frac{1}{768} \, d^{2}{\left (\frac{37 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{27 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{4 \,{\left (11 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 7 \, \sqrt{d x^{3} + c} c\right )}}{c d^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(d*x^3 + c)^(3/2)/((d*x^3 - 8*c)*x^7),x, algorithm="giac")

[Out]

1/768*d^2*(37*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c) - 27*arctan(1/3*sqrt
(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c) - 4*(11*(d*x^3 + c)^(3/2) - 7*sqrt(d*x^3 + c)
*c)/(c*d^2*x^6))