3.331 \(\int \frac{1}{x^7 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=128 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{20736 c^{9/2}}-\frac{109 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{9/2}}+\frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}} \]

[Out]

(245*d^2)/(1728*c^4*Sqrt[c + d*x^3]) - 1/(48*c^2*x^6*Sqrt[c + d*x^3]) + (3*d)/(6
4*c^3*x^3*Sqrt[c + d*x^3]) + (d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(20736*c
^(9/2)) - (109*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(768*c^(9/2))

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Rubi [A]  time = 0.487813, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{20736 c^{9/2}}-\frac{109 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{9/2}}+\frac{245 d^2}{1728 c^4 \sqrt{c+d x^3}}+\frac{3 d}{64 c^3 x^3 \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]

[Out]

(245*d^2)/(1728*c^4*Sqrt[c + d*x^3]) - 1/(48*c^2*x^6*Sqrt[c + d*x^3]) + (3*d)/(6
4*c^3*x^3*Sqrt[c + d*x^3]) + (d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(20736*c
^(9/2)) - (109*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(768*c^(9/2))

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Rubi in Sympy [A]  time = 70.4029, size = 119, normalized size = 0.93 \[ - \frac{1}{48 c^{2} x^{6} \sqrt{c + d x^{3}}} + \frac{3 d}{64 c^{3} x^{3} \sqrt{c + d x^{3}}} + \frac{245 d^{2}}{1728 c^{4} \sqrt{c + d x^{3}}} + \frac{d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{20736 c^{\frac{9}{2}}} - \frac{109 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{768 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)

[Out]

-1/(48*c**2*x**6*sqrt(c + d*x**3)) + 3*d/(64*c**3*x**3*sqrt(c + d*x**3)) + 245*d
**2/(1728*c**4*sqrt(c + d*x**3)) + d**2*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(207
36*c**(9/2)) - 109*d**2*atanh(sqrt(c + d*x**3)/sqrt(c))/(768*c**(9/2))

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Mathematica [C]  time = 0.362232, size = 336, normalized size = 2.62 \[ \frac{-\frac{1960 c d^3 x^9 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{3270 c d^3 x^9 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-36 c^2+81 c d x^3+245 d^2 x^6}{1728 c^4 x^6 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^7*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]

[Out]

(-36*c^2 + 81*c*d*x^3 + 245*d^2*x^6 - (1960*c*d^3*x^9*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)]))) + (3270*c*d^3*x^9*Ap
pellF1[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)])))/(1728*c^4*x^6*Sqrt[c + d*x^3])

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Maple [C]  time = 0.044, size = 636, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x)

[Out]

1/8/c*(-1/6*(d*x^3+c)^(1/2)/c^2/x^6+7/12*d*(d*x^3+c)^(1/2)/c^3/x^3+2/3*d^2/c^3/(
(x^3+c/d)*d)^(1/2)-5/4*d^2*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(7/2))+1/64*d/c^2*
(-1/3*(d*x^3+c)^(1/2)/c^2/x^3-2/3*d/c^2/((x^3+c/d)*d)^(1/2)+d*arctanh((d*x^3+c)^
(1/2)/c^(1/2))/c^(5/2))+1/512*d^2/c^3*(2/3/c/((x^3+c/d)*d)^(1/2)-2/3*arctanh((d*
x^3+c)^(1/2)/c^(1/2))/c^(3/2))-1/512*d^3/c^3*(2/27/d/c/((x^3+c/d)*d)^(1/2)+1/243
*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)*_a
lpha*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. \[ -\int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^7),x, algorithm="maxima")

[Out]

-integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^7), x)

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Fricas [A]  time = 0.243703, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d x^{3} + c} d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 2943 \, \sqrt{d x^{3} + c} d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 24 \,{\left (245 \, d^{2} x^{6} + 81 \, c d x^{3} - 36 \, c^{2}\right )} \sqrt{c}}{41472 \, \sqrt{d x^{3} + c} c^{\frac{9}{2}} x^{6}}, -\frac{\sqrt{d x^{3} + c} d^{2} x^{6} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 2943 \, \sqrt{d x^{3} + c} d^{2} x^{6} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 12 \,{\left (245 \, d^{2} x^{6} + 81 \, c d x^{3} - 36 \, c^{2}\right )} \sqrt{-c}}{20736 \, \sqrt{d x^{3} + c} \sqrt{-c} c^{4} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^7),x, algorithm="fricas")

[Out]

[1/41472*(sqrt(d*x^3 + c)*d^2*x^6*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c
)*c)/(d*x^3 - 8*c)) + 2943*sqrt(d*x^3 + c)*d^2*x^6*log(((d*x^3 + 2*c)*sqrt(c) -
2*sqrt(d*x^3 + c)*c)/x^3) + 24*(245*d^2*x^6 + 81*c*d*x^3 - 36*c^2)*sqrt(c))/(sqr
t(d*x^3 + c)*c^(9/2)*x^6), -1/20736*(sqrt(d*x^3 + c)*d^2*x^6*arctan(3*c/(sqrt(d*
x^3 + c)*sqrt(-c))) - 2943*sqrt(d*x^3 + c)*d^2*x^6*arctan(c/(sqrt(d*x^3 + c)*sqr
t(-c))) - 12*(245*d^2*x^6 + 81*c*d*x^3 - 36*c^2)*sqrt(-c))/(sqrt(d*x^3 + c)*sqrt
(-c)*c^4*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(1/x**7/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219562, size = 146, normalized size = 1.14 \[ \frac{1}{20736} \, d^{2}{\left (\frac{2943 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{4}} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{4}} + \frac{1536}{\sqrt{d x^{3} + c} c^{4}} + \frac{108 \,{\left (13 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 17 \, \sqrt{d x^{3} + c} c\right )}}{c^{4} d^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^7),x, algorithm="giac")

[Out]

1/20736*d^2*(2943*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) - arctan(1/3*s
qrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) + 1536/(sqrt(d*x^3 + c)*c^4) + 108*(13*(
d*x^3 + c)^(3/2) - 17*sqrt(d*x^3 + c)*c)/(c^4*d^2*x^6))