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

Optimal. Leaf size=185 \[ \frac{13 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{497664 c^{11/2}}-\frac{33 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{11/2}}+\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

[Out]

(665*d^2)/(41472*c^5*Sqrt[c + d*x^3]) - (71*d^2)/(13824*c^4*(8*c - d*x^3)*Sqrt[c
+ d*x^3]) - 1/(48*c^2*x^6*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (17*d)/(384*c^3*x^3*
(8*c - d*x^3)*Sqrt[c + d*x^3]) + (13*d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(
497664*c^(11/2)) - (33*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(2048*c^(11/2))

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Rubi [A]  time = 0.649548, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ \frac{13 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{497664 c^{11/2}}-\frac{33 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{11/2}}+\frac{665 d^2}{41472 c^5 \sqrt{c+d x^3}}-\frac{71 d^2}{13824 c^4 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\frac{17 d}{384 c^3 x^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{1}{48 c^2 x^6 \left (8 c-d x^3\right ) \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(665*d^2)/(41472*c^5*Sqrt[c + d*x^3]) - (71*d^2)/(13824*c^4*(8*c - d*x^3)*Sqrt[c
+ d*x^3]) - 1/(48*c^2*x^6*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (17*d)/(384*c^3*x^3*
(8*c - d*x^3)*Sqrt[c + d*x^3]) + (13*d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(
497664*c^(11/2)) - (33*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(2048*c^(11/2))

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Rubi in Sympy [A]  time = 97.0921, size = 160, normalized size = 0.86 \[ - \frac{1}{48 c^{2} x^{6} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} + \frac{11 d}{3456 c^{3} x^{3} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} + \frac{71 d}{13824 c^{4} x^{3} \sqrt{c + d x^{3}}} + \frac{665 d^{2}}{41472 c^{5} \sqrt{c + d x^{3}}} + \frac{13 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{497664 c^{\frac{11}{2}}} - \frac{33 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{2048 c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(48*c**2*x**6*sqrt(c + d*x**3)*(8*c - d*x**3)) + 11*d/(3456*c**3*x**3*sqrt(c
+ d*x**3)*(8*c - d*x**3)) + 71*d/(13824*c**4*x**3*sqrt(c + d*x**3)) + 665*d**2/(
41472*c**5*sqrt(c + d*x**3)) + 13*d**2*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(4976
64*c**(11/2)) - 33*d**2*atanh(sqrt(c + d*x**3)/sqrt(c))/(2048*c**(11/2))

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Mathematica [C]  time = 0.462981, size = 349, normalized size = 1.89 \[ \frac{\frac{\frac{8910 c d^3 x^9 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}+864 c^3-1836 c^2 d x^3-5107 c d^2 x^6+665 d^3 x^9}{d x^3-8 c}-\frac{5320 c d^3 x^9 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}}{41472 c^5 x^6 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-5320*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*(AppellF
1[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)]))) + (864*c^3 - 1836*c^2*d*x^3 - 5107*c*d^2*x^6 + 665*d^3*x
^9 + (8910*c*d^3*x^9*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))/(41472*c^5*x^6*Sqrt[c + d*x^3])

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Maple [C]  time = 0.018, size = 1106, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64/c^2*(-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))+3/4096/c
^4*d^2*(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))+
1/256/c^3*d*(-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*arctan
h((d*x^3+c)^(1/2)/c^(1/2))/c^(5/2))+1/512*d^3/c^3*(-1/243/d/c^2*(d*x^3+c)^(1/2)/
(d*x^3-8*c)-2/243/d/c^2/((x^3+c/d)*d)^(1/2)-1/1458*I/d^3/c^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)),_alp
ha=RootOf(_Z^3*d-8*c)))-3/4096*d^3/c^4*(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)*_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. \[ \int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )}^{2} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232975, size = 1, normalized size = 0.01 \[ \left [\frac{13 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 8019 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 24 \,{\left (665 \, d^{3} x^{9} - 5107 \, c d^{2} x^{6} - 1836 \, c^{2} d x^{3} + 864 \, c^{3}\right )} \sqrt{c}}{995328 \,{\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )} \sqrt{d x^{3} + c} \sqrt{c}}, -\frac{13 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{d x^{3} + c} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 8019 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{d x^{3} + c} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 12 \,{\left (665 \, d^{3} x^{9} - 5107 \, c d^{2} x^{6} - 1836 \, c^{2} d x^{3} + 864 \, c^{3}\right )} \sqrt{-c}}{497664 \,{\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )} \sqrt{d x^{3} + c} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/995328*(13*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(d*x^3 + c)*log(((d*x^3 + 10*c)*sqrt(c
) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 - 8*c)) + 8019*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(d*x^
3 + c)*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3 + c)*c)/x^3) + 24*(665*d^3*x^9
- 5107*c*d^2*x^6 - 1836*c^2*d*x^3 + 864*c^3)*sqrt(c))/((c^5*d*x^9 - 8*c^6*x^6)*s
qrt(d*x^3 + c)*sqrt(c)), -1/497664*(13*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(d*x^3 + c)*a
rctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) - 8019*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(d*x^3
+ c)*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) - 12*(665*d^3*x^9 - 5107*c*d^2*x^6 - 1
836*c^2*d*x^3 + 864*c^3)*sqrt(-c))/((c^5*d*x^9 - 8*c^6*x^6)*sqrt(d*x^3 + c)*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(1/x**7/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222187, size = 180, normalized size = 0.97 \[ \frac{1}{497664} \, d^{2}{\left (\frac{8019 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{5}} - \frac{13 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{5}} + \frac{12 \,{\left (341 \, d x^{3} - 2731 \, c\right )}}{{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )} c^{5}} + \frac{1296 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 4 \, \sqrt{d x^{3} + c} c\right )}}{c^{5} d^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/497664*d^2*(8019*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^5) - 13*arctan(1
/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^5) + 12*(341*d*x^3 - 2731*c)/(((d*x^3 +
c)^(3/2) - 9*sqrt(d*x^3 + c)*c)*c^5) + 1296*(3*(d*x^3 + c)^(3/2) - 4*sqrt(d*x^3
+ c)*c)/(c^5*d^2*x^6))