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

Optimal. Leaf size=164 \[ \frac{31 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{165888 c^{9/2}}-\frac{19 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{6144 c^{9/2}}-\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )} \]

[Out]

(-35*d^2*Sqrt[c + d*x^3])/(13824*c^4*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(48*c^2*x^
6*(8*c - d*x^3)) + (3*d*Sqrt[c + d*x^3])/(128*c^3*x^3*(8*c - d*x^3)) + (31*d^2*A
rcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(165888*c^(9/2)) - (19*d^2*ArcTanh[Sqrt[c +
d*x^3]/Sqrt[c]])/(6144*c^(9/2))

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Rubi [A]  time = 0.508699, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{31 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{165888 c^{9/2}}-\frac{19 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{6144 c^{9/2}}-\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-35*d^2*Sqrt[c + d*x^3])/(13824*c^4*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(48*c^2*x^
6*(8*c - d*x^3)) + (3*d*Sqrt[c + d*x^3])/(128*c^3*x^3*(8*c - d*x^3)) + (31*d^2*A
rcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(165888*c^(9/2)) - (19*d^2*ArcTanh[Sqrt[c +
d*x^3]/Sqrt[c]])/(6144*c^(9/2))

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Rubi in Sympy [A]  time = 75.0742, size = 138, normalized size = 0.84 \[ - \frac{\sqrt{c + d x^{3}}}{48 c^{2} x^{6} \left (8 c - d x^{3}\right )} + \frac{11 d \sqrt{c + d x^{3}}}{3456 c^{3} x^{3} \left (8 c - d x^{3}\right )} + \frac{35 d \sqrt{c + d x^{3}}}{13824 c^{4} x^{3}} + \frac{31 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{165888 c^{\frac{9}{2}}} - \frac{19 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{6144 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)/(48*c**2*x**6*(8*c - d*x**3)) + 11*d*sqrt(c + d*x**3)/(3456*c*
*3*x**3*(8*c - d*x**3)) + 35*d*sqrt(c + d*x**3)/(13824*c**4*x**3) + 31*d**2*atan
h(sqrt(c + d*x**3)/(3*sqrt(c)))/(165888*c**(9/2)) - 19*d**2*atanh(sqrt(c + d*x**
3)/sqrt(c))/(6144*c**(9/2))

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Mathematica [C]  time = 0.416793, size = 349, normalized size = 2.13 \[ \frac{\frac{\frac{570 c d^3 x^9 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}+288 c^3-36 c^2 d x^3-289 c d^2 x^6+35 d^3 x^9}{d x^3-8 c}-\frac{280 c d^3 x^9 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}}{13824 c^4 x^6 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-280*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)]))) + (288*c^3 - 36*c^2*d*x^3 - 289*c*d^2*x^6 + 35*d^3*x^9 +
(570*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))/(13824*c^4*x^6*Sqrt[c + d*x^3])

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Maple [C]  time = 0.019, size = 989, 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)^(1/2),x)

[Out]

1/64/c^2*(-1/6*(d*x^3+c)^(1/2)/c/x^6+1/4*d*(d*x^3+c)^(1/2)/c^2/x^3-1/4*d^2*arcta
nh((d*x^3+c)^(1/2)/c^(1/2))/c^(5/2))-1/2048*d^2*arctanh((d*x^3+c)^(1/2)/c^(1/2))
/c^(9/2)+1/256/c^3*d*(-1/3*(d*x^3+c)^(1/2)/c/x^3+1/3*d*arctanh((d*x^3+c)^(1/2)/c
^(1/2))/c^(3/2))+1/512*d^3/c^3*(-1/27/d/c*(d*x^3+c)^(1/2)/(d*x^3-8*c)-1/486*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)))-1/36864*I/c^5*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)),_a
lpha=RootOf(_Z^3*d-8*c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )}^{2} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243214, size = 1, normalized size = 0.01 \[ \left [\frac{24 \,{\left (35 \, d^{2} x^{6} - 324 \, c d x^{3} + 288 \, c^{2}\right )} \sqrt{d x^{3} + c} \sqrt{c} + 31 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 513 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{331776 \,{\left (c^{4} d x^{9} - 8 \, c^{5} x^{6}\right )} \sqrt{c}}, \frac{12 \,{\left (35 \, d^{2} x^{6} - 324 \, c d x^{3} + 288 \, c^{2}\right )} \sqrt{d x^{3} + c} \sqrt{-c} - 31 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 513 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{165888 \,{\left (c^{4} d x^{9} - 8 \, c^{5} x^{6}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/331776*(24*(35*d^2*x^6 - 324*c*d*x^3 + 288*c^2)*sqrt(d*x^3 + c)*sqrt(c) + 31*
(d^3*x^9 - 8*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)) + 513*(d^3*x^9 - 8*c*d^2*x^6)*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^
3 + c)*c)/x^3))/((c^4*d*x^9 - 8*c^5*x^6)*sqrt(c)), 1/165888*(12*(35*d^2*x^6 - 32
4*c*d*x^3 + 288*c^2)*sqrt(d*x^3 + c)*sqrt(-c) - 31*(d^3*x^9 - 8*c*d^2*x^6)*arcta
n(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) + 513*(d^3*x^9 - 8*c*d^2*x^6)*arctan(c/(sqrt(d
*x^3 + c)*sqrt(-c))))/((c^4*d*x^9 - 8*c^5*x^6)*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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220342, size = 158, normalized size = 0.96 \[ \frac{1}{165888} \, d^{2}{\left (\frac{513 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{4}} - \frac{31 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{4}} - \frac{12 \, \sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} c^{4}} + \frac{432 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 2 \, \sqrt{d x^{3} + c} c\right )}}{c^{4} d^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/165888*d^2*(513*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) - 31*arctan(1/
3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) - 12*sqrt(d*x^3 + c)/((d*x^3 - 8*c)*c
^4) + 432*((d*x^3 + c)^(3/2) - 2*sqrt(d*x^3 + c)*c)/(c^4*d^2*x^6))