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

Optimal. Leaf size=107 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2304 c^{7/2}}-\frac{7 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{7/2}}+\frac{5 d \sqrt{c+d x^3}}{192 c^3 x^3}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6} \]

[Out]

-Sqrt[c + d*x^3]/(48*c^2*x^6) + (5*d*Sqrt[c + d*x^3])/(192*c^3*x^3) + (d^2*ArcTa
nh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(2304*c^(7/2)) - (7*d^2*ArcTanh[Sqrt[c + d*x^3]
/Sqrt[c]])/(256*c^(7/2))

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Rubi [A]  time = 0.37379, antiderivative size = 107, 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 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2304 c^{7/2}}-\frac{7 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{7/2}}+\frac{5 d \sqrt{c+d x^3}}{192 c^3 x^3}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c + d*x^3]/(48*c^2*x^6) + (5*d*Sqrt[c + d*x^3])/(192*c^3*x^3) + (d^2*ArcTa
nh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(2304*c^(7/2)) - (7*d^2*ArcTanh[Sqrt[c + d*x^3]
/Sqrt[c]])/(256*c^(7/2))

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Rubi in Sympy [A]  time = 50.6554, size = 97, normalized size = 0.91 \[ - \frac{\sqrt{c + d x^{3}}}{48 c^{2} x^{6}} + \frac{5 d \sqrt{c + d x^{3}}}{192 c^{3} x^{3}} + \frac{d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{2304 c^{\frac{7}{2}}} - \frac{7 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{256 c^{\frac{7}{2}}} \]

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)**(1/2),x)

[Out]

-sqrt(c + d*x**3)/(48*c**2*x**6) + 5*d*sqrt(c + d*x**3)/(192*c**3*x**3) + d**2*a
tanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(2304*c**(7/2)) - 7*d**2*atanh(sqrt(c + d*x**
3)/sqrt(c))/(256*c**(7/2))

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Mathematica [C]  time = 0.316924, size = 332, normalized size = 3.1 \[ \frac{-\frac{40 c d^3 x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{70 c d^3 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{\left (d x^3-8 c\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-\frac{4 c^2}{x^6}+\frac{c d}{x^3}+5 d^2}{192 c^3 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*d^2 - (4*c^2)/x^6 + (c*d)/x^3 - (40*c*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)]))) + (70*c*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*AppellF
1[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)])))/(192*c^3*Sqrt[c + d*x^3])

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Maple [C]  time = 0.037, size = 540, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/c*(-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*arctanh(
(d*x^3+c)^(1/2)/c^(1/2))/c^(5/2))+1/64*d/c^2*(-1/3*(d*x^3+c)^(1/2)/c/x^3+1/3*d*a
rctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))-1/768*d^2*arctanh((d*x^3+c)^(1/2)/c^(1/
2))/c^(7/2)-1/13824*I/c^4*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*_al
pha^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. \[ -\int \frac{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.252336, size = 1, normalized size = 0.01 \[ \left [\frac{d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 63 \, d^{2} x^{6} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right ) + 24 \,{\left (5 \, d x^{3} - 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{c}}{4608 \, c^{\frac{7}{2}} x^{6}}, -\frac{d^{2} x^{6} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 63 \, d^{2} x^{6} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 12 \,{\left (5 \, d x^{3} - 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{-c}}{2304 \, \sqrt{-c} c^{3} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4608*(d^2*x^6*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 - 8*c
)) + 63*d^2*x^6*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3 + c)*c)/x^3) + 24*(5*d
*x^3 - 4*c)*sqrt(d*x^3 + c)*sqrt(c))/(c^(7/2)*x^6), -1/2304*(d^2*x^6*arctan(3*c/
(sqrt(d*x^3 + c)*sqrt(-c))) - 63*d^2*x^6*arctan(c/(sqrt(d*x^3 + c)*sqrt(-c))) -
12*(5*d*x^3 - 4*c)*sqrt(d*x^3 + c)*sqrt(-c))/(sqrt(-c)*c^3*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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221481, size = 127, normalized size = 1.19 \[ \frac{1}{2304} \, d^{2}{\left (\frac{63 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} - \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{12 \,{\left (5 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )}}{c^{3} d^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2304*d^2*(63*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) - arctan(1/3*sqrt
(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) + 12*(5*(d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 +
c)*c)/(c^3*d^2*x^6))