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

Optimal. Leaf size=107 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{5/2}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{5/2}}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}-\frac{\sqrt{c+d x^3}}{48 c x^6} \]

[Out]

-Sqrt[c + d*x^3]/(48*c*x^6) - (d*Sqrt[c + d*x^3])/(64*c^2*x^3) + (d^2*ArcTanh[Sq
rt[c + d*x^3]/(3*Sqrt[c])])/(256*c^(5/2)) + (d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]
])/(256*c^(5/2))

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Rubi [A]  time = 0.402872, 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 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{5/2}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{5/2}}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}-\frac{\sqrt{c+d x^3}}{48 c x^6} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c + d*x^3]/(48*c*x^6) - (d*Sqrt[c + d*x^3])/(64*c^2*x^3) + (d^2*ArcTanh[Sq
rt[c + d*x^3]/(3*Sqrt[c])])/(256*c^(5/2)) + (d^2*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]
])/(256*c^(5/2))

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Rubi in Sympy [A]  time = 50.7096, size = 92, normalized size = 0.86 \[ - \frac{\sqrt{c + d x^{3}}}{48 c x^{6}} - \frac{d \sqrt{c + d x^{3}}}{64 c^{2} x^{3}} + \frac{d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{256 c^{\frac{5}{2}}} + \frac{d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{256 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)/(48*c*x**6) - d*sqrt(c + d*x**3)/(64*c**2*x**3) + d**2*atanh(s
qrt(c + d*x**3)/(3*sqrt(c)))/(256*c**(5/2)) + d**2*atanh(sqrt(c + d*x**3)/sqrt(c
))/(256*c**(5/2))

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Mathematica [C]  time = 0.366423, size = 341, normalized size = 3.19 \[ \frac{\frac{12 d^3 x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{c \left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}+\frac{5 d^3 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{c \left (8 c-d x^3\right ) \left (5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )\right )}-\frac{3 d^2}{2 c^2}-\frac{7 d}{2 c x^3}-\frac{2}{x^6}}{96 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-3*d^2)/(2*c^2) - 2/x^6 - (7*d)/(2*c*x^3) + (12*d^3*x^3*AppellF1[1, 1/2, 1, 2,
-((d*x^3)/c), (d*x^3)/(8*c)])/(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)]))) + (5*d^3*x^3*A
ppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), (8*c)/(d*x^3)])/(c*(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)])))/(96*Sqrt[c + d*x^3])

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Maple [C]  time = 0.037, size = 574, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8/c*(-1/6*(d*x^3+c)^(1/2)/x^6-1/12*d*(d*x^3+c)^(1/2)/c/x^3+1/12*d^2*arctanh((d
*x^3+c)^(1/2)/c^(1/2))/c^(3/2))+1/64*d/c^2*(-1/3*(d*x^3+c)^(1/2)/x^3-1/3*d*arcta
nh((d*x^3+c)^(1/2)/c^(1/2))/c^(1/2))+1/512*d^2/c^3*(2/3*(d*x^3+c)^(1/2)-2/3*arct
anh((d*x^3+c)^(1/2)/c^(1/2))*c^(1/2))-1/512*d^3/c^3*(2/3*(d*x^3+c)^(1/2)/d+1/3*I
/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. \[ -\int \frac{\sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.252486, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, d^{2} x^{6} \log \left (\frac{8 \,{\left (c d x^{3} + 4 \, c^{2}\right )} \sqrt{d x^{3} + c} +{\left (d^{2} x^{6} + 24 \, c d x^{3} + 32 \, c^{2}\right )} \sqrt{c}}{d x^{6} - 8 \, c x^{3}}\right ) - 8 \,{\left (3 \, d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{c}}{1536 \, c^{\frac{5}{2}} x^{6}}, \frac{3 \, d^{2} x^{6} \arctan \left (\frac{{\left (d x^{3} + 4 \, c\right )} \sqrt{-c}}{4 \, \sqrt{d x^{3} + c} c}\right ) - 4 \,{\left (3 \, d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{-c}}{768 \, \sqrt{-c} c^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1536*(3*d^2*x^6*log((8*(c*d*x^3 + 4*c^2)*sqrt(d*x^3 + c) + (d^2*x^6 + 24*c*d*
x^3 + 32*c^2)*sqrt(c))/(d*x^6 - 8*c*x^3)) - 8*(3*d*x^3 + 4*c)*sqrt(d*x^3 + c)*sq
rt(c))/(c^(5/2)*x^6), 1/768*(3*d^2*x^6*arctan(1/4*(d*x^3 + 4*c)*sqrt(-c)/(sqrt(d
*x^3 + c)*c)) - 4*(3*d*x^3 + 4*c)*sqrt(d*x^3 + c)*sqrt(-c))/(sqrt(-c)*c^2*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)**(1/2)/x**7/(-d*x**3+8*c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220168, size = 126, normalized size = 1.18 \[ -\frac{1}{768} \, d^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{3 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{4 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} + \sqrt{d x^{3} + c} c\right )}}{c^{2} d^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/768*d^2*(3*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) + 3*arctan(1/3*sqr
t(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) + 4*(3*(d*x^3 + c)^(3/2) + sqrt(d*x^3 + c)
*c)/(c^2*d^2*x^6))