3.269 \(\int \frac{\sqrt{c+d x^3}}{x^3 \left (4 c+d x^3\right )} \, dx\)

Optimal. Leaf size=66 \[ -\frac{\sqrt{c+d x^3} F_1\left (-\frac{2}{3};1,-\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{8 c x^2 \sqrt{\frac{d x^3}{c}+1}} \]

[Out]

-(Sqrt[c + d*x^3]*AppellF1[-2/3, 1, -1/2, 1/3, -(d*x^3)/(4*c), -((d*x^3)/c)])/(8
*c*x^2*Sqrt[1 + (d*x^3)/c])

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Rubi [A]  time = 0.205045, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\sqrt{c+d x^3} F_1\left (-\frac{2}{3};1,-\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{8 c x^2 \sqrt{\frac{d x^3}{c}+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^3]/(x^3*(4*c + d*x^3)),x]

[Out]

-(Sqrt[c + d*x^3]*AppellF1[-2/3, 1, -1/2, 1/3, -(d*x^3)/(4*c), -((d*x^3)/c)])/(8
*c*x^2*Sqrt[1 + (d*x^3)/c])

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Rubi in Sympy [A]  time = 25.3777, size = 56, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},- \frac{1}{2},1,\frac{1}{3},- \frac{d x^{3}}{c},- \frac{d x^{3}}{4 c} \right )}}{8 c x^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c)**(1/2)/x**3/(d*x**3+4*c),x)

[Out]

-sqrt(c + d*x**3)*appellf1(-2/3, -1/2, 1, 1/3, -d*x**3/c, -d*x**3/(4*c))/(8*c*x*
*2*sqrt(1 + d*x**3/c))

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Mathematica [B]  time = 0.452843, size = 344, normalized size = 5.21 \[ \frac{\frac{7 d^2 x^6 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (3 d x^3 \left (F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )-28 c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )}+\frac{128 c d x^3 F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (16 c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )}-\frac{2 d x^3}{c}-2}{16 x^2 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[c + d*x^3]/(x^3*(4*c + d*x^3)),x]

[Out]

(-2 - (2*d*x^3)/c + (128*c*d*x^3*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -(d*x^
3)/(4*c)])/((4*c + d*x^3)*(16*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), -(d*x^3
)/(4*c)] - 3*d*x^3*(AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)] + 2
*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)]))) + (7*d^2*x^6*Appell
F1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)])/((4*c + d*x^3)*(-28*c*Appell
F1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -(d*x^3)/(4*c)] + 3*d*x^3*(AppellF1[7/3, 1/2,
2, 10/3, -((d*x^3)/c), -(d*x^3)/(4*c)] + 2*AppellF1[7/3, 3/2, 1, 10/3, -((d*x^3
)/c), -(d*x^3)/(4*c)]))))/(16*x^2*Sqrt[c + d*x^3])

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Maple [C]  time = 0.033, size = 1002, normalized size = 15.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^(1/2)/x^3/(d*x^3+4*c),x)

[Out]

1/4/c*(-1/2/x^2*(d*x^3+c)^(1/2)-1/2*I*3^(1/2)*(-c*d^2)^(1/3)*(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)*((x-1/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)*(-
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)/(d*x^3+c)^(1/2)*EllipticF(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),(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)))-1/4*d/c*(
-2/3*I*3^(1/2)/d*(-c*d^2)^(1/3)*(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)*((x-1/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)*(-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)/(d*x^3+c)^(1/2)*Ell
ipticF(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),(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))+1/3*I/d^3*2^(1/2)*sum(1/_alpha^2*(-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),1/6/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)),_alph
a=RootOf(_Z^3*d+4*c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{3} + c}}{{\left (d x^{3} + 4 \, c\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x^{3} + c}}{d x^{6} + 4 \, c x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^3 + c)/(d*x^6 + 4*c*x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{3}}}{x^{3} \left (4 c + d x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**3+c)**(1/2)/x**3/(d*x**3+4*c),x)

[Out]

Integral(sqrt(c + d*x**3)/(x**3*(4*c + d*x**3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{3} + c}}{{\left (d x^{3} + 4 \, c\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x^3), x)