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

Optimal. Leaf size=66 \[ \frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{4}{3};1,\frac{1}{2};\frac{7}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{16 c \sqrt{c+d x^3}} \]

[Out]

(x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1, 1/2, 7/3, -(d*x^3)/(4*c), -((d*x^3)/c)
])/(16*c*Sqrt[c + d*x^3])

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Rubi [A]  time = 0.205822, 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 \[ \frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{4}{3};1,\frac{1}{2};\frac{7}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{16 c \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1, 1/2, 7/3, -(d*x^3)/(4*c), -((d*x^3)/c)
])/(16*c*Sqrt[c + d*x^3])

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Rubi in Sympy [A]  time = 28.4695, size = 53, normalized size = 0.8 \[ \frac{x^{4} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{4}{3},\frac{1}{2},1,\frac{7}{3},- \frac{d x^{3}}{c},- \frac{d x^{3}}{4 c} \right )}}{16 c^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*sqrt(c + d*x**3)*appellf1(4/3, 1/2, 1, 7/3, -d*x**3/c, -d*x**3/(4*c))/(16*c
**2*sqrt(1 + d*x**3/c))

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Mathematica [B]  time = 0.0908576, size = 167, normalized size = 2.53 \[ \frac{7 c x^4 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\sqrt{c+d x^3} \left (4 c+d x^3\right ) \left (28 c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.05, size = 696, normalized size = 10.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*I/d^2*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)*E
llipticF(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))+4/9*I/d^4*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)),_al
pha=RootOf(_Z^3*d+4*c))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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