3.712 \(\int \frac{x^3}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=64 \[ \frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 \sqrt{c+d x^6}} \]

[Out]

(x^4*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 2, 1/2, 5/3, -((b*x^6)/a), -((d*x^6)/c)])
/(4*a^2*Sqrt[c + d*x^6])

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Rubi [A]  time = 0.246309, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x^4*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 2, 1/2, 5/3, -((b*x^6)/a), -((d*x^6)/c)])
/(4*a^2*Sqrt[c + d*x^6])

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Rubi in Sympy [A]  time = 26.2414, size = 53, normalized size = 0.83 \[ \frac{x^{4} \sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},2,\frac{5}{3},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{4 a^{2} c \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

x**4*sqrt(c + d*x**6)*appellf1(2/3, 1/2, 2, 5/3, -d*x**6/c, -b*x**6/a)/(4*a**2*c
*sqrt(1 + d*x**6/c))

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Mathematica [B]  time = 0.529893, size = 342, normalized size = 5.34 \[ \frac{x^4 \left (-\frac{8 b c d x^6 F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}+\frac{25 c (b c-3 a d) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}-\frac{5 b \left (c+d x^6\right )}{a}\right )}{30 \left (a+b x^6\right ) \sqrt{c+d x^6} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^3/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x^4*((-5*b*(c + d*x^6))/a + (25*c*(b*c - 3*a*d)*AppellF1[2/3, 1/2, 1, 5/3, -((d
*x^6)/c), -((b*x^6)/a)])/(-10*a*c*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*
x^6)/a)] + 3*x^6*(2*b*c*AppellF1[5/3, 1/2, 2, 8/3, -((d*x^6)/c), -((b*x^6)/a)] +
a*d*AppellF1[5/3, 3/2, 1, 8/3, -((d*x^6)/c), -((b*x^6)/a)])) - (8*b*c*d*x^6*App
ellF1[5/3, 1/2, 1, 8/3, -((d*x^6)/c), -((b*x^6)/a)])/(-16*a*c*AppellF1[5/3, 1/2,
1, 8/3, -((d*x^6)/c), -((b*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[8/3, 1/2, 2, 11/3,
-((d*x^6)/c), -((b*x^6)/a)] + a*d*AppellF1[8/3, 3/2, 1, 11/3, -((d*x^6)/c), -((b
*x^6)/a)]))))/(30*(-(b*c) + a*d)*(a + b*x^6)*Sqrt[c + d*x^6])

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

Timed out

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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(x**3/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)