3.711 \(\int \frac{x^4}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=64 \[ \frac{x^5 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{5}{6};2,\frac{1}{2};\frac{11}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{5 a^2 \sqrt{c+d x^6}} \]

[Out]

(x^5*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6, 2, 1/2, 11/6, -((b*x^6)/a), -((d*x^6)/c)]
)/(5*a^2*Sqrt[c + d*x^6])

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Rubi [A]  time = 0.200847, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{x^5 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{5}{6};2,\frac{1}{2};\frac{11}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{5 a^2 \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x^5*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6, 2, 1/2, 11/6, -((b*x^6)/a), -((d*x^6)/c)]
)/(5*a^2*Sqrt[c + d*x^6])

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Rubi in Sympy [A]  time = 23.2677, size = 53, normalized size = 0.83 \[ \frac{x^{5} \sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (\frac{5}{6},\frac{1}{2},2,\frac{11}{6},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{5 a^{2} c \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

x**5*sqrt(c + d*x**6)*appellf1(5/6, 1/2, 2, 11/6, -d*x**6/c, -b*x**6/a)/(5*a**2*
c*sqrt(1 + d*x**6/c))

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Mathematica [B]  time = 0.520959, size = 342, normalized size = 5.34 \[ \frac{x^5 \left (-\frac{170 b c d x^6 F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{17}{6};\frac{1}{2},2;\frac{23}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{17}{6};\frac{3}{2},1;\frac{23}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-17 a c F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}+\frac{121 c (b c-6 a d) F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{11}{6};\frac{1}{2},2;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{11}{6};\frac{3}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-11 a c F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}-\frac{55 b \left (c+d x^6\right )}{a}\right )}{330 \left (a+b x^6\right ) \sqrt{c+d x^6} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(x^5*((-55*b*(c + d*x^6))/a + (121*c*(b*c - 6*a*d)*AppellF1[5/6, 1/2, 1, 11/6, -
((d*x^6)/c), -((b*x^6)/a)])/(-11*a*c*AppellF1[5/6, 1/2, 1, 11/6, -((d*x^6)/c), -
((b*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[11/6, 1/2, 2, 17/6, -((d*x^6)/c), -((b*x^6)
/a)] + a*d*AppellF1[11/6, 3/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])) - (170*b*c
*d*x^6*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])/(-17*a*c*Appell
F1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[17/6,
1/2, 2, 23/6, -((d*x^6)/c), -((b*x^6)/a)] + a*d*AppellF1[17/6, 3/2, 1, 23/6, -(
(d*x^6)/c), -((b*x^6)/a)]))))/(330*(-(b*c) + a*d)*(a + b*x^6)*Sqrt[c + d*x^6])

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(x^4/(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. \[ \int \frac{x^{4}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

integrate(x^4/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b^{2} x^{12} + 2 \, a b x^{6} + a^{2}\right )} \sqrt{d x^{6} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

integral(x^4/((b^2*x^12 + 2*a*b*x^6 + a^2)*sqrt(d*x^6 + c)), x)

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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**4/(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. \[ \int \frac{x^{4}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

integrate(x^4/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)