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

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

[Out]

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

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Rubi [A]  time = 0.162972, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{x^2 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{1}{3};2,\frac{1}{2};\frac{4}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{2 a^2 \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.1844, size = 53, normalized size = 0.83 \[ \frac{x^{2} \sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{2},2,\frac{4}{3},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{2 a^{2} c \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.555039, size = 343, normalized size = 5.36 \[ \frac{x^2 \left (\frac{7 b c d x^6 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-14 a c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}+\frac{32 c (2 b c-3 a d) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}-\frac{4 b \left (c+d x^6\right )}{a}\right )}{24 \left (a+b x^6\right ) \sqrt{c+d x^6} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x^2*((-4*b*(c + d*x^6))/a + (32*c*(2*b*c - 3*a*d)*AppellF1[1/3, 1/2, 1, 4/3, -(
(d*x^6)/c), -((b*x^6)/a)])/(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^6)/c), -((b
*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^6)/c), -((b*x^6)/a)]
+ a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)])) + (7*b*c*d*x^6*Ap
pellF1[4/3, 1/2, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)])/(-14*a*c*AppellF1[4/3, 1/2
, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)] + 3*x^6*(2*b*c*AppellF1[7/3, 1/2, 2, 10/3,
-((d*x^6)/c), -((b*x^6)/a)] + a*d*AppellF1[7/3, 3/2, 1, 10/3, -((d*x^6)/c), -((
b*x^6)/a)]))))/(24*(-(b*c) + a*d)*(a + b*x^6)*Sqrt[c + d*x^6])

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{x}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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