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

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

[Out]

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

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Rubi [A]  time = 0.161897, 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 \[ \frac{x^2 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{1}{3};1,\frac{1}{2};\frac{4}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{2 a \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.8379, size = 51, normalized size = 0.8 \[ \frac{x^{2} \sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{2},1,\frac{4}{3},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{2 a c \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.288023, size = 163, normalized size = 2.55 \[ -\frac{4 a c x^2 F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{\left (a+b x^6\right ) \sqrt{c+d x^6} \left (3 x^6 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*a*c*x^2*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^6)/c), -((b*x^6)/a)])/((a + b*x^6)
*Sqrt[c + d*x^6]*(-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*Appe
llF1[4/3, 3/2, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)])))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{x}{b{x}^{6}+a}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/((a + b*x**6)*sqrt(c + d*x**6)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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