3.698 \(\int \frac{1}{x^2 \left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}} \]

[Out]

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

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Rubi [A]  time = 0.189551, antiderivative size = 62, 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 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.2079, size = 51, normalized size = 0.82 \[ - \frac{\sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (- \frac{1}{6},\frac{1}{2},1,\frac{5}{6},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{a c x \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.721137, size = 344, normalized size = 5.55 \[ \frac{\frac{121 x^6 (b c-2 a d) F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{\left (a+b x^6\right ) \left (3 x^6 \left (2 b c F_1\left (\frac{11}{6};\frac{1}{2},2;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{11}{6};\frac{3}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-11 a c F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )}-\frac{170 b d x^{12} F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{\left (a+b x^6\right ) \left (3 x^6 \left (2 b c F_1\left (\frac{17}{6};\frac{1}{2},2;\frac{23}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{17}{6};\frac{3}{2},1;\frac{23}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-17 a c F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )}-\frac{55 \left (c+d x^6\right )}{a c}}{55 x \sqrt{c+d x^6}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-55*(c + d*x^6))/(a*c) + (121*(b*c - 2*a*d)*x^6*AppellF1[5/6, 1/2, 1, 11/6, -(
(d*x^6)/c), -((b*x^6)/a)])/((a + b*x^6)*(-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*d*x^12*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])/((a
+ b*x^6)*(-17*a*c*AppellF1[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*Appell
F1[17/6, 3/2, 1, 23/6, -((d*x^6)/c), -((b*x^6)/a)]))))/(55*x*Sqrt[c + d*x^6])

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Maple [F]  time = 0.11, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( b{x}^{6}+a \right ) }{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{8} + a x^{2}\right )} \sqrt{d x^{6} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^2),x, algorithm="fricas")

[Out]

integral(1/((b*x^8 + a*x^2)*sqrt(d*x^6 + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^2),x, algorithm="giac")

[Out]

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