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

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

[Out]

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

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Rubi [A]  time = 0.192979, 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 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};2,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a^2 x \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.2535, size = 53, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (- \frac{1}{6},\frac{1}{2},2,\frac{5}{6},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{a^{2} c x \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 1.32019, size = 399, normalized size = 6.44 \[ \frac{-\frac{121 a x^6 \left (12 a^2 d^2-24 a b c d+7 b^2 c^2\right ) F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{11}{6};\frac{1}{2},2;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{11}{6};\frac{3}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-11 a c F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}+\frac{55 \left (c+d x^6\right ) \left (-6 a^2 d+6 a b \left (c-d x^6\right )+7 b^2 c x^6\right )}{c}+\frac{170 a b d x^{12} (7 b c-6 a d) F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{17}{6};\frac{1}{2},2;\frac{23}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{17}{6};\frac{3}{2},1;\frac{23}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-17 a c F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}}{330 a^2 x \left (a+b x^6\right ) \sqrt{c+d x^6} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

((55*(c + d*x^6)*(-6*a^2*d + 7*b^2*c*x^6 + 6*a*b*(c - d*x^6)))/c - (121*a*(7*b^2
*c^2 - 24*a*b*c*d + 12*a^2*d^2)*x^6*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*Appe
llF1[11/6, 3/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])) + (170*a*b*d*(7*b*c - 6*a
*d)*x^12*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])/(-17*a*c*Appe
llF1[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*a^2*(-(b*c) + a*d)*x*(a + b*x^6)*Sqrt[c + d*
x^6])

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Maple [F]  time = 0.121, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x^2/(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. \[ \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{14} + 2 \, a b x^{8} + a^{2} x^{2}\right )} \sqrt{d x^{6} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((b^2*x^14 + 2*a*b*x^8 + a^2*x^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(1/x**2/(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. \[ \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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