3.717 \(\int \frac{1}{x^5 \left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.281496, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{2}{3};2,\frac{1}{2};\frac{1}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 x^4 \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.3719, size = 56, normalized size = 0.88 \[ - \frac{\sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{1}{2},2,\frac{1}{3},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{4 a^{2} c x^{4} \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

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

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Mathematica [B]  time = 1.19247, size = 399, normalized size = 6.23 \[ \frac{\frac{16 a x^6 \left (3 a^2 d^2+21 a b c d-20 b^2 c^2\right ) F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}+\frac{4 \left (c+d x^6\right ) \left (-3 a^2 d+3 a b \left (c-d x^6\right )+5 b^2 c x^6\right )}{c}+\frac{7 a b d x^{12} (3 a d-5 b c) F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{3 x^6 \left (2 b c F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-14 a c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}}{48 a^2 x^4 \left (a+b x^6\right ) \sqrt{c+d x^6} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

((4*(c + d*x^6)*(-3*a^2*d + 5*b^2*c*x^6 + 3*a*b*(c - d*x^6)))/c + (16*a*(-20*b^2
*c^2 + 21*a*b*c*d + 3*a^2*d^2)*x^6*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*a*b*d*(-5*b*c + 3*a*d)*x^12*
AppellF1[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)])))/(48*a^2*(-(b*c) + a*d)*x^4*(a + b*x^6)*Sqrt[c + d*x^6])

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5} \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^5/(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. \[ \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^5), 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(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^5),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(1/x**5/(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. \[ \int \frac{1}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^5),x, algorithm="giac")

[Out]

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