3.758 \(\int \frac{1}{x^2 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{1}{8};2,\frac{1}{2};\frac{7}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 x \sqrt{c+d x^8}} \]

[Out]

-((Sqrt[1 + (d*x^8)/c]*AppellF1[-1/8, 2, 1/2, 7/8, -((b*x^8)/a), -((d*x^8)/c)])/
(a^2*x*Sqrt[c + d*x^8]))

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Rubi [A]  time = 0.204087, 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 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{1}{8};2,\frac{1}{2};\frac{7}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a^2 x \sqrt{c+d x^8}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-((Sqrt[1 + (d*x^8)/c]*AppellF1[-1/8, 2, 1/2, 7/8, -((b*x^8)/a), -((d*x^8)/c)])/
(a^2*x*Sqrt[c + d*x^8]))

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Rubi in Sympy [A]  time = 25.2912, size = 53, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{8}} \operatorname{appellf_{1}}{\left (- \frac{1}{8},\frac{1}{2},2,\frac{7}{8},- \frac{d x^{8}}{c},- \frac{b x^{8}}{a} \right )}}{a^{2} c x \sqrt{1 + \frac{d x^{8}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

-sqrt(c + d*x**8)*appellf1(-1/8, 1/2, 2, 7/8, -d*x**8/c, -b*x**8/a)/(a**2*c*x*sq
rt(1 + d*x**8/c))

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Mathematica [B]  time = 1.45121, size = 399, normalized size = 6.44 \[ \frac{-\frac{75 a x^8 \left (24 a^2 d^2-40 a b c d+9 b^2 c^2\right ) F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{4 x^8 \left (2 b c F_1\left (\frac{15}{8};\frac{1}{2},2;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{15}{8};\frac{3}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-15 a c F_1\left (\frac{7}{8};\frac{1}{2},1;\frac{15}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+\frac{35 \left (c+d x^8\right ) \left (-8 a^2 d+8 a b \left (c-d x^8\right )+9 b^2 c x^8\right )}{c}+\frac{161 a b d x^{16} (9 b c-8 a d) F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{4 x^8 \left (2 b c F_1\left (\frac{23}{8};\frac{1}{2},2;\frac{31}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{23}{8};\frac{3}{2},1;\frac{31}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-23 a c F_1\left (\frac{15}{8};\frac{1}{2},1;\frac{23}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}}{280 a^2 x \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^2*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

((35*(c + d*x^8)*(-8*a^2*d + 9*b^2*c*x^8 + 8*a*b*(c - d*x^8)))/c - (75*a*(9*b^2*
c^2 - 40*a*b*c*d + 24*a^2*d^2)*x^8*AppellF1[7/8, 1/2, 1, 15/8, -((d*x^8)/c), -((
b*x^8)/a)])/(-15*a*c*AppellF1[7/8, 1/2, 1, 15/8, -((d*x^8)/c), -((b*x^8)/a)] + 4
*x^8*(2*b*c*AppellF1[15/8, 1/2, 2, 23/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*Appel
lF1[15/8, 3/2, 1, 23/8, -((d*x^8)/c), -((b*x^8)/a)])) + (161*a*b*d*(9*b*c - 8*a*
d)*x^16*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^8)/c), -((b*x^8)/a)])/(-23*a*c*Appel
lF1[15/8, 1/2, 1, 23/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*x^8*(2*b*c*AppellF1[23/8
, 1/2, 2, 31/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[23/8, 3/2, 1, 31/8, -
((d*x^8)/c), -((b*x^8)/a)])))/(280*a^2*(-(b*c) + a*d)*x*(a + b*x^8)*Sqrt[c + d*x
^8])

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Maple [F]  time = 0.119, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{18} + 2 \, a b x^{10} + a^{2} x^{2}\right )} \sqrt{d x^{8} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^2),x, algorithm="fricas")

[Out]

integral(1/((b^2*x^18 + 2*a*b*x^10 + a^2*x^2)*sqrt(d*x^8 + 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**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^2),x, algorithm="giac")

[Out]

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