3.759 \(\int \frac{1}{x^4 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=64 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a^2 x^3 \sqrt{c+d x^8}} \]

[Out]

-(Sqrt[1 + (d*x^8)/c]*AppellF1[-3/8, 2, 1/2, 5/8, -((b*x^8)/a), -((d*x^8)/c)])/(
3*a^2*x^3*Sqrt[c + d*x^8])

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Rubi [A]  time = 0.201814, antiderivative size = 64, 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 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{3}{8};2,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a^2 x^3 \sqrt{c+d x^8}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-(Sqrt[1 + (d*x^8)/c]*AppellF1[-3/8, 2, 1/2, 5/8, -((b*x^8)/a), -((d*x^8)/c)])/(
3*a^2*x^3*Sqrt[c + d*x^8])

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Rubi in Sympy [A]  time = 25.1768, size = 56, normalized size = 0.88 \[ - \frac{\sqrt{c + d x^{8}} \operatorname{appellf_{1}}{\left (- \frac{3}{8},\frac{1}{2},2,\frac{5}{8},- \frac{d x^{8}}{c},- \frac{b x^{8}}{a} \right )}}{3 a^{2} c x^{3} \sqrt{1 + \frac{d x^{8}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

-sqrt(c + d*x**8)*appellf1(-3/8, 1/2, 2, 5/8, -d*x**8/c, -b*x**8/a)/(3*a**2*c*x*
*3*sqrt(1 + d*x**8/c))

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Mathematica [B]  time = 1.52038, size = 399, normalized size = 6.23 \[ \frac{-\frac{169 a x^8 \left (8 a^2 d^2-56 a b c d+33 b^2 c^2\right ) F_1\left (\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{4 x^8 \left (2 b c F_1\left (\frac{13}{8};\frac{1}{2},2;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{13}{8};\frac{3}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-13 a c F_1\left (\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+\frac{65 \left (c+d x^8\right ) \left (-8 a^2 d+8 a b \left (c-d x^8\right )+11 b^2 c x^8\right )}{c}+\frac{105 a b d x^{16} (11 b c-8 a d) F_1\left (\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{4 x^8 \left (2 b c F_1\left (\frac{21}{8};\frac{1}{2},2;\frac{29}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{21}{8};\frac{3}{2},1;\frac{29}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-21 a c F_1\left (\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}}{1560 a^2 x^3 \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^4*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

((65*(c + d*x^8)*(-8*a^2*d + 11*b^2*c*x^8 + 8*a*b*(c - d*x^8)))/c - (169*a*(33*b
^2*c^2 - 56*a*b*c*d + 8*a^2*d^2)*x^8*AppellF1[5/8, 1/2, 1, 13/8, -((d*x^8)/c), -
((b*x^8)/a)])/(-13*a*c*AppellF1[5/8, 1/2, 1, 13/8, -((d*x^8)/c), -((b*x^8)/a)] +
4*x^8*(2*b*c*AppellF1[13/8, 1/2, 2, 21/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*App
ellF1[13/8, 3/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)])) + (105*a*b*d*(11*b*c - 8
*a*d)*x^16*AppellF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)])/(-21*a*c*Ap
pellF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*x^8*(2*b*c*AppellF1[2
1/8, 1/2, 2, 29/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[21/8, 3/2, 1, 29/8
, -((d*x^8)/c), -((b*x^8)/a)])))/(1560*a^2*(-(b*c) + a*d)*x^3*(a + b*x^8)*Sqrt[c
+ d*x^8])

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Maple [F]  time = 0.117, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^4/(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. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^4), 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^8 + a)^2*sqrt(d*x^8 + c)*x^4),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**4/(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. \[ \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^4),x, algorithm="giac")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^4), x)