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

Optimal. Leaf size=64 \[ \frac{x^5 \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{5}{8};2,\frac{1}{2};\frac{13}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{5 a^2 \sqrt{c+d x^8}} \]

[Out]

(x^5*Sqrt[1 + (d*x^8)/c]*AppellF1[5/8, 2, 1/2, 13/8, -((b*x^8)/a), -((d*x^8)/c)]
)/(5*a^2*Sqrt[c + d*x^8])

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Rubi [A]  time = 0.211893, 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 \[ \frac{x^5 \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{5}{8};2,\frac{1}{2};\frac{13}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{5 a^2 \sqrt{c+d x^8}} \]

Antiderivative was successfully verified.

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

[Out]

(x^5*Sqrt[1 + (d*x^8)/c]*AppellF1[5/8, 2, 1/2, 13/8, -((b*x^8)/a), -((d*x^8)/c)]
)/(5*a^2*Sqrt[c + d*x^8])

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Rubi in Sympy [A]  time = 24.4298, size = 53, normalized size = 0.83 \[ \frac{x^{5} \sqrt{c + d x^{8}} \operatorname{appellf_{1}}{\left (\frac{5}{8},\frac{1}{2},2,\frac{13}{8},- \frac{d x^{8}}{c},- \frac{b x^{8}}{a} \right )}}{5 a^{2} c \sqrt{1 + \frac{d x^{8}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*sqrt(c + d*x**8)*appellf1(5/8, 1/2, 2, 13/8, -d*x**8/c, -b*x**8/a)/(5*a**2*
c*sqrt(1 + d*x**8/c))

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Mathematica [B]  time = 0.69892, size = 343, normalized size = 5.36 \[ \frac{x^5 \left (-\frac{105 b c d x^8 F_1\left (\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{4 x^8 \left (2 b c F_1\left (\frac{21}{8};\frac{1}{2},2;\frac{29}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{21}{8};\frac{3}{2},1;\frac{29}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-21 a c F_1\left (\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}+\frac{169 c (3 b c-8 a d) F_1\left (\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{4 x^8 \left (2 b c F_1\left (\frac{13}{8};\frac{1}{2},2;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{13}{8};\frac{3}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-13 a c F_1\left (\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}-\frac{65 b \left (c+d x^8\right )}{a}\right )}{520 \left (a+b x^8\right ) \sqrt{c+d x^8} (a d-b c)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x^5*((-65*b*(c + d*x^8))/a + (169*c*(3*b*c - 8*a*d)*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*AppellF1[13/8, 3/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)])) - (105*b
*c*d*x^8*AppellF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)])/(-21*a*c*Appe
llF1[13/8, 1/2, 1, 21/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*x^8*(2*b*c*AppellF1[21/
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)]))))/(520*(-(b*c) + a*d)*(a + b*x^8)*Sqrt[c + d*x^8])

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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