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

Optimal. Leaf size=59 \[ \frac{x \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{1}{8};1,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a \sqrt{c+d x^8}} \]

[Out]

(x*Sqrt[1 + (d*x^8)/c]*AppellF1[1/8, 1, 1/2, 9/8, -((b*x^8)/a), -((d*x^8)/c)])/(
a*Sqrt[c + d*x^8])

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Rubi [A]  time = 0.0923407, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{x \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{1}{8};1,\frac{1}{2};\frac{9}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{a \sqrt{c+d x^8}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[1 + (d*x^8)/c]*AppellF1[1/8, 1, 1/2, 9/8, -((b*x^8)/a), -((d*x^8)/c)])/(
a*Sqrt[c + d*x^8])

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Rubi in Sympy [A]  time = 21.9986, size = 48, normalized size = 0.81 \[ \frac{x \sqrt{c + d x^{8}} \operatorname{appellf_{1}}{\left (\frac{1}{8},\frac{1}{2},1,\frac{9}{8},- \frac{d x^{8}}{c},- \frac{b x^{8}}{a} \right )}}{a c \sqrt{1 + \frac{d x^{8}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(c + d*x**8)*appellf1(1/8, 1/2, 1, 9/8, -d*x**8/c, -b*x**8/a)/(a*c*sqrt(1
+ d*x**8/c))

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Mathematica [B]  time = 0.272379, size = 161, normalized size = 2.73 \[ -\frac{9 a c x F_1\left (\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{\left (a+b x^8\right ) \sqrt{c+d x^8} \left (4 x^8 \left (2 b c F_1\left (\frac{9}{8};\frac{1}{2},2;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+a d F_1\left (\frac{9}{8};\frac{3}{2},1;\frac{17}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )-9 a c F_1\left (\frac{1}{8};\frac{1}{2},1;\frac{9}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-9*a*c*x*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)])/((a + b*x^8)*S
qrt[c + d*x^8]*(-9*a*c*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)] +
4*x^8*(2*b*c*AppellF1[9/8, 1/2, 2, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*Appel
lF1[9/8, 3/2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)])))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x**8)*sqrt(c + d*x**8)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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