3.725 \(\int \frac{x^3}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=54 \[ \frac{\tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 \sqrt{a} \sqrt{b c-a d}} \]

[Out]

ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])]/(4*Sqrt[a]*Sqrt[b*c - a*
d])

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Rubi [A]  time = 0.13969, antiderivative size = 54, 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 \[ \frac{\tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 \sqrt{a} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])]/(4*Sqrt[a]*Sqrt[b*c - a*
d])

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Rubi in Sympy [A]  time = 19.228, size = 46, normalized size = 0.85 \[ \frac{\operatorname{atanh}{\left (\frac{x^{4} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{8}}} \right )}}{4 \sqrt{a} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

atanh(x**4*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**8)))/(4*sqrt(a)*sqrt(a*d - b*c
))

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Mathematica [A]  time = 0.0408359, size = 54, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{4 \sqrt{a} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])]/(4*Sqrt[a]*Sqrt[b*c - a*
d])

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(x^3/(b*x^8+a)/(d*x^8+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270894, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{12} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{4}\right )} \sqrt{d x^{8} + c} +{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{16} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{8} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{16} + 2 \, a b x^{8} + a^{2}}\right )}{16 \, \sqrt{-a b c + a^{2} d}}, \frac{\arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{8} - a c}{2 \, \sqrt{d x^{8} + c} \sqrt{a b c - a^{2} d} x^{4}}\right )}{8 \, \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

[1/16*log((4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^12 - (a^2*b*c^2 - a^3*c*d)
*x^4)*sqrt(d*x^8 + c) + ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^16 - 2*(3*a*b*c^2 -
4*a^2*c*d)*x^8 + a^2*c^2)*sqrt(-a*b*c + a^2*d))/(b^2*x^16 + 2*a*b*x^8 + a^2))/s
qrt(-a*b*c + a^2*d), 1/8*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)/(sqrt(d*x^8 + c)*s
qrt(a*b*c - a^2*d)*x^4))/sqrt(a*b*c - a^2*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b x^{8}\right ) \sqrt{c + d x^{8}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Integral(x**3/((a + b*x**8)*sqrt(c + d*x**8)), x)

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GIAC/XCAS [A]  time = 0.22934, size = 97, normalized size = 1.8 \[ -\frac{\sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x^{4} - \sqrt{d x^{8} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{4 \, \sqrt{a b c d - a^{2} d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

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