3.1452 \(\int \frac{x^3}{a+b x^8} \, dx\)

Optimal. Leaf size=29 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b}} \]

[Out]

ArcTan[(Sqrt[b]*x^4)/Sqrt[a]]/(4*Sqrt[a]*Sqrt[b])

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Rubi [A]  time = 0.0391394, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^8),x]

[Out]

ArcTan[(Sqrt[b]*x^4)/Sqrt[a]]/(4*Sqrt[a]*Sqrt[b])

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Rubi in Sympy [A]  time = 5.36744, size = 26, normalized size = 0.9 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{4}}{\sqrt{a}} \right )}}{4 \sqrt{a} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**8+a),x)

[Out]

atan(sqrt(b)*x**4/sqrt(a))/(4*sqrt(a)*sqrt(b))

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Mathematica [A]  time = 0.0118336, size = 29, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^4}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x^8),x]

[Out]

ArcTan[(Sqrt[b]*x^4)/Sqrt[a]]/(4*Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.003, size = 19, normalized size = 0.7 \[{\frac{1}{4}\arctan \left ({b{x}^{4}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^8+a),x)

[Out]

1/4/(a*b)^(1/2)*arctan(x^4*b/(a*b)^(1/2))

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220754, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, a b x^{4} +{\left (b x^{8} - a\right )} \sqrt{-a b}}{b x^{8} + a}\right )}{8 \, \sqrt{-a b}}, -\frac{\arctan \left (\frac{a}{\sqrt{a b} x^{4}}\right )}{4 \, \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*log((2*a*b*x^4 + (b*x^8 - a)*sqrt(-a*b))/(b*x^8 + a))/sqrt(-a*b), -1/4*arct
an(a/(sqrt(a*b)*x^4))/sqrt(a*b)]

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Sympy [A]  time = 0.658494, size = 56, normalized size = 1.93 \[ - \frac{\sqrt{- \frac{1}{a b}} \log{\left (- a \sqrt{- \frac{1}{a b}} + x^{4} \right )}}{8} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (a \sqrt{- \frac{1}{a b}} + x^{4} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**8+a),x)

[Out]

-sqrt(-1/(a*b))*log(-a*sqrt(-1/(a*b)) + x**4)/8 + sqrt(-1/(a*b))*log(a*sqrt(-1/(
a*b)) + x**4)/8

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GIAC/XCAS [A]  time = 0.2275, size = 24, normalized size = 0.83 \[ \frac{\arctan \left (\frac{b x^{4}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*arctan(b*x^4/sqrt(a*b))/sqrt(a*b)