3.183 \(\int \left (a+b x^4\right )^p \, dx\)

Optimal. Leaf size=44 \[ x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right ) \]

[Out]

(x*(a + b*x^4)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^
p

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Rubi [A]  time = 0.0227041, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^p,x]

[Out]

(x*(a + b*x^4)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^
p

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Rubi in Sympy [A]  time = 3.98662, size = 34, normalized size = 0.77 \[ x \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**p,x)

[Out]

x*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*hyper((-p, 1/4), (5/4,), -b*x**4/a)

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Mathematica [A]  time = 0.0100062, size = 44, normalized size = 1. \[ x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^p,x]

[Out]

(x*(a + b*x^4)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^
p

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^p,x)

[Out]

int((b*x^4+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^p,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^p,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^p, x)

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Sympy [A]  time = 38.5955, size = 34, normalized size = 0.77 \[ \frac{a^{p} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, - p \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**p,x)

[Out]

a**p*x*gamma(1/4)*hyper((1/4, -p), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(5/
4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^p,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^p, x)