3.185 \(\int (1+b x^4)^p \, dx\)

Optimal. Leaf size=18 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]

[Out]

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

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Rubi [A]  time = 0.0042675, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {245} \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + b*x^4)^p,x]

[Out]

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

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (1+b x^4\right )^p \, dx &=x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0016625, size = 18, normalized size = 1. \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + b*x^4)^p,x]

[Out]

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

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Maple [A]  time = 0.021, size = 17, normalized size = 0.9 \begin{align*} x{\mbox{$_2$F$_1$}({\frac{1}{4}},-p;\,{\frac{5}{4}};\,-b{x}^{4})} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+1)^p,x)

[Out]

x*hypergeom([1/4,-p],[5/4],-b*x^4)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + 1\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+1)^p,x, algorithm="maxima")

[Out]

integrate((b*x^4 + 1)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{4} + 1\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+1)^p,x, algorithm="fricas")

[Out]

integral((b*x^4 + 1)^p, x)

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Sympy [C]  time = 9.93296, size = 29, normalized size = 1.61 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, - p \\ \frac{5}{4} \end{matrix}\middle |{b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+1)**p,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + 1\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+1)^p,x, algorithm="giac")

[Out]

integrate((b*x^4 + 1)^p, x)