3.2.85 \(\int (1+b x^4)^p \, dx\) [185]

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

[Out]

x*hypergeom([1/4, -p],[5/4],-b*x^4)

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {251} \begin {gather*} x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + b*x^4)^p,x]

[Out]

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

Rule 251

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*}

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Mathematica [A]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} x \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-b x^4\right ) \end {gather*}

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 = 0.12, size = 17, normalized size = 0.94

method	result	size
meijerg	\(x \hypergeom \left (\left [\frac {1}{4}, -p \right ], \left [\frac {5}{4}\right ], -b \,x^{4}\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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] Result contains complex when optimal does not.
time = 3.23, size = 29, normalized size = 1.61 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.07, size = 15, normalized size = 0.83 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},-p;\ \frac {5}{4};\ -b\,x^4\right ) \end {gather*}

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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