3.13.66 \(\int \frac {(a+b x^4)^p}{x} \, dx\) [1266]

Optimal. Leaf size=41 \[ -\frac {\left (a+b x^4\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^4}{a}\right )}{4 a (1+p)} \]

[Out]

-1/4*(b*x^4+a)^(1+p)*hypergeom([1, 1+p],[2+p],1+b*x^4/a)/a/(1+p)

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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 67} \begin {gather*} -\frac {\left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b x^4}{a}+1\right )}{4 a (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^p/x,x]

[Out]

-1/4*((a + b*x^4)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^4)/a])/(a*(1 + p))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^4\right )^p}{x} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,x^4\right )\\ &=-\frac {\left (a+b x^4\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^4}{a}\right )}{4 a (1+p)}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 41, normalized size = 1.00 \begin {gather*} -\frac {\left (a+b x^4\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^4}{a}\right )}{4 a (1+p)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^p/x,x]

[Out]

-1/4*((a + b*x^4)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*x^4)/a])/(a*(1 + p))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{p}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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+a)^p/x,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^p/x, x)

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Fricas [F]
time = 0.38, size = 15, normalized size = 0.37 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{p}}{x}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^4 + a)^p/x, x)

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Sympy [C] Result contains complex when optimal does not.
time = 2.52, size = 39, normalized size = 0.95 \begin {gather*} - \frac {b^{p} x^{4 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (1 - p\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**p/x,x)

[Out]

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

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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+a)^p/x,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^p/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^p}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^p/x,x)

[Out]

int((a + b*x^4)^p/x, x)

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