3.1266 \(\int \frac {(a+b x^4)^p}{x} \, dx\)

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

[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 = {266, 65} \[ -\frac {\left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b x^4}{a}+1\right )}{4 a (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^p/x,x]

[Out]

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

Rule 65

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

Rule 266

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} \operatorname {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 = 0.02, size = 41, normalized size = 1.00 \[ -\frac {\left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b x^4}{a}+1\right )}{4 a (p+1)} \]

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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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{p}}{x}, x\right ) \]

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

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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maple [F]  time = 0.14, 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 \[ \int \frac {{\left (b x^{4} + a\right )}^{p}}{x}\,{d x} \]

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

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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sympy [C]  time = 7.82, size = 39, normalized size = 0.95 \[ - \frac {b^{p} x^{4 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (1 - p\right )} \]

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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