3.1264 \(\int \frac{\left (a+b x^4\right )^p}{x} \, dx\)

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]

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

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Rubi [A]  time = 0.051428, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b x^4}{a}+1\right )}{4 a (p+1)} \]

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))

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Rubi in Sympy [A]  time = 5.64421, size = 31, normalized size = 0.76 \[ - \frac{\left (a + b x^{4}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b x^{4}}{a}} \right )}}{4 a \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**p/x,x)

[Out]

-(a + b*x**4)**(p + 1)*hyper((1, p + 1), (p + 2,), 1 + b*x**4/a)/(4*a*(p + 1))

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Mathematica [A]  time = 0.026868, size = 51, normalized size = 1.24 \[ \frac{\left (\frac{a}{b x^4}+1\right )^{-p} \left (a+b x^4\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{a}{b x^4}\right )}{4 p} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^p/x,x]

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{4}+a \right ) ^{p}}{x}}\, dx \]

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

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

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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Sympy [A]  time = 48.4367, size = 39, normalized size = 0.95 \[ - \frac{b^{p} x^{4 p} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (- p + 1\right )} \]

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

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

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)