3.802 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x} \, dx\)

Optimal. Leaf size=63 \[ -\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac{b x^2}{a}+1\right )}{2 a (2 p+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0946251, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);\frac{b x^2}{a}+1\right )}{2 a (2 p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.6159, size = 76, normalized size = 1.21 \[ - \frac{\left (a b + b^{2} x^{2}\right )^{- 2 p} \left (a b + b^{2} x^{2}\right )^{2 p + 1} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} 1, 2 p + 1 \\ 2 p + 2 \end{matrix}\middle |{1 + \frac{b x^{2}}{a}} \right )}}{2 a b \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**p/x,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0241706, size = 53, normalized size = 0.84 \[ \frac{\left (\frac{a}{b x^2}+1\right )^{-2 p} \left (\left (a+b x^2\right )^2\right )^p \, _2F_1\left (-2 p,-2 p;1-2 p;-\frac{a}{b x^2}\right )}{4 p} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/x,x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{ \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^p/x,x)

[Out]

int((b^2*x^4+2*a*b*x^2+a^2)^p/x,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/x,x, algorithm="maxima")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/x, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/x,x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)^p/x, x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**p/x,x)

[Out]

Integral(((a + b*x**2)**2)**p/x, x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/x,x, algorithm="giac")

[Out]

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/x, x)