3.548 \(\int x^{-1+2 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p \, dx\)

Optimal. Leaf size=103 \[ \frac{a^2 \left (\frac{b x^n}{a}+1\right )^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 b^2 n (p+1)}-\frac{a^2 \left (\frac{b x^n}{a}+1\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{b^2 n (2 p+1)} \]

[Out]

-((a^2*(1 + (b*x^n)/a)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p)/(b^2*n*(1 + 2*p))) + (
a^2*(1 + (b*x^n)/a)^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p)/(2*b^2*n*(1 + p))

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Rubi [A]  time = 0.139361, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a^2 \left (\frac{b x^n}{a}+1\right )^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 b^2 n (p+1)}-\frac{a^2 \left (\frac{b x^n}{a}+1\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{b^2 n (2 p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p,x]

[Out]

-((a^2*(1 + (b*x^n)/a)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p)/(b^2*n*(1 + 2*p))) + (
a^2*(1 + (b*x^n)/a)^2*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p)/(2*b^2*n*(1 + p))

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Rubi in Sympy [A]  time = 18.8872, size = 78, normalized size = 0.76 \[ - \frac{a \left (2 a + 2 b x^{n}\right ) \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{p}}{2 b^{2} n \left (2 p + 1\right )} + \frac{\left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{p + 1}}{2 b^{2} n \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)*(a**2+2*a*b*x**n+b**2*x**(2*n))**p,x)

[Out]

-a*(2*a + 2*b*x**n)*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**p/(2*b**2*n*(2*p + 1))
+ (a**2 + 2*a*b*x**n + b**2*x**(2*n))**(p + 1)/(2*b**2*n*(p + 1))

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Mathematica [A]  time = 0.0541331, size = 54, normalized size = 0.52 \[ \frac{\left (a+b x^n\right ) \left (\left (a+b x^n\right )^2\right )^p \left (b (2 p+1) x^n-a\right )}{2 b^2 n (p+1) (2 p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 2*n)*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p,x]

[Out]

((a + b*x^n)*((a + b*x^n)^2)^p*(-a + b*(1 + 2*p)*x^n))/(2*b^2*n*(1 + p)*(1 + 2*p
))

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Maple [C]  time = 0.109, size = 148, normalized size = 1.4 \[ -{\frac{-2\,{b}^{2}p \left ({x}^{n} \right ) ^{2}-2\,ap{x}^{n}b-{b}^{2} \left ({x}^{n} \right ) ^{2}+{a}^{2}}{ \left ( 2+4\,p \right ) \left ( 1+p \right ) n{b}^{2}}{{\rm e}^{-{\frac{p \left ( i\pi \, \left ({\it csgn} \left ( i \left ( a+b{x}^{n} \right ) ^{2} \right ) \right ) ^{3}-2\,i\pi \, \left ({\it csgn} \left ( i \left ( a+b{x}^{n} \right ) ^{2} \right ) \right ) ^{2}{\it csgn} \left ( i \left ( a+b{x}^{n} \right ) \right ) +i\pi \,{\it csgn} \left ( i \left ( a+b{x}^{n} \right ) ^{2} \right ) \left ({\it csgn} \left ( i \left ( a+b{x}^{n} \right ) \right ) \right ) ^{2}-4\,\ln \left ( a+b{x}^{n} \right ) \right ) }{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)*(a^2+2*a*b*x^n+b^2*x^(2*n))^p,x)

[Out]

-1/2*(-2*b^2*p*(x^n)^2-2*a*p*x^n*b-b^2*(x^n)^2+a^2)/(1+2*p)/(1+p)/n/b^2*exp(-1/2
*p*(I*Pi*csgn(I*(a+b*x^n)^2)^3-2*I*Pi*csgn(I*(a+b*x^n)^2)^2*csgn(I*(a+b*x^n))+I*
Pi*csgn(I*(a+b*x^n)^2)*csgn(I*(a+b*x^n))^2-4*ln(a+b*x^n)))

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Maxima [A]  time = 0.775058, size = 80, normalized size = 0.78 \[ \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{2 \, n} + 2 \, a b p x^{n} - a^{2}\right )}{\left (b x^{n} + a\right )}^{2 \, p}}{2 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*x^(2*n - 1),x, algorithm="maxima")

[Out]

1/2*(b^2*(2*p + 1)*x^(2*n) + 2*a*b*p*x^n - a^2)*(b*x^n + a)^(2*p)/((2*p^2 + 3*p
+ 1)*b^2*n)

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Fricas [A]  time = 0.271336, size = 105, normalized size = 1.02 \[ \frac{{\left (2 \, a b p x^{n} - a^{2} +{\left (2 \, b^{2} p + b^{2}\right )} x^{2 \, n}\right )}{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p}}{2 \,{\left (2 \, b^{2} n p^{2} + 3 \, b^{2} n p + b^{2} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*x^(2*n - 1),x, algorithm="fricas")

[Out]

1/2*(2*a*b*p*x^n - a^2 + (2*b^2*p + b^2)*x^(2*n))*(b^2*x^(2*n) + 2*a*b*x^n + a^2
)^p/(2*b^2*n*p^2 + 3*b^2*n*p + b^2*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+2*n)*(a**2+2*a*b*x**n+b**2*x**(2*n))**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p} x^{2 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*x^(2*n - 1),x, algorithm="giac")

[Out]

integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^p*x^(2*n - 1), x)