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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]