Optimal. Leaf size=103 \[ \frac{a^2 \left (\frac{b x^n}{a}+1\right )^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 b^2 n (p+1)}-\frac{a^2 \left (\frac{b x^n}{a}+1\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{b^2 n (2 p+1)} \]
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Rubi [A] time = 0.0647758, 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, Rules used = {1356, 266, 43} \[ \frac{a^2 \left (\frac{b x^n}{a}+1\right )^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 b^2 n (p+1)}-\frac{a^2 \left (\frac{b x^n}{a}+1\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{b^2 n (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 1356
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+2 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p \, dx &=\left (\left (1+\frac{b x^n}{a}\right )^{-2 p} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p\right ) \int x^{-1+2 n} \left (1+\frac{b x^n}{a}\right )^{2 p} \, dx\\ &=\frac{\left (\left (1+\frac{b x^n}{a}\right )^{-2 p} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p\right ) \operatorname{Subst}\left (\int x \left (1+\frac{b x}{a}\right )^{2 p} \, dx,x,x^n\right )}{n}\\ &=\frac{\left (\left (1+\frac{b x^n}{a}\right )^{-2 p} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p\right ) \operatorname{Subst}\left (\int \left (-\frac{a \left (1+\frac{b x}{a}\right )^{2 p}}{b}+\frac{a \left (1+\frac{b x}{a}\right )^{1+2 p}}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^2 \left (1+\frac{b x^n}{a}\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{b^2 n (1+2 p)}+\frac{a^2 \left (1+\frac{b x^n}{a}\right )^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 b^2 n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0288713, 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.
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Maple [C] time = 0.071, size = 148, normalized size = 1.4 \begin{align*} -{\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}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00132, size = 80, normalized size = 0.78 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6522, size = 161, normalized size = 1.56 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p} x^{2 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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