Optimal. Leaf size=117 \[ \frac{(d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{p+1}}{2 a^2 d n (p+1) (2 p+1)}-\frac{\left (a+b x^n\right ) (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{a d n (2 p+1)} \]
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Rubi [A] time = 0.0642789, antiderivative size = 124, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {1356, 273, 264} \[ \frac{\left (\frac{b x^n}{a}+1\right )^2 (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 d n \left (2 p^2+3 p+1\right )}-\frac{\left (\frac{b x^n}{a}+1\right ) (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{d n (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 1356
Rule 273
Rule 264
Rubi steps
\begin{align*} \int (d x)^{-1-2 n (1+p)} \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 (d x)^{-1-2 n (1+p)} \left (1+\frac{b x^n}{a}\right )^{2 p} \, dx\\ &=-\frac{(d x)^{-2 n (1+p)} \left (1+\frac{b x^n}{a}\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{d n (1+2 p)}+\frac{\left ((-2 n (1+p)+n (1+2 p)) \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 (d x)^{-1-2 n (1+p)} \left (1+\frac{b x^n}{a}\right )^{1+2 p} \, dx}{n (1+2 p)}\\ &=-\frac{(d x)^{-2 n (1+p)} \left (1+\frac{b x^n}{a}\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{d n (1+2 p)}+\frac{(d x)^{-2 n (1+p)} \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 d n \left (1+3 p+2 p^2\right )}\\ \end{align*}
Mathematica [C] time = 0.0327773, size = 75, normalized size = 0.64 \[ -\frac{x (d x)^{-2 n (p+1)-1} \left (\left (a+b x^n\right )^2\right )^p \left (\frac{b x^n}{a}+1\right )^{-2 p} \, _2F_1\left (-2 p,-2 (p+1);1-2 (p+1);-\frac{b x^n}{a}\right )}{2 n (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{-1-2\,n \left ( 1+p \right ) } \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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} \left (d x\right )^{-2 \, n{\left (p + 1\right )} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62185, size = 401, normalized size = 3.43 \begin{align*} -\frac{{\left (2 \, a b p x x^{n} e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) -{\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )} - b^{2} x x^{2 \, n} e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) -{\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )} +{\left (2 \, a^{2} p + a^{2}\right )} x e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) -{\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )}\right )}{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p}}{2 \,{\left (2 \, a^{2} n p^{2} + 3 \, a^{2} n p + a^{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} \left (d x\right )^{-2 \, n{\left (p + 1\right )} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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