Optimal. Leaf size=49 \[ \frac{\left (a+b x^n\right )^{p+2}}{b^2 n (p+2)}-\frac{a \left (a+b x^n\right )^{p+1}}{b^2 n (p+1)} \]
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Rubi [A] time = 0.0316899, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac{\left (a+b x^n\right )^{p+2}}{b^2 n (p+2)}-\frac{a \left (a+b x^n\right )^{p+1}}{b^2 n (p+1)} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+2 n} \left (a+b x^n\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int x (a+b x)^p \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^p}{b}+\frac{(a+b x)^{1+p}}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a \left (a+b x^n\right )^{1+p}}{b^2 n (1+p)}+\frac{\left (a+b x^n\right )^{2+p}}{b^2 n (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0231275, size = 40, normalized size = 0.82 \[ \frac{\left (a+b x^n\right )^{p+1} \left (b (p+1) x^n-a\right )}{b^2 n (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 61, normalized size = 1.2 \begin{align*} -{\frac{ \left ( -{b}^{2}p \left ({x}^{n} \right ) ^{2}-ap{x}^{n}b-{b}^{2} \left ({x}^{n} \right ) ^{2}+{a}^{2} \right ) \left ( a+b{x}^{n} \right ) ^{p}}{ \left ( 1+p \right ) \left ( 2+p \right ) n{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0158, size = 69, normalized size = 1.41 \begin{align*} \frac{{\left (b^{2}{\left (p + 1\right )} x^{2 \, n} + a b p x^{n} - a^{2}\right )}{\left (b x^{n} + a\right )}^{p}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3653, size = 123, normalized size = 2.51 \begin{align*} \frac{{\left (a b p x^{n} - a^{2} +{\left (b^{2} p + b^{2}\right )} x^{2 \, n}\right )}{\left (b x^{n} + a\right )}^{p}}{b^{2} n p^{2} + 3 \, b^{2} n p + 2 \, b^{2} n} \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 x^{n} + a\right )}^{p} x^{2 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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