Optimal. Leaf size=18 \[ x^{m+q+1} \left (a+b x^n\right )^{p+1} \]
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Rubi [A] time = 0.0375639, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {1584, 449} \[ x^{m+q+1} \left (a+b x^n\right )^{p+1} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 449
Rubi steps
\begin{align*} \int x^m \left (a+b x^n\right )^p \left (a (1+m+q) x^q+b (1+m+n (1+p)+q) x^{n+q}\right ) \, dx &=\int x^{m+q} \left (a+b x^n\right )^p \left (a (1+m+q)+b (1+m+n (1+p)+q) x^n\right ) \, dx\\ &=x^{1+m+q} \left (a+b x^n\right )^{1+p}\\ \end{align*}
Mathematica [C] time = 0.170488, size = 116, normalized size = 6.44 \[ x^{m+q+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (\frac{b x^n (m+n p+n+q+1) \, _2F_1\left (-p,\frac{m+n+q+1}{n};\frac{m+2 n+q+1}{n};-\frac{b x^n}{a}\right )}{m+n+q+1}+a \, _2F_1\left (-p,\frac{m+q+1}{n};\frac{m+n+q+1}{n};-\frac{b x^n}{a}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.745, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( a \left ( 1+m+q \right ){x}^{q}+b \left ( 1+m+n \left ( 1+p \right ) +q \right ){x}^{n+q} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.43783, size = 50, normalized size = 2.78 \begin{align*}{\left (a x x^{m} + b x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + q \log \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.54495, size = 88, normalized size = 4.89 \begin{align*}{\left (b x x^{m} x^{n + q} + a x x^{m} x^{q}\right )} \left (\frac{b x^{n + q} + a x^{q}}{x^{q}}\right )^{p} \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 [B] time = 1.38679, size = 65, normalized size = 3.61 \begin{align*}{\left (b x^{n} + a\right )}^{p} b x x^{n} e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )} +{\left (b x^{n} + a\right )}^{p} a x e^{\left (m \log \left (x\right ) + q \log \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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