Optimal. Leaf size=63 \[ \frac{x^{m+1} \left (c x^2\right )^p (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]
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Rubi [A] time = 0.0188879, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {15, 66, 64} \[ \frac{x^{m+1} \left (c x^2\right )^p (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 66
Rule 64
Rubi steps
\begin{align*} \int x^m \left (c x^2\right )^p (a+b x)^n \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{m+2 p} (a+b x)^n \, dx\\ &=\left (x^{-2 p} \left (c x^2\right )^p (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n}\right ) \int x^{m+2 p} \left (1+\frac{b x}{a}\right )^n \, dx\\ &=\frac{x^{1+m} \left (c x^2\right )^p (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n} \, _2F_1\left (-n,1+m+2 p;2+m+2 p;-\frac{b x}{a}\right )}{1+m+2 p}\\ \end{align*}
Mathematica [A] time = 0.0125635, size = 63, normalized size = 1. \[ \frac{x^{m+1} \left (c x^2\right )^p (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (-n,m+2 p+1;m+2 p+2;-\frac{b x}{a}\right )}{m+2 p+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( c{x}^{2} \right ) ^{p} \left ( bx+a \right ) ^{n}\, 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 (c x^{2}\right )^{p}{\left (b x + a\right )}^{n} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (c x^{2}\right )^{p}{\left (b x + a\right )}^{n} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \left (c x^{2}\right )^{p} \left (a + b x\right )^{n}\, dx \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 (c x^{2}\right )^{p}{\left (b x + a\right )}^{n} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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