Optimal. Leaf size=85 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{e (m+1)} \]
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Rubi [A] time = 0.0543152, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {646, 70, 69} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} (d+e x)^m \, dx\\ &=\left (\left (\frac{e \left (a b+b^2 x\right )}{-b^2 d+a b e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int (d+e x)^m \left (-\frac{a e}{b d-a e}-\frac{b e x}{b d-a e}\right )^{2 p} \, dx\\ &=\frac{\left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1+m,-2 p;2+m;\frac{b (d+e x)}{b d-a e}\right )}{e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0350256, size = 75, normalized size = 0.88 \[ \frac{\left ((a+b x)^2\right )^p (d+e x)^{m+1} \left (\frac{e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.316, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \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} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{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 (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{m}, x\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} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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