Optimal. Leaf size=174 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} (A b e (m+2 p+2)-B (a e (m+1)+b (2 d p+d))) \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{b e^2 (m+1) (m+2 p+2)}+\frac{B (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1}}{b e (m+2 p+2)} \]
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Rubi [A] time = 0.186949, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {770, 80, 70, 69} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} (A b e (m+2 p+2)-B (a e (m+1)+b (2 d p+d))) \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{b e^2 (m+1) (m+2 p+2)}+\frac{B (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1}}{b e (m+2 p+2)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 80
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (A+B x) (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} (A+B x) (d+e x)^m \, dx\\ &=\frac{B (a+b x) (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p}{b e (2+m+2 p)}+\left (\left (A-\frac{B (a e (1+m)+b (d+2 d p))}{b e (2+m+2 p)}\right ) \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\\ &=\frac{B (a+b x) (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p}{b e (2+m+2 p)}+\left (\left (A-\frac{B (a e (1+m)+b (d+2 d p))}{b e (2+m+2 p)}\right ) \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{B (a+b x) (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p}{b e (2+m+2 p)}+\frac{\left (A-\frac{B (a e (1+m)+b (d+2 d p))}{b e (2+m+2 p)}\right ) \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 = 0.128592, size = 125, normalized size = 0.72 \[ \frac{\left ((a+b x)^2\right )^p (d+e x)^{m+1} \left (B e (a+b x)-\frac{\left (\frac{e (a+b x)}{a e-b d}\right )^{-2 p} (a B e (m+1)-A b e (m+2 p+2)+b B (2 d p+d)) \, _2F_1\left (m+1,-2 p;m+2;\frac{b (d+e x)}{b d-a e}\right )}{m+1}\right )}{b e^2 (m+2 p+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int \left ( Bx+A \right ) \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 x + A\right )}{\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 x + A\right )}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (d + e x\right )^{m} \left (\left (a + b x\right )^{2}\right )^{p}\, 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 (B x + A\right )}{\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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