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 = 0.19, antiderivative size = 174, normalized size of antiderivative = 1.00, 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 69
Rule 70
Rule 80
Rule 770
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*}
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Mathematica [A] time = 0.12, 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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fricas [F] time = 1.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B x + A\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B x + A\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.42, size = 0, normalized size = 0.00 \[ \int \left (B x +A \right ) \left (e x +d \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B x + A\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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