Optimal. Leaf size=47 \[ \frac {B (d+e x)^{m+2}}{e^2 (m+2)}-\frac {(B d-A e) (d+e x)^{m+1}}{e^2 (m+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ \frac {B (d+e x)^{m+2}}{e^2 (m+2)}-\frac {(B d-A e) (d+e x)^{m+1}}{e^2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^m \, dx &=\int \left (\frac {(-B d+A e) (d+e x)^m}{e}+\frac {B (d+e x)^{1+m}}{e}\right ) \, dx\\ &=-\frac {(B d-A e) (d+e x)^{1+m}}{e^2 (1+m)}+\frac {B (d+e x)^{2+m}}{e^2 (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.87 \[ \frac {(d+e x)^{m+1} (A e (m+2)-B d+B e (m+1) x)}{e^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 83, normalized size = 1.77 \[ \frac {{\left (A d e m - B d^{2} + 2 \, A d e + {\left (B e^{2} m + B e^{2}\right )} x^{2} + {\left (2 \, A e^{2} + {\left (B d e + A e^{2}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{2} m^{2} + 3 \, e^{2} m + 2 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 136, normalized size = 2.89 \[ \frac {{\left (x e + d\right )}^{m} B m x^{2} e^{2} + {\left (x e + d\right )}^{m} B d m x e + {\left (x e + d\right )}^{m} A m x e^{2} + {\left (x e + d\right )}^{m} B x^{2} e^{2} + {\left (x e + d\right )}^{m} A d m e - {\left (x e + d\right )}^{m} B d^{2} + 2 \, {\left (x e + d\right )}^{m} A x e^{2} + 2 \, {\left (x e + d\right )}^{m} A d e}{m^{2} e^{2} + 3 \, m e^{2} + 2 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 46, normalized size = 0.98 \[ \frac {\left (B e m x +A e m +B e x +2 A e -B d \right ) \left (e x +d \right )^{m +1}}{\left (m^{2}+3 m +2\right ) e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 63, normalized size = 1.34 \[ \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A}{e {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 88, normalized size = 1.87 \[ {\left (d+e\,x\right )}^m\,\left (\frac {x\,\left (2\,A\,e^2+A\,e^2\,m+B\,d\,e\,m\right )}{e^2\,\left (m^2+3\,m+2\right )}+\frac {B\,x^2\,\left (m+1\right )}{m^2+3\,m+2}+\frac {d\,\left (2\,A\,e-B\,d+A\,e\,m\right )}{e^2\,\left (m^2+3\,m+2\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 377, normalized size = 8.02 \[ \begin {cases} d^{m} \left (A x + \frac {B x^{2}}{2}\right ) & \text {for}\: e = 0 \\- \frac {A e}{d e^{2} + e^{3} x} + \frac {B d \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac {B d}{d e^{2} + e^{3} x} + \frac {B e x \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text {for}\: m = -2 \\\frac {A \log {\left (\frac {d}{e} + x \right )}}{e} - \frac {B d \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {B x}{e} & \text {for}\: m = -1 \\\frac {A d e m \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {2 A d e \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {A e^{2} m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {2 A e^{2} x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} - \frac {B d^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {B d e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {B e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} + \frac {B e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 3 e^{2} m + 2 e^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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