Optimal. Leaf size=60 \[ \frac {(b d-a e) (B d-A e) \log (d+e x)}{e^3}+\frac {B (a+b x)^2}{2 b e}-\frac {b x (B d-A e)}{e^2} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \frac {(b d-a e) (B d-A e) \log (d+e x)}{e^3}+\frac {B (a+b x)^2}{2 b e}-\frac {b x (B d-A e)}{e^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{d+e x} \, dx &=\int \left (\frac {b (-B d+A e)}{e^2}+\frac {B (a+b x)}{e}+\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {b (B d-A e) x}{e^2}+\frac {B (a+b x)^2}{2 b e}+\frac {(b d-a e) (B d-A e) \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.93 \[ \frac {e x (2 a B e+b (2 A e-2 B d+B e x))+2 (b d-a e) (B d-A e) \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 68, normalized size = 1.13 \[ \frac {B b e^{2} x^{2} - 2 \, {\left (B b d e - {\left (B a + A b\right )} e^{2}\right )} x + 2 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 71, normalized size = 1.18 \[ {\left (B b d^{2} - B a d e - A b d e + A a e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B b x^{2} e - 2 \, B b d x + 2 \, B a x e + 2 \, A b x e\right )} e^{\left (-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 90, normalized size = 1.50 \[ \frac {B b \,x^{2}}{2 e}+\frac {A a \ln \left (e x +d \right )}{e}-\frac {A b d \ln \left (e x +d \right )}{e^{2}}+\frac {A b x}{e}-\frac {B a d \ln \left (e x +d \right )}{e^{2}}+\frac {B a x}{e}+\frac {B b \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {B b d x}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 66, normalized size = 1.10 \[ \frac {B b e x^{2} - 2 \, {\left (B b d - {\left (B a + A b\right )} e\right )} x}{2 \, e^{2}} + \frac {{\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 68, normalized size = 1.13 \[ x\,\left (\frac {A\,b+B\,a}{e}-\frac {B\,b\,d}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (A\,a\,e^2+B\,b\,d^2-A\,b\,d\,e-B\,a\,d\,e\right )}{e^3}+\frac {B\,b\,x^2}{2\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 53, normalized size = 0.88 \[ \frac {B b x^{2}}{2 e} + x \left (\frac {A b}{e} + \frac {B a}{e} - \frac {B b d}{e^{2}}\right ) - \frac {\left (- A e + B d\right ) \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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