Optimal. Leaf size=87 \[ -\frac {d (B d-A e) (c d-b e) \log (d+e x)}{e^4}+\frac {x (B d-A e) (c d-b e)}{e^3}-\frac {x^2 (-A c e-b B e+B c d)}{2 e^2}+\frac {B c x^3}{3 e} \]
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Rubi [A] time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {771} \begin {gather*} -\frac {x^2 (-A c e-b B e+B c d)}{2 e^2}+\frac {x (B d-A e) (c d-b e)}{e^3}-\frac {d (B d-A e) (c d-b e) \log (d+e x)}{e^4}+\frac {B c x^3}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{d+e x} \, dx &=\int \left (\frac {(-B d+A e) (-c d+b e)}{e^3}+\frac {(-B c d+b B e+A c e) x}{e^2}+\frac {B c x^2}{e}-\frac {d (B d-A e) (c d-b e)}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {(B d-A e) (c d-b e) x}{e^3}-\frac {(B c d-b B e-A c e) x^2}{2 e^2}+\frac {B c x^3}{3 e}-\frac {d (B d-A e) (c d-b e) \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 88, normalized size = 1.01 \begin {gather*} \frac {e x \left (3 A e (2 b e-2 c d+c e x)+3 b B e (e x-2 d)+B c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 d (B d-A e) (c d-b e) \log (d+e x)}{6 e^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (b x+c x^2\right )}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 106, normalized size = 1.22 \begin {gather*} \frac {2 \, B c e^{3} x^{3} - 3 \, {\left (B c d e^{2} - {\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + A b e^{3} - {\left (B b + A c\right )} d e^{2}\right )} x - 6 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 117, normalized size = 1.34 \begin {gather*} -{\left (B c d^{3} - B b d^{2} e - A c d^{2} e + A b d e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, B c x^{3} e^{2} - 3 \, B c d x^{2} e + 6 \, B c d^{2} x + 3 \, B b x^{2} e^{2} + 3 \, A c x^{2} e^{2} - 6 \, B b d x e - 6 \, A c d x e + 6 \, A b x e^{2}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 138, normalized size = 1.59 \begin {gather*} \frac {B c \,x^{3}}{3 e}+\frac {A c \,x^{2}}{2 e}+\frac {B b \,x^{2}}{2 e}-\frac {B c d \,x^{2}}{2 e^{2}}-\frac {A b d \ln \left (e x +d \right )}{e^{2}}+\frac {A b x}{e}+\frac {A c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {A c d x}{e^{2}}+\frac {B b \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {B b d x}{e^{2}}-\frac {B c \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {B c \,d^{2} x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 105, normalized size = 1.21 \begin {gather*} \frac {2 \, B c e^{2} x^{3} - 3 \, {\left (B c d e - {\left (B b + A c\right )} e^{2}\right )} x^{2} + 6 \, {\left (B c d^{2} + A b e^{2} - {\left (B b + A c\right )} d e\right )} x}{6 \, e^{3}} - \frac {{\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} \log \left (e x + d\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 113, normalized size = 1.30 \begin {gather*} x^2\,\left (\frac {A\,c+B\,b}{2\,e}-\frac {B\,c\,d}{2\,e^2}\right )-x\,\left (\frac {d\,\left (\frac {A\,c+B\,b}{e}-\frac {B\,c\,d}{e^2}\right )}{e}-\frac {A\,b}{e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (B\,c\,d^3+A\,b\,d\,e^2-A\,c\,d^2\,e-B\,b\,d^2\,e\right )}{e^4}+\frac {B\,c\,x^3}{3\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 95, normalized size = 1.09 \begin {gather*} \frac {B c x^{3}}{3 e} + \frac {d \left (- A e + B d\right ) \left (b e - c d\right ) \log {\left (d + e x \right )}}{e^{4}} + x^{2} \left (\frac {A c}{2 e} + \frac {B b}{2 e} - \frac {B c d}{2 e^{2}}\right ) + x \left (\frac {A b}{e} - \frac {A c d}{e^{2}} - \frac {B b d}{e^{2}} + \frac {B c d^{2}}{e^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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