Optimal. Leaf size=77 \[ \frac {(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}+\frac {e x (A c e-b B e+2 B c d)}{c^2}+\frac {A d^2 \log (x)}{b}+\frac {B e^2 x^2}{2 c} \]
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Rubi [A] time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {e x (A c e-b B e+2 B c d)}{c^2}+\frac {(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}+\frac {A d^2 \log (x)}{b}+\frac {B e^2 x^2}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{b x+c x^2} \, dx &=\int \left (\frac {e (2 B c d-b B e+A c e)}{c^2}+\frac {A d^2}{b x}+\frac {B e^2 x}{c}+\frac {(b B-A c) (-c d+b e)^2}{b c^2 (b+c x)}\right ) \, dx\\ &=\frac {e (2 B c d-b B e+A c e) x}{c^2}+\frac {B e^2 x^2}{2 c}+\frac {A d^2 \log (x)}{b}+\frac {(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.96 \begin {gather*} \frac {b c e x (2 A c e+B (-2 b e+4 c d+c e x))+2 (b B-A c) (c d-b e)^2 \log (b+c x)+2 A c^3 d^2 \log (x)}{2 b c^3} \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) (d+e x)^2}{b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 125, normalized size = 1.62 \begin {gather*} \frac {B b c^{2} e^{2} x^{2} + 2 \, A c^{3} d^{2} \log \relax (x) + 2 \, {\left (2 \, B b c^{2} d e - {\left (B b^{2} c - A b c^{2}\right )} e^{2}\right )} x + 2 \, {\left ({\left (B b c^{2} - A c^{3}\right )} d^{2} - 2 \, {\left (B b^{2} c - A b c^{2}\right )} d e + {\left (B b^{3} - A b^{2} c\right )} e^{2}\right )} \log \left (c x + b\right )}{2 \, b c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 117, normalized size = 1.52 \begin {gather*} \frac {A d^{2} \log \left ({\left | x \right |}\right )}{b} + \frac {B c x^{2} e^{2} + 4 \, B c d x e - 2 \, B b x e^{2} + 2 \, A c x e^{2}}{2 \, c^{2}} + \frac {{\left (B b c^{2} d^{2} - A c^{3} d^{2} - 2 \, B b^{2} c d e + 2 \, A b c^{2} d e + B b^{3} e^{2} - A b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 144, normalized size = 1.87 \begin {gather*} \frac {B \,e^{2} x^{2}}{2 c}-\frac {A b \,e^{2} \ln \left (c x +b \right )}{c^{2}}+\frac {A \,d^{2} \ln \relax (x )}{b}-\frac {A \,d^{2} \ln \left (c x +b \right )}{b}+\frac {2 A d e \ln \left (c x +b \right )}{c}+\frac {A \,e^{2} x}{c}+\frac {B \,b^{2} e^{2} \ln \left (c x +b \right )}{c^{3}}-\frac {2 B b d e \ln \left (c x +b \right )}{c^{2}}-\frac {B b \,e^{2} x}{c^{2}}+\frac {B \,d^{2} \ln \left (c x +b \right )}{c}+\frac {2 B d e x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 115, normalized size = 1.49 \begin {gather*} \frac {A d^{2} \log \relax (x)}{b} + \frac {B c e^{2} x^{2} + 2 \, {\left (2 \, B c d e - {\left (B b - A c\right )} e^{2}\right )} x}{2 \, c^{2}} + \frac {{\left ({\left (B b c^{2} - A c^{3}\right )} d^{2} - 2 \, {\left (B b^{2} c - A b c^{2}\right )} d e + {\left (B b^{3} - A b^{2} c\right )} e^{2}\right )} \log \left (c x + b\right )}{b c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 122, normalized size = 1.58 \begin {gather*} x\,\left (\frac {A\,e^2+2\,B\,d\,e}{c}-\frac {B\,b\,e^2}{c^2}\right )-\ln \left (b+c\,x\right )\,\left (\frac {A\,d^2}{b}-\frac {c^2\,\left (B\,b\,d^2+2\,A\,b\,e\,d\right )-c\,\left (A\,b^2\,e^2+2\,B\,d\,b^2\,e\right )+B\,b^3\,e^2}{b\,c^3}\right )+\frac {A\,d^2\,\ln \relax (x)}{b}+\frac {B\,e^2\,x^2}{2\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.80, size = 163, normalized size = 2.12 \begin {gather*} \frac {A d^{2} \log {\relax (x )}}{b} + \frac {B e^{2} x^{2}}{2 c} + x \left (\frac {A e^{2}}{c} - \frac {B b e^{2}}{c^{2}} + \frac {2 B d e}{c}\right ) + \frac {\left (- A c + B b\right ) \left (b e - c d\right )^{2} \log {\left (x + \frac {- A b c^{2} d^{2} + \frac {b \left (- A c + B b\right ) \left (b e - c d\right )^{2}}{c}}{- A b^{2} c e^{2} + 2 A b c^{2} d e - 2 A c^{3} d^{2} + B b^{3} e^{2} - 2 B b^{2} c d e + B b c^{2} d^{2}} \right )}}{b c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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