Optimal. Leaf size=108 \[ \frac {d \log (x) (2 A b e-2 A c d+b B d)}{b^3}+\frac {(b B-A c) (c d-b e)^2}{b^2 c^2 (b+c x)}-\frac {A d^2}{b^2 x}-\frac {(c d-b e) \log (b+c x) \left (-2 A c^2 d+b^2 B e+b B c d\right )}{b^3 c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 108, 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 {(b B-A c) (c d-b e)^2}{b^2 c^2 (b+c x)}-\frac {(c d-b e) \log (b+c x) \left (-2 A c^2 d+b^2 B e+b B c d\right )}{b^3 c^2}+\frac {d \log (x) (2 A b e-2 A c d+b B d)}{b^3}-\frac {A d^2}{b^2 x} \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}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {A d^2}{b^2 x^2}+\frac {d (b B d-2 A c d+2 A b e)}{b^3 x}-\frac {(b B-A c) (-c d+b e)^2}{b^2 c (b+c x)^2}+\frac {(-c d+b e) \left (b B c d-2 A c^2 d+b^2 B e\right )}{b^3 c (b+c x)}\right ) \, dx\\ &=-\frac {A d^2}{b^2 x}+\frac {(b B-A c) (c d-b e)^2}{b^2 c^2 (b+c x)}+\frac {d (b B d-2 A c d+2 A b e) \log (x)}{b^3}-\frac {(c d-b e) \left (b B c d-2 A c^2 d+b^2 B e\right ) \log (b+c x)}{b^3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 101, normalized size = 0.94 \begin {gather*} \frac {\frac {(b e-c d) \log (b+c x) \left (-2 A c^2 d+b^2 B e+b B c d\right )}{c^2}+\frac {b (b B-A c) (c d-b e)^2}{c^2 (b+c x)}+d \log (x) (2 A b e-2 A c d+b B d)-\frac {A b d^2}{x}}{b^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}{\left (b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 258, normalized size = 2.39 \begin {gather*} -\frac {A b^{2} c^{2} d^{2} - {\left ({\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{2} - 2 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e + {\left (B b^{4} - A b^{3} c\right )} e^{2}\right )} x + {\left ({\left (2 \, A b c^{3} d e - B b^{3} c e^{2} + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2}\right )} x^{2} + {\left (2 \, A b^{2} c^{2} d e - B b^{4} e^{2} + {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{2}\right )} x\right )} \log \left (c x + b\right ) - {\left ({\left (2 \, A b c^{3} d e + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2}\right )} x^{2} + {\left (2 \, A b^{2} c^{2} d e + {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{2}\right )} x\right )} \log \relax (x)}{b^{3} c^{3} x^{2} + b^{4} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 167, normalized size = 1.55 \begin {gather*} \frac {{\left (B b d^{2} - 2 \, A c d^{2} + 2 \, A b d e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {{\left (B b c^{2} d^{2} - 2 \, A c^{3} d^{2} + 2 \, A b c^{2} d e - B b^{3} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{2}} - \frac {A b c^{2} d^{2} - {\left (B b c^{2} d^{2} - 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 )} x}{{\left (c x + b\right )} b^{2} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 199, normalized size = 1.84 \begin {gather*} \frac {2 A d e}{\left (c x +b \right ) b}-\frac {A c \,d^{2}}{\left (c x +b \right ) b^{2}}+\frac {2 A d e \ln \relax (x )}{b^{2}}-\frac {2 A d e \ln \left (c x +b \right )}{b^{2}}-\frac {2 A c \,d^{2} \ln \relax (x )}{b^{3}}+\frac {2 A c \,d^{2} \ln \left (c x +b \right )}{b^{3}}-\frac {A \,e^{2}}{\left (c x +b \right ) c}+\frac {B b \,e^{2}}{\left (c x +b \right ) c^{2}}+\frac {B \,d^{2}}{\left (c x +b \right ) b}+\frac {B \,d^{2} \ln \relax (x )}{b^{2}}-\frac {B \,d^{2} \ln \left (c x +b \right )}{b^{2}}-\frac {2 B d e}{\left (c x +b \right ) c}+\frac {B \,e^{2} \ln \left (c x +b \right )}{c^{2}}-\frac {A \,d^{2}}{b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 165, normalized size = 1.53 \begin {gather*} -\frac {A b c^{2} d^{2} - {\left ({\left (B b c^{2} - 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 )} x}{b^{2} c^{3} x^{2} + b^{3} c^{2} x} + \frac {{\left (2 \, A b d e + {\left (B b - 2 \, A c\right )} d^{2}\right )} \log \relax (x)}{b^{3}} - \frac {{\left (2 \, A b c^{2} d e - B b^{3} e^{2} + {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2}\right )} \log \left (c x + b\right )}{b^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 154, normalized size = 1.43 \begin {gather*} \frac {\ln \relax (x)\,\left (b\,\left (B\,d^2+2\,A\,e\,d\right )-2\,A\,c\,d^2\right )}{b^3}-\frac {\frac {A\,d^2}{b}+\frac {x\,\left (-B\,b^3\,e^2+2\,B\,b^2\,c\,d\,e+A\,b^2\,c\,e^2-B\,b\,c^2\,d^2-2\,A\,b\,c^2\,d\,e+2\,A\,c^3\,d^2\right )}{b^2\,c^2}}{c\,x^2+b\,x}+\frac {\ln \left (b+c\,x\right )\,\left (b\,e-c\,d\right )\,\left (B\,e\,b^2+B\,d\,b\,c-2\,A\,d\,c^2\right )}{b^3\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.86, size = 367, normalized size = 3.40 \begin {gather*} \frac {- A b c^{2} d^{2} + x \left (- 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^{3} c^{2} x + b^{2} c^{3} x^{2}} + \frac {d \left (2 A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {- 2 A b^{2} c d e + 2 A b c^{2} d^{2} - B b^{2} c d^{2} + b c d \left (2 A b e - 2 A c d + B b d\right )}{- 4 A b c^{2} d e + 4 A c^{3} d^{2} + B b^{3} e^{2} - 2 B b c^{2} d^{2}} \right )}}{b^{3}} + \frac {\left (b e - c d\right ) \left (- 2 A c^{2} d + B b^{2} e + B b c d\right ) \log {\left (x + \frac {- 2 A b^{2} c d e + 2 A b c^{2} d^{2} - B b^{2} c d^{2} + \frac {b \left (b e - c d\right ) \left (- 2 A c^{2} d + B b^{2} e + B b c d\right )}{c}}{- 4 A b c^{2} d e + 4 A c^{3} d^{2} + B b^{3} e^{2} - 2 B b c^{2} d^{2}} \right )}}{b^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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