Optimal. Leaf size=147 \[ \frac {\log (x) (-A b e-2 A c d+b B d)}{b^3 d^2}+\frac {c (b B-A c)}{b^2 (b+c x) (c d-b e)}-\frac {A}{b^2 d x}+\frac {c \log (b+c x) \left (-b c (3 A e+B d)+2 A c^2 d+2 b^2 B e\right )}{b^3 (c d-b e)^2}-\frac {e^2 (B d-A e) \log (d+e x)}{d^2 (c d-b e)^2} \]
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Rubi [A] time = 0.20, antiderivative size = 147, 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} \[ \frac {c \log (b+c x) \left (-b c (3 A e+B d)+2 A c^2 d+2 b^2 B e\right )}{b^3 (c d-b e)^2}+\frac {\log (x) (-A b e-2 A c d+b B d)}{b^3 d^2}+\frac {c (b B-A c)}{b^2 (b+c x) (c d-b e)}-\frac {A}{b^2 d x}-\frac {e^2 (B d-A e) \log (d+e x)}{d^2 (c d-b e)^2} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {A}{b^2 d x^2}+\frac {b B d-2 A c d-A b e}{b^3 d^2 x}+\frac {c^2 (b B-A c)}{b^2 (-c d+b e) (b+c x)^2}+\frac {c^2 \left (2 A c^2 d+2 b^2 B e-b c (B d+3 A e)\right )}{b^3 (c d-b e)^2 (b+c x)}-\frac {e^3 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)}\right ) \, dx\\ &=-\frac {A}{b^2 d x}+\frac {c (b B-A c)}{b^2 (c d-b e) (b+c x)}+\frac {(b B d-2 A c d-A b e) \log (x)}{b^3 d^2}+\frac {c \left (2 A c^2 d+2 b^2 B e-b c (B d+3 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^2}-\frac {e^2 (B d-A e) \log (d+e x)}{d^2 (c d-b e)^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 146, normalized size = 0.99 \[ \frac {\log (x) (-A b e-2 A c d+b B d)}{b^3 d^2}+\frac {c (A c-b B)}{b^2 (b+c x) (b e-c d)}-\frac {A}{b^2 d x}+\frac {c \log (b+c x) \left (-b c (3 A e+B d)+2 A c^2 d+2 b^2 B e\right )}{b^3 (c d-b e)^2}+\frac {e^2 (A e-B d) \log (d+e x)}{d^2 (c d-b e)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 67.63, size = 450, normalized size = 3.06 \[ -\frac {A b^{2} c^{2} d^{3} - 2 \, A b^{3} c d^{2} e + A b^{4} d e^{2} + {\left (A b^{3} c d e^{2} - {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} + {\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x + {\left ({\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - {\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} d^{2} e\right )} x^{2} + {\left ({\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} - {\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (B b^{3} c d e^{2} - A b^{3} c e^{3}\right )} x^{2} + {\left (B b^{4} d e^{2} - A b^{4} e^{3}\right )} x\right )} \log \left (e x + d\right ) - {\left ({\left (B b^{3} c d e^{2} - A b^{3} c e^{3} + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - {\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} d^{2} e\right )} x^{2} + {\left (B b^{4} d e^{2} - A b^{4} e^{3} + {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} - {\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x\right )} \log \relax (x)}{{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e + b^{5} c d^{2} e^{2}\right )} x^{2} + {\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 268, normalized size = 1.82 \[ -\frac {{\left (B b c^{3} d - 2 \, A c^{4} d - 2 \, B b^{2} c^{2} e + 3 \, A b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3} d^{2} - 2 \, b^{4} c^{2} d e + b^{5} c e^{2}} - \frac {{\left (B d e^{3} - A e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}} + \frac {{\left (B b d - 2 \, A c d - A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{2}} - \frac {A b c^{2} d^{3} - 2 \, A b^{2} c d^{2} e + A b^{3} d e^{2} - {\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - B b^{2} c d^{2} e + 3 \, A b c^{2} d^{2} e - A b^{2} c d e^{2}\right )} x}{{\left (c d - b e\right )}^{2} {\left (c x + b\right )} b^{2} d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 248, normalized size = 1.69 \[ -\frac {3 A \,c^{2} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{2} b^{2}}+\frac {2 A \,c^{3} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{2} b^{3}}+\frac {A \,e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{2} d^{2}}+\frac {2 B c e \ln \left (c x +b \right )}{\left (b e -c d \right )^{2} b}-\frac {B \,c^{2} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{2} b^{2}}-\frac {B \,e^{2} \ln \left (e x +d \right )}{\left (b e -c d \right )^{2} d}+\frac {A \,c^{2}}{\left (b e -c d \right ) \left (c x +b \right ) b^{2}}-\frac {B c}{\left (b e -c d \right ) \left (c x +b \right ) b}-\frac {A e \ln \relax (x )}{b^{2} d^{2}}-\frac {2 A c \ln \relax (x )}{b^{3} d}+\frac {B \ln \relax (x )}{b^{2} d}-\frac {A}{b^{2} d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 227, normalized size = 1.54 \[ -\frac {{\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}} - \frac {{\left (B d e^{2} - A e^{3}\right )} \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}} - \frac {A b c d - A b^{2} e - {\left (A b c e + {\left (B b c - 2 \, A c^{2}\right )} d\right )} x}{{\left (b^{2} c^{2} d^{2} - b^{3} c d e\right )} x^{2} + {\left (b^{3} c d^{2} - b^{4} d e\right )} x} - \frac {{\left (A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \relax (x)}{b^{3} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 201, normalized size = 1.37 \[ \frac {\ln \left (b+c\,x\right )\,\left (d\,\left (2\,A\,c^3-B\,b\,c^2\right )-3\,A\,b\,c^2\,e+2\,B\,b^2\,c\,e\right )}{b^5\,e^2-2\,b^4\,c\,d\,e+b^3\,c^2\,d^2}-\frac {\frac {A}{b\,d}+\frac {x\,\left (A\,b\,c\,e-2\,A\,c^2\,d+B\,b\,c\,d\right )}{b^2\,d\,\left (b\,e-c\,d\right )}}{c\,x^2+b\,x}+\frac {\ln \left (d+e\,x\right )\,\left (A\,e^3-B\,d\,e^2\right )}{b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4}-\frac {\ln \relax (x)\,\left (b\,\left (A\,e-B\,d\right )+2\,A\,c\,d\right )}{b^3\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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