Optimal. Leaf size=201 \[ \frac {\log (x) (-2 A b e-2 A c d+b B d)}{b^3 d^3}+\frac {c^2 (b B-A c)}{b^2 (b+c x) (c d-b e)^2}-\frac {A}{b^2 d^2 x}+\frac {c^2 \log (b+c x) \left (-b c (4 A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^3 (c d-b e)^3}+\frac {e^2 \log (d+e x) (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3}+\frac {e^2 (B d-A e)}{d^2 (d+e x) (c d-b e)^2} \]
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Rubi [A] time = 0.33, antiderivative size = 201, 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 {c^2 (b B-A c)}{b^2 (b+c x) (c d-b e)^2}+\frac {c^2 \log (b+c x) \left (-b c (4 A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^3 (c d-b e)^3}+\frac {\log (x) (-2 A b e-2 A c d+b B d)}{b^3 d^3}-\frac {A}{b^2 d^2 x}+\frac {e^2 (B d-A e)}{d^2 (d+e x) (c d-b e)^2}+\frac {e^2 \log (d+e x) (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3} \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}{b^2 d^2 x^2}+\frac {b B d-2 A c d-2 A b e}{b^3 d^3 x}-\frac {c^3 (b B-A c)}{b^2 (-c d+b e)^2 (b+c x)^2}+\frac {c^3 \left (2 A c^2 d+3 b^2 B e-b c (B d+4 A e)\right )}{b^3 (c d-b e)^3 (b+c x)}-\frac {e^3 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)^2}+\frac {e^3 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {A}{b^2 d^2 x}+\frac {c^2 (b B-A c)}{b^2 (c d-b e)^2 (b+c x)}+\frac {e^2 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {(b B d-2 A c d-2 A b e) \log (x)}{b^3 d^3}+\frac {c^2 \left (2 A c^2 d+3 b^2 B e-b c (B d+4 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^3}+\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e)) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 201, normalized size = 1.00 \begin {gather*} \frac {\log (x) (-2 A b e-2 A c d+b B d)}{b^3 d^3}+\frac {c^2 (b B-A c)}{b^2 (b+c x) (c d-b e)^2}-\frac {A}{b^2 d^2 x}-\frac {c^2 \log (b+c x) \left (-b c (4 A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^3 (b e-c d)^3}-\frac {e^2 \log (d+e x) (2 A e (b e-2 c d)+B d (3 c d-b e))}{d^3 (c d-b e)^3}+\frac {e^2 (B d-A e)}{d^2 (d+e x) (c d-b e)^2} \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 = 134.90, size = 1034, normalized size = 5.14 \begin {gather*} -\frac {A b^{2} c^{3} d^{5} - 3 \, A b^{3} c^{2} d^{4} e + 3 \, A b^{4} c d^{3} e^{2} - A b^{5} d^{2} e^{3} - {\left (4 \, A b^{2} c^{3} d^{3} e^{2} + 2 \, A b^{4} c d e^{4} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{4} e - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d^{2} e^{3}\right )} x^{2} - {\left (B b^{4} c d^{3} e^{2} + 2 \, A b^{5} d e^{4} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} - {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d^{4} e - {\left (B b^{5} + 3 \, A b^{4} c\right )} d^{2} e^{3}\right )} x + {\left ({\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} e - {\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} d^{3} e^{2}\right )} x^{3} + {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{5} - 2 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{4} e - {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{3} e^{2}\right )} x^{2} + {\left ({\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} - {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{4} e\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (3 \, B b^{3} c^{2} d^{2} e^{3} + 2 \, A b^{4} c e^{5} - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d e^{4}\right )} x^{3} + {\left (3 \, B b^{3} c^{2} d^{3} e^{2} + 2 \, A b^{5} e^{5} + 2 \, {\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2} e^{3} - {\left (B b^{5} + 2 \, A b^{4} c\right )} d e^{4}\right )} x^{2} + {\left (3 \, B b^{4} c d^{3} e^{2} + 2 \, A b^{5} d e^{4} - {\left (B b^{5} + 4 \, A b^{4} c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right ) - {\left ({\left (3 \, B b^{3} c^{2} d^{2} e^{3} + 2 \, A b^{4} c e^{5} + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} e - {\left (3 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} d^{3} e^{2} - {\left (B b^{4} c + 4 \, A b^{3} c^{2}\right )} d e^{4}\right )} x^{3} + {\left (4 \, A b^{2} c^{3} d^{3} e^{2} + 2 \, A b^{5} e^{5} + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{5} - 2 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{4} e + 2 \, {\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2} e^{3} - {\left (B b^{5} + 2 \, A b^{4} c\right )} d e^{4}\right )} x^{2} + {\left (3 \, B b^{4} c d^{3} e^{2} + 2 \, A b^{5} d e^{4} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{5} - {\left (3 \, B b^{3} c^{2} - 4 \, A b^{2} c^{3}\right )} d^{4} e - {\left (B b^{5} + 4 \, A b^{4} c\right )} d^{2} e^{3}\right )} x\right )} \log \relax (x)}{{\left (b^{3} c^{4} d^{6} e - 3 \, b^{4} c^{3} d^{5} e^{2} + 3 \, b^{5} c^{2} d^{4} e^{3} - b^{6} c d^{3} e^{4}\right )} x^{3} + {\left (b^{3} c^{4} d^{7} - 2 \, b^{4} c^{3} d^{6} e + 2 \, b^{6} c d^{4} e^{3} - b^{7} d^{3} e^{4}\right )} x^{2} + {\left (b^{4} c^{3} d^{7} - 3 \, b^{5} c^{2} d^{6} e + 3 \, b^{6} c d^{5} e^{2} - b^{7} d^{4} e^{3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 670, normalized size = 3.33 \begin {gather*} \frac {{\left (2 \, B b c^{3} d^{4} e^{2} - 4 \, A c^{4} d^{4} e^{2} - 6 \, B b^{2} c^{2} d^{3} e^{3} + 8 \, A b c^{3} d^{3} e^{3} + 3 \, B b^{3} c d^{2} e^{4} - B b^{4} d e^{5} - 4 \, A b^{3} c d e^{5} + 2 \, A b^{4} e^{6}\right )} e^{\left (-2\right )} \log \left (\frac {{\left | 2 \, c d e - \frac {2 \, c d^{2} e}{x e + d} - b e^{2} + \frac {2 \, b d e^{2}}{x e + d} - {\left | b \right |} e^{2} \right |}}{{\left | 2 \, c d e - \frac {2 \, c d^{2} e}{x e + d} - b e^{2} + \frac {2 \, b d e^{2}}{x e + d} + {\left | b \right |} e^{2} \right |}}\right )}{2 \, {\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} {\left | b \right |}} + \frac {{\left (3 \, B c d^{2} e^{2} - B b d e^{3} - 4 \, A c d e^{3} + 2 \, A b e^{4}\right )} \log \left ({\left | c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}} \right |}\right )}{2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}} + \frac {\frac {B d e^{6}}{x e + d} - \frac {A e^{7}}{x e + d}}{c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}} + \frac {\frac {B b c^{3} d^{3} e - 2 \, A c^{4} d^{3} e + 3 \, A b c^{3} d^{2} e^{2} - 3 \, A b^{2} c^{2} d e^{3} + A b^{3} c e^{4}}{c d^{2} - b d e} - \frac {{\left (B b c^{3} d^{4} e^{2} - 2 \, A c^{4} d^{4} e^{2} + 4 \, A b c^{3} d^{3} e^{3} - 6 \, A b^{2} c^{2} d^{2} e^{4} + 4 \, A b^{3} c d e^{5} - A b^{4} e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e\right )} {\left (x e + d\right )}}}{{\left (c d - b e\right )}^{2} b^{2} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}\right )} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 357, normalized size = 1.78 \begin {gather*} \frac {2 A b \,e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}+\frac {4 A \,c^{3} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{2}}-\frac {2 A \,c^{4} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {4 A c \,e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{2}}-\frac {B b \,e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{2}}-\frac {3 B \,c^{2} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b}+\frac {B \,c^{3} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{2}}+\frac {3 B c \,e^{2} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d}-\frac {A \,c^{3}}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{2}}-\frac {A \,e^{3}}{\left (b e -c d \right )^{2} \left (e x +d \right ) d^{2}}+\frac {B \,c^{2}}{\left (b e -c d \right )^{2} \left (c x +b \right ) b}+\frac {B \,e^{2}}{\left (b e -c d \right )^{2} \left (e x +d \right ) d}-\frac {2 A e \ln \relax (x )}{b^{2} d^{3}}-\frac {2 A c \ln \relax (x )}{b^{3} d^{2}}+\frac {B \ln \relax (x )}{b^{2} d^{2}}-\frac {A}{b^{2} d^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 467, normalized size = 2.32 \begin {gather*} -\frac {{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d - {\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} - \frac {{\left (3 \, B c d^{2} e^{2} + 2 \, A b e^{4} - {\left (B b + 4 \, A c\right )} d e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {A b c^{2} d^{3} - 2 \, A b^{2} c d^{2} e + A b^{3} d e^{2} + {\left (2 \, A b^{2} c e^{3} - {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} e - {\left (B b^{2} c + 2 \, A b c^{2}\right )} d e^{2}\right )} x^{2} - {\left (A b c^{2} d^{2} e - 2 \, A b^{3} e^{3} + {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{3} + {\left (B b^{3} + A b^{2} c\right )} d e^{2}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} + {\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} + {\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac {{\left (2 \, A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \relax (x)}{b^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 410, normalized size = 2.04 \begin {gather*} \frac {\frac {x\,\left (B\,b^3\,d\,e^2-2\,A\,b^3\,e^3+A\,b^2\,c\,d\,e^2+B\,b\,c^2\,d^3+A\,b\,c^2\,d^2\,e-2\,A\,c^3\,d^3\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}-\frac {A}{b\,d}+\frac {x^2\,\left (B\,b^2\,c\,d\,e^2-2\,A\,b^2\,c\,e^3+B\,b\,c^2\,d^2\,e+2\,A\,b\,c^2\,d\,e^2-2\,A\,c^3\,d^2\,e\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+b\,d\,x}-\frac {\ln \left (b+c\,x\right )\,\left (e\,\left (3\,B\,b^2\,c^2-4\,A\,b\,c^3\right )+d\,\left (2\,A\,c^4-B\,b\,c^3\right )\right )}{b^6\,e^3-3\,b^5\,c\,d\,e^2+3\,b^4\,c^2\,d^2\,e-b^3\,c^3\,d^3}-\frac {\ln \left (d+e\,x\right )\,\left (c\,\left (3\,B\,d^2\,e^2-4\,A\,d\,e^3\right )+b\,\left (2\,A\,e^4-B\,d\,e^3\right )\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}-\frac {\ln \relax (x)\,\left (d\,\left (2\,A\,c-B\,b\right )+2\,A\,b\,e\right )}{b^3\,d^3} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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