Optimal. Leaf size=283 \[ \frac {\log (x) (-3 A b e-2 A c d+b B d)}{b^3 d^4}+\frac {c^3 (b B-A c)}{b^2 (b+c x) (c d-b e)^3}-\frac {e^2 \log (d+e x) \left (B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )-A e \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )\right )}{d^4 (c d-b e)^4}-\frac {A}{b^2 d^3 x}+\frac {c^3 \log (b+c x) \left (-b c (5 A e+B d)+2 A c^2 d+4 b^2 B e\right )}{b^3 (c d-b e)^4}-\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (d+e x) (c d-b e)^3}+\frac {e^2 (B d-A e)}{2 d^2 (d+e x)^2 (c d-b e)^2} \]
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Rubi [A] time = 0.46, antiderivative size = 283, 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^2 \log (d+e x) \left (B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )-A e \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )\right )}{d^4 (c d-b e)^4}+\frac {c^3 (b B-A c)}{b^2 (b+c x) (c d-b e)^3}+\frac {c^3 \log (b+c x) \left (-b c (5 A e+B d)+2 A c^2 d+4 b^2 B e\right )}{b^3 (c d-b e)^4}+\frac {\log (x) (-3 A b e-2 A c d+b B d)}{b^3 d^4}-\frac {A}{b^2 d^3 x}-\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (d+e x) (c d-b e)^3}+\frac {e^2 (B d-A e)}{2 d^2 (d+e x)^2 (c d-b e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {A}{b^2 d^3 x^2}+\frac {b B d-2 A c d-3 A b e}{b^3 d^4 x}+\frac {c^4 (b B-A c)}{b^2 (-c d+b e)^3 (b+c x)^2}+\frac {c^4 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right )}{b^3 (c d-b e)^4 (b+c x)}-\frac {e^3 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)^3}+\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)^2}+\frac {e^3 \left (-B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right )}{d^4 (c d-b e)^4 (d+e x)}\right ) \, dx\\ &=-\frac {A}{b^2 d^3 x}+\frac {c^3 (b B-A c)}{b^2 (c d-b e)^3 (b+c x)}+\frac {e^2 (B d-A e)}{2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e^2 (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)}+\frac {(b B d-2 A c d-3 A b e) \log (x)}{b^3 d^4}+\frac {c^3 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^4}-\frac {e^2 \left (B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 279, normalized size = 0.99 \begin {gather*} \frac {\log (x) (-3 A b e-2 A c d+b B d)}{b^3 d^4}+\frac {c^3 (A c-b B)}{b^2 (b+c x) (b e-c d)^3}+\frac {e^2 \log (d+e x) \left (A e \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )-B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{d^4 (c d-b e)^4}-\frac {A}{b^2 d^3 x}+\frac {c^3 \log (b+c x) \left (-b c (5 A e+B d)+2 A c^2 d+4 b^2 B e\right )}{b^3 (c d-b e)^4}+\frac {e^2 (2 A e (b e-2 c d)+B d (3 c d-b e))}{d^3 (d+e x) (c d-b e)^3}+\frac {e^2 (B d-A e)}{2 d^2 (d+e x)^2 (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)^3 \left (b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [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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giac [B] time = 0.20, size = 745, normalized size = 2.63 \begin {gather*} -\frac {{\left (B b c^{5} d - 2 \, A c^{6} d - 4 \, B b^{2} c^{4} e + 5 \, A b c^{5} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{5} d^{4} - 4 \, b^{4} c^{4} d^{3} e + 6 \, b^{5} c^{3} d^{2} e^{2} - 4 \, b^{6} c^{2} d e^{3} + b^{7} c e^{4}} - \frac {{\left (6 \, B c^{2} d^{3} e^{3} - 4 \, B b c d^{2} e^{4} - 10 \, A c^{2} d^{2} e^{4} + B b^{2} d e^{5} + 10 \, A b c d e^{5} - 3 \, A b^{2} e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e - 4 \, b c^{3} d^{7} e^{2} + 6 \, b^{2} c^{2} d^{6} e^{3} - 4 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5}} + \frac {{\left (B b d - 2 \, A c d - 3 \, A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{4}} - \frac {2 \, A b c^{4} d^{7} - 8 \, A b^{2} c^{3} d^{6} e + 12 \, A b^{3} c^{2} d^{5} e^{2} - 8 \, A b^{4} c d^{4} e^{3} + 2 \, A b^{5} d^{3} e^{4} - 2 \, {\left (B b c^{4} d^{5} e^{2} - 2 \, A c^{5} d^{5} e^{2} + 2 \, B b^{2} c^{3} d^{4} e^{3} + 5 \, A b c^{4} d^{4} e^{3} - 4 \, B b^{3} c^{2} d^{3} e^{4} - 10 \, A b^{2} c^{3} d^{3} e^{4} + B b^{4} c d^{2} e^{5} + 10 \, A b^{3} c^{2} d^{2} e^{5} - 3 \, A b^{4} c d e^{6}\right )} x^{3} - {\left (4 \, B b c^{4} d^{6} e - 8 \, A c^{5} d^{6} e + 3 \, B b^{2} c^{3} d^{5} e^{2} + 18 \, A b c^{4} d^{5} e^{2} - 4 \, B b^{3} c^{2} d^{4} e^{3} - 25 \, A b^{2} c^{3} d^{4} e^{3} - 5 \, B b^{4} c d^{3} e^{4} + 10 \, A b^{3} c^{2} d^{3} e^{4} + 2 \, B b^{5} d^{2} e^{5} + 11 \, A b^{4} c d^{2} e^{5} - 6 \, A b^{5} d e^{6}\right )} x^{2} - {\left (2 \, B b c^{4} d^{7} - 4 \, A c^{5} d^{7} - 2 \, B b^{2} c^{3} d^{6} e + 6 \, A b c^{4} d^{6} e + 7 \, B b^{3} c^{2} d^{5} e^{2} + 4 \, A b^{2} c^{3} d^{5} e^{2} - 10 \, B b^{4} c d^{4} e^{3} - 25 \, A b^{3} c^{2} d^{4} e^{3} + 3 \, B b^{5} d^{3} e^{4} + 28 \, A b^{4} c d^{3} e^{4} - 9 \, A b^{5} d^{2} e^{5}\right )} x}{2 \, {\left (c d - b e\right )}^{4} {\left (c x + b\right )} {\left (x e + d\right )}^{2} b^{2} d^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 528, normalized size = 1.87 \begin {gather*} \frac {3 A \,b^{2} e^{5} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{4}}-\frac {10 A b c \,e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{3}}-\frac {5 A \,c^{4} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{2}}+\frac {2 A \,c^{5} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{3}}+\frac {10 A \,c^{2} e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{2}}-\frac {B \,b^{2} e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{3}}+\frac {4 B b c \,e^{3} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{2}}+\frac {4 B \,c^{3} e \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b}-\frac {B \,c^{4} d \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{2}}-\frac {6 B \,c^{2} e^{2} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d}-\frac {2 A b \,e^{4}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d^{3}}+\frac {A \,c^{4}}{\left (b e -c d \right )^{3} \left (c x +b \right ) b^{2}}+\frac {4 A c \,e^{3}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d^{2}}+\frac {B b \,e^{3}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d^{2}}-\frac {B \,c^{3}}{\left (b e -c d \right )^{3} \left (c x +b \right ) b}-\frac {3 B c \,e^{2}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d}-\frac {A \,e^{3}}{2 \left (b e -c d \right )^{2} \left (e x +d \right )^{2} d^{2}}+\frac {B \,e^{2}}{2 \left (b e -c d \right )^{2} \left (e x +d \right )^{2} d}-\frac {3 A e \ln \relax (x )}{b^{2} d^{4}}-\frac {2 A c \ln \relax (x )}{b^{3} d^{3}}+\frac {B \ln \relax (x )}{b^{2} d^{3}}-\frac {A}{b^{2} d^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.11, size = 813, normalized size = 2.87 \begin {gather*} -\frac {{\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d - {\left (4 \, B b^{2} c^{3} - 5 \, A b c^{4}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{4} d^{4} - 4 \, b^{4} c^{3} d^{3} e + 6 \, b^{5} c^{2} d^{2} e^{2} - 4 \, b^{6} c d e^{3} + b^{7} e^{4}} - \frac {{\left (6 \, B c^{2} d^{3} e^{2} - 3 \, A b^{2} e^{5} - 2 \, {\left (2 \, B b c + 5 \, A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 10 \, A b c\right )} d e^{4}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} - \frac {2 \, A b c^{3} d^{5} - 6 \, A b^{2} c^{2} d^{4} e + 6 \, A b^{3} c d^{3} e^{2} - 2 \, A b^{4} d^{2} e^{3} - 2 \, {\left (3 \, A b^{3} c e^{5} + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} c^{2} + A b c^{3}\right )} d^{2} e^{3} - {\left (B b^{3} c + 7 \, A b^{2} c^{2}\right )} d e^{4}\right )} x^{3} - {\left (6 \, A b^{4} e^{5} + 4 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{4} e + {\left (7 \, B b^{2} c^{2} + 10 \, A b c^{3}\right )} d^{3} e^{2} + 3 \, {\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} d^{2} e^{3} - {\left (2 \, B b^{4} + 5 \, A b^{3} c\right )} d e^{4}\right )} x^{2} - {\left (2 \, A b c^{3} d^{4} e + 9 \, A b^{4} d e^{4} + 2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{5} + {\left (7 \, B b^{3} c + 6 \, A b^{2} c^{2}\right )} d^{3} e^{2} - {\left (3 \, B b^{4} + 19 \, A b^{3} c\right )} d^{2} e^{3}\right )} x}{2 \, {\left ({\left (b^{2} c^{4} d^{6} e^{2} - 3 \, b^{3} c^{3} d^{5} e^{3} + 3 \, b^{4} c^{2} d^{4} e^{4} - b^{5} c d^{3} e^{5}\right )} x^{4} + {\left (2 \, b^{2} c^{4} d^{7} e - 5 \, b^{3} c^{3} d^{6} e^{2} + 3 \, b^{4} c^{2} d^{5} e^{3} + b^{5} c d^{4} e^{4} - b^{6} d^{3} e^{5}\right )} x^{3} + {\left (b^{2} c^{4} d^{8} - b^{3} c^{3} d^{7} e - 3 \, b^{4} c^{2} d^{6} e^{2} + 5 \, b^{5} c d^{5} e^{3} - 2 \, b^{6} d^{4} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{8} - 3 \, b^{4} c^{2} d^{7} e + 3 \, b^{5} c d^{6} e^{2} - b^{6} d^{5} e^{3}\right )} x\right )}} - \frac {{\left (3 \, A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \relax (x)}{b^{3} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.94, size = 726, normalized size = 2.57 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (\left (3\,A\,e^5-B\,d\,e^4\right )\,b^2+\left (4\,B\,d^2\,e^3-10\,A\,d\,e^4\right )\,b\,c+\left (10\,A\,d^2\,e^3-6\,B\,d^3\,e^2\right )\,c^2\right )}{b^4\,d^4\,e^4-4\,b^3\,c\,d^5\,e^3+6\,b^2\,c^2\,d^6\,e^2-4\,b\,c^3\,d^7\,e+c^4\,d^8}-\frac {\frac {A}{b\,d}+\frac {x^2\,\left (-2\,B\,b^4\,d\,e^4+6\,A\,b^4\,e^5+3\,B\,b^3\,c\,d^2\,e^3-5\,A\,b^3\,c\,d\,e^4+7\,B\,b^2\,c^2\,d^3\,e^2-15\,A\,b^2\,c^2\,d^2\,e^3+4\,B\,b\,c^3\,d^4\,e+10\,A\,b\,c^3\,d^3\,e^2-8\,A\,c^4\,d^4\,e\right )}{2\,b^2\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {x\,\left (-3\,B\,b^4\,d\,e^3+9\,A\,b^4\,e^4+7\,B\,b^3\,c\,d^2\,e^2-19\,A\,b^3\,c\,d\,e^3+6\,A\,b^2\,c^2\,d^2\,e^2+2\,B\,b\,c^3\,d^4+2\,A\,b\,c^3\,d^3\,e-4\,A\,c^4\,d^4\right )}{2\,b^2\,d^2\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {c\,e^2\,x^3\,\left (-B\,b^3\,d\,e^2+3\,A\,b^3\,e^3+3\,B\,b^2\,c\,d^2\,e-7\,A\,b^2\,c\,d\,e^2+B\,b\,c^2\,d^3+3\,A\,b\,c^2\,d^2\,e-2\,A\,c^3\,d^3\right )}{b^2\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{x^2\,\left (c\,d^2+2\,b\,e\,d\right )+x^3\,\left (b\,e^2+2\,c\,d\,e\right )+c\,e^2\,x^4+b\,d^2\,x}+\frac {\ln \left (b+c\,x\right )\,\left (e\,\left (4\,B\,b^2\,c^3-5\,A\,b\,c^4\right )+d\,\left (2\,A\,c^5-B\,b\,c^4\right )\right )}{b^7\,e^4-4\,b^6\,c\,d\,e^3+6\,b^5\,c^2\,d^2\,e^2-4\,b^4\,c^3\,d^3\,e+b^3\,c^4\,d^4}-\frac {\ln \relax (x)\,\left (d\,\left (2\,A\,c-B\,b\right )+3\,A\,b\,e\right )}{b^3\,d^4} \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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