Optimal. Leaf size=279 \[ -\frac {-A b e-3 A c d+b B d}{b^4 d^2 x}-\frac {c^2 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)}-\frac {A}{2 b^3 d x^2}+\frac {\log (x) \left (b^2 (-e) (B d-A e)-3 b c d (B d-A e)+6 A c^2 d^2\right )}{b^5 d^3}+\frac {c^2 \left (-2 b c (2 A e+B d)+3 A c^2 d+3 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^2}-\frac {c^2 \log (b+c x) \left (2 b^2 c e (5 A e+4 B d)-3 b c^2 d (5 A e+B d)+6 A c^3 d^2-6 b^3 B e^2\right )}{b^5 (c d-b e)^3}-\frac {e^4 (B d-A e) \log (d+e x)}{d^3 (c d-b e)^3} \]
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Rubi [A] time = 0.50, antiderivative size = 279, 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 \log (b+c x) \left (2 b^2 c e (5 A e+4 B d)-3 b c^2 d (5 A e+B d)+6 A c^3 d^2-6 b^3 B e^2\right )}{b^5 (c d-b e)^3}+\frac {\log (x) \left (b^2 (-e) (B d-A e)-3 b c d (B d-A e)+6 A c^2 d^2\right )}{b^5 d^3}+\frac {c^2 \left (-2 b c (2 A e+B d)+3 A c^2 d+3 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^2}-\frac {c^2 (b B-A c)}{2 b^3 (b+c x)^2 (c d-b e)}-\frac {-A b e-3 A c d+b B d}{b^4 d^2 x}-\frac {A}{2 b^3 d x^2}-\frac {e^4 (B d-A e) \log (d+e x)}{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) \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {A}{b^3 d x^3}+\frac {b B d-3 A c d-A b e}{b^4 d^2 x^2}+\frac {6 A c^2 d^2-3 b c d (B d-A e)-b^2 e (B d-A e)}{b^5 d^3 x}-\frac {c^3 (b B-A c)}{b^3 (-c d+b e) (b+c x)^3}+\frac {c^3 \left (-3 A c^2 d-3 b^2 B e+2 b c (B d+2 A e)\right )}{b^4 (c d-b e)^2 (b+c x)^2}+\frac {c^3 \left (-6 A c^3 d^2+6 b^3 B e^2+3 b c^2 d (B d+5 A e)-2 b^2 c e (4 B d+5 A e)\right )}{b^5 (c d-b e)^3 (b+c x)}-\frac {e^5 (B d-A e)}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {A}{2 b^3 d x^2}-\frac {b B d-3 A c d-A b e}{b^4 d^2 x}-\frac {c^2 (b B-A c)}{2 b^3 (c d-b e) (b+c x)^2}+\frac {c^2 \left (3 A c^2 d+3 b^2 B e-2 b c (B d+2 A e)\right )}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 A c^2 d^2-3 b c d (B d-A e)-b^2 e (B d-A e)\right ) \log (x)}{b^5 d^3}-\frac {c^2 \left (6 A c^3 d^2-6 b^3 B e^2-3 b c^2 d (B d+5 A e)+2 b^2 c e (4 B d+5 A e)\right ) \log (b+c x)}{b^5 (c d-b e)^3}-\frac {e^4 (B d-A e) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 276, normalized size = 0.99 \begin {gather*} \frac {A b e+3 A c d-b B d}{b^4 d^2 x}+\frac {c^2 (b B-A c)}{2 b^3 (b+c x)^2 (b e-c d)}-\frac {A}{2 b^3 d x^2}-\frac {\log (x) \left (b^2 e (B d-A e)+3 b c d (B d-A e)-6 A c^2 d^2\right )}{b^5 d^3}+\frac {c^2 \left (-2 b c (2 A e+B d)+3 A c^2 d+3 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^2}+\frac {c^2 \log (b+c x) \left (2 b^2 c e (5 A e+4 B d)-3 b c^2 d (5 A e+B d)+6 A c^3 d^2-6 b^3 B e^2\right )}{b^5 (b e-c d)^3}+\frac {e^4 (A e-B d) \log (d+e x)}{d^3 (c d-b e)^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) \left (b x+c x^2\right )^3} \, 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 = 0.24, size = 649, normalized size = 2.33 \begin {gather*} \frac {{\left (3 \, B b c^{5} d^{2} - 6 \, A c^{6} d^{2} - 8 \, B b^{2} c^{4} d e + 15 \, A b c^{5} d e + 6 \, B b^{3} c^{3} e^{2} - 10 \, A b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} - \frac {{\left (B d e^{5} - A e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} - \frac {{\left (3 \, B b c d^{2} - 6 \, A c^{2} d^{2} + B b^{2} d e - 3 \, A b c d e - A b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac {A b^{3} c^{3} d^{5} - 3 \, A b^{4} c^{2} d^{4} e + 3 \, A b^{5} c d^{3} e^{2} - A b^{6} d^{2} e^{3} + 2 \, {\left (3 \, B b c^{5} d^{5} - 6 \, A c^{6} d^{5} - 8 \, B b^{2} c^{4} d^{4} e + 15 \, A b c^{5} d^{4} e + 6 \, B b^{3} c^{3} d^{3} e^{2} - 10 \, A b^{2} c^{4} d^{3} e^{2} - B b^{4} c^{2} d^{2} e^{3} + A b^{4} c^{2} d e^{4}\right )} x^{3} + {\left (9 \, B b^{2} c^{4} d^{5} - 18 \, A b c^{5} d^{5} - 24 \, B b^{3} c^{3} d^{4} e + 45 \, A b^{2} c^{4} d^{4} e + 19 \, B b^{4} c^{2} d^{3} e^{2} - 30 \, A b^{3} c^{3} d^{3} e^{2} - 4 \, B b^{5} c d^{2} e^{3} - A b^{4} c^{2} d^{2} e^{3} + 4 \, A b^{5} c d e^{4}\right )} x^{2} + 2 \, {\left (B b^{3} c^{3} d^{5} - 2 \, A b^{2} c^{4} d^{5} - 3 \, B b^{4} c^{2} d^{4} e + 5 \, A b^{3} c^{3} d^{4} e + 3 \, B b^{5} c d^{3} e^{2} - 3 \, A b^{4} c^{2} d^{3} e^{2} - B b^{6} d^{2} e^{3} - A b^{5} c d^{2} e^{3} + A b^{6} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 490, normalized size = 1.76 \begin {gather*} \frac {10 A \,c^{3} e^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {15 A \,c^{4} d e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{4}}+\frac {6 A \,c^{5} d^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{5}}-\frac {A \,e^{5} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}-\frac {6 B \,c^{2} e^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{2}}+\frac {8 B \,c^{3} d e \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {3 B \,c^{4} d^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{4}}+\frac {B \,e^{4} \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{2}}-\frac {4 A \,c^{3} e}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{3}}+\frac {3 A \,c^{4} d}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{4}}+\frac {3 B \,c^{2} e}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{2}}-\frac {2 B \,c^{3} d}{\left (b e -c d \right )^{2} \left (c x +b \right ) b^{3}}-\frac {A \,c^{3}}{2 \left (b e -c d \right ) \left (c x +b \right )^{2} b^{3}}+\frac {B \,c^{2}}{2 \left (b e -c d \right ) \left (c x +b \right )^{2} b^{2}}+\frac {A \,e^{2} \ln \relax (x )}{b^{3} d^{3}}+\frac {3 A c e \ln \relax (x )}{b^{4} d^{2}}+\frac {6 A \,c^{2} \ln \relax (x )}{b^{5} d}-\frac {B e \ln \relax (x )}{b^{3} d^{2}}-\frac {3 B c \ln \relax (x )}{b^{4} d}+\frac {A e}{b^{3} d^{2} x}+\frac {3 A c}{b^{4} d x}-\frac {B}{b^{3} d x}-\frac {A}{2 b^{3} d \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 612, normalized size = 2.19 \begin {gather*} \frac {{\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} - {\left (8 \, B b^{2} c^{3} - 15 \, A b c^{4}\right )} d e + 2 \, {\left (3 \, B b^{3} c^{2} - 5 \, A b^{2} c^{3}\right )} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac {{\left (B d e^{4} - A e^{5}\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^{3} c^{2} d^{3} - 2 \, A b^{4} c d^{2} e + A b^{5} d e^{2} - 2 \, {\left (A b^{3} c^{2} e^{3} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + {\left (5 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} d^{2} e - {\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d e^{2}\right )} x^{3} - {\left (4 \, A b^{4} c e^{3} - 9 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + 3 \, {\left (5 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} d^{2} e - {\left (4 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2}\right )} x^{2} + 2 \, {\left (B b^{5} d e^{2} - A b^{5} e^{3} + {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3} - {\left (2 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} d^{2} e\right )} x}{2 \, {\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac {{\left (A b^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} - {\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \relax (x)}{b^{5} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.61, size = 512, normalized size = 1.84 \begin {gather*} \frac {\frac {x\,\left (A\,b\,e+2\,A\,c\,d-B\,b\,d\right )}{b^2\,d^2}-\frac {A}{2\,b\,d}+\frac {x^3\,\left (-B\,b^3\,c^2\,d\,e^2+A\,b^3\,c^2\,e^3+5\,B\,b^2\,c^3\,d^2\,e+A\,b^2\,c^3\,d\,e^2-3\,B\,b\,c^4\,d^3-9\,A\,b\,c^4\,d^2\,e+6\,A\,c^5\,d^3\right )}{b^4\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x^2\,\left (-4\,B\,b^3\,c\,d\,e^2+4\,A\,b^3\,c\,e^3+15\,B\,b^2\,c^2\,d^2\,e+3\,A\,b^2\,c^2\,d\,e^2-9\,B\,b\,c^3\,d^3-27\,A\,b\,c^3\,d^2\,e+18\,A\,c^4\,d^3\right )}{2\,b^3\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (d+e\,x\right )\,\left (A\,e^5-B\,d\,e^4\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 \left (b+c\,x\right )\,\left (d^2\,\left (6\,A\,c^5-3\,B\,b\,c^4\right )-d\,\left (15\,A\,b\,c^4\,e-8\,B\,b^2\,c^3\,e\right )+10\,A\,b^2\,c^3\,e^2-6\,B\,b^3\,c^2\,e^2\right )}{b^8\,e^3-3\,b^7\,c\,d\,e^2+3\,b^6\,c^2\,d^2\,e-b^5\,c^3\,d^3}+\frac {\ln \relax (x)\,\left (d^2\,\left (6\,A\,c^2-3\,B\,b\,c\right )-d\,\left (B\,b^2\,e-3\,A\,b\,c\,e\right )+A\,b^2\,e^2\right )}{b^5\,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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