Optimal. Leaf size=198 \[ -\frac {d (2 A b e-3 A c d+b B d)}{b^4 x}-\frac {(c d-b e) (A b e-3 A c d+2 b B d)}{b^4 (b+c x)}-\frac {(b B-A c) (c d-b e)^2}{2 b^3 c (b+c x)^2}-\frac {A d^2}{2 b^3 x^2}+\frac {\log (x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac {\log (b+c x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{b^5} \]
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Rubi [A] time = 0.24, antiderivative size = 198, 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 {\log (x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac {\log (b+c x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac {d (2 A b e-3 A c d+b B d)}{b^4 x}-\frac {(c d-b e) (A b e-3 A c d+2 b B d)}{b^4 (b+c x)}-\frac {(b B-A c) (c d-b e)^2}{2 b^3 c (b+c x)^2}-\frac {A d^2}{2 b^3 x^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)^2}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {A d^2}{b^3 x^3}+\frac {d (b B d-3 A c d+2 A b e)}{b^4 x^2}+\frac {6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)}{b^5 x}+\frac {(b B-A c) (-c d+b e)^2}{b^3 (b+c x)^3}-\frac {c (-c d+b e) (2 b B d-3 A c d+A b e)}{b^4 (b+c x)^2}+\frac {c \left (-6 A c^2 d^2-b^2 e (2 B d+A e)+3 b c d (B d+2 A e)\right )}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {A d^2}{2 b^3 x^2}-\frac {d (b B d-3 A c d+2 A b e)}{b^4 x}-\frac {(b B-A c) (c d-b e)^2}{2 b^3 c (b+c x)^2}-\frac {(c d-b e) (2 b B d-3 A c d+A b e)}{b^4 (b+c x)}+\frac {\left (6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)\right ) \log (x)}{b^5}-\frac {\left (6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)\right ) \log (b+c x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 190, normalized size = 0.96 \begin {gather*} -\frac {-2 \log (x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )+2 \log (b+c x) \left (b^2 e (A e+2 B d)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )+\frac {b^2 (b B-A c) (c d-b e)^2}{c (b+c x)^2}+\frac {A b^2 d^2}{x^2}+\frac {2 b d (2 A b e-3 A c d+b B d)}{x}-\frac {2 b (b e-c d) (A b e-3 A c d+2 b B d)}{b+c x}}{2 b^5} \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 )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 558, normalized size = 2.82 \begin {gather*} -\frac {A b^{4} c d^{2} - 2 \, {\left (A b^{3} c^{2} e^{2} - 3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} + {\left (9 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} - 6 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e + {\left (B b^{5} - 3 \, A b^{4} c\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, A b^{4} c d e + {\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2}\right )} x + 2 \, {\left ({\left (A b^{2} c^{3} e^{2} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} + 2 \, {\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d e\right )} x^{4} + 2 \, {\left (A b^{3} c^{2} e^{2} - 3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} + {\left (A b^{4} c e^{2} - 3 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} + 2 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (A b^{2} c^{3} e^{2} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} + 2 \, {\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d e\right )} x^{4} + 2 \, {\left (A b^{3} c^{2} e^{2} - 3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} + {\left (A b^{4} c e^{2} - 3 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} + 2 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{3} x^{4} + 2 \, b^{6} c^{2} x^{3} + b^{7} c x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 324, normalized size = 1.64 \begin {gather*} -\frac {{\left (3 \, B b c d^{2} - 6 \, A c^{2} d^{2} - 2 \, B b^{2} d e + 6 \, A b c d e - A b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {{\left (3 \, B b c^{2} d^{2} - 6 \, A c^{3} d^{2} - 2 \, B b^{2} c d e + 6 \, A b c^{2} d e - A b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {6 \, B b c^{3} d^{2} x^{3} - 12 \, A c^{4} d^{2} x^{3} - 4 \, B b^{2} c^{2} d x^{3} e + 12 \, A b c^{3} d x^{3} e + 9 \, B b^{2} c^{2} d^{2} x^{2} - 18 \, A b c^{3} d^{2} x^{2} - 2 \, A b^{2} c^{2} x^{3} e^{2} - 6 \, B b^{3} c d x^{2} e + 18 \, A b^{2} c^{2} d x^{2} e + 2 \, B b^{3} c d^{2} x - 4 \, A b^{2} c^{2} d^{2} x + B b^{4} x^{2} e^{2} - 3 \, A b^{3} c x^{2} e^{2} + 4 \, A b^{3} c d x e + A b^{3} c d^{2}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 365, normalized size = 1.84 \begin {gather*} \frac {A \,e^{2}}{2 \left (c x +b \right )^{2} b}-\frac {A c d e}{\left (c x +b \right )^{2} b^{2}}+\frac {A \,c^{2} d^{2}}{2 \left (c x +b \right )^{2} b^{3}}+\frac {B d e}{\left (c x +b \right )^{2} b}-\frac {B c \,d^{2}}{2 \left (c x +b \right )^{2} b^{2}}-\frac {B \,e^{2}}{2 \left (c x +b \right )^{2} c}+\frac {A \,e^{2}}{\left (c x +b \right ) b^{2}}-\frac {4 A c d e}{\left (c x +b \right ) b^{3}}+\frac {A \,e^{2} \ln \relax (x )}{b^{3}}-\frac {A \,e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 A \,c^{2} d^{2}}{\left (c x +b \right ) b^{4}}-\frac {6 A c d e \ln \relax (x )}{b^{4}}+\frac {6 A c d e \ln \left (c x +b \right )}{b^{4}}+\frac {6 A \,c^{2} d^{2} \ln \relax (x )}{b^{5}}-\frac {6 A \,c^{2} d^{2} \ln \left (c x +b \right )}{b^{5}}+\frac {2 B d e}{\left (c x +b \right ) b^{2}}-\frac {2 B c \,d^{2}}{\left (c x +b \right ) b^{3}}+\frac {2 B d e \ln \relax (x )}{b^{3}}-\frac {2 B d e \ln \left (c x +b \right )}{b^{3}}-\frac {3 B c \,d^{2} \ln \relax (x )}{b^{4}}+\frac {3 B c \,d^{2} \ln \left (c x +b \right )}{b^{4}}-\frac {2 A d e}{b^{3} x}+\frac {3 A c \,d^{2}}{b^{4} x}-\frac {B \,d^{2}}{b^{3} x}-\frac {A \,d^{2}}{2 b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 293, normalized size = 1.48 \begin {gather*} -\frac {A b^{3} c d^{2} - 2 \, {\left (A b^{2} c^{2} e^{2} - 3 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} + 2 \, {\left (B b^{2} c^{2} - 3 \, A b c^{3}\right )} d e\right )} x^{3} + {\left (9 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{2} - 6 \, {\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} d e + {\left (B b^{4} - 3 \, A b^{3} c\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, A b^{3} c d e + {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d^{2}\right )} x}{2 \, {\left (b^{4} c^{3} x^{4} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x^{2}\right )}} - \frac {{\left (A b^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \left (c x + b\right )}{b^{5}} + \frac {{\left (A b^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \relax (x)}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 319, normalized size = 1.61 \begin {gather*} -\frac {\frac {A\,d^2}{2\,b}-\frac {c\,x^3\,\left (2\,B\,b^2\,d\,e+A\,b^2\,e^2-3\,B\,b\,c\,d^2-6\,A\,b\,c\,d\,e+6\,A\,c^2\,d^2\right )}{b^4}+\frac {d\,x\,\left (2\,A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^2}-\frac {x^2\,\left (-B\,b^3\,e^2+6\,B\,b^2\,c\,d\,e+3\,A\,b^2\,c\,e^2-9\,B\,b\,c^2\,d^2-18\,A\,b\,c^2\,d\,e+18\,A\,c^3\,d^2\right )}{2\,b^3\,c}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {\left (b+2\,c\,x\right )\,\left (b^2\,\left (A\,e^2+2\,B\,d\,e\right )-b\,\left (3\,B\,c\,d^2+6\,A\,c\,e\,d\right )+6\,A\,c^2\,d^2\right )}{b\,\left (2\,B\,b^2\,d\,e+A\,b^2\,e^2-3\,B\,b\,c\,d^2-6\,A\,b\,c\,d\,e+6\,A\,c^2\,d^2\right )}\right )\,\left (b^2\,\left (A\,e^2+2\,B\,d\,e\right )-b\,\left (3\,B\,c\,d^2+6\,A\,c\,e\,d\right )+6\,A\,c^2\,d^2\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 10.67, size = 660, normalized size = 3.33 \begin {gather*} \frac {- A b^{3} c d^{2} + x^{3} \left (2 A b^{2} c^{2} e^{2} - 12 A b c^{3} d e + 12 A c^{4} d^{2} + 4 B b^{2} c^{2} d e - 6 B b c^{3} d^{2}\right ) + x^{2} \left (3 A b^{3} c e^{2} - 18 A b^{2} c^{2} d e + 18 A b c^{3} d^{2} - B b^{4} e^{2} + 6 B b^{3} c d e - 9 B b^{2} c^{2} d^{2}\right ) + x \left (- 4 A b^{3} c d e + 4 A b^{2} c^{2} d^{2} - 2 B b^{3} c d^{2}\right )}{2 b^{6} c x^{2} + 4 b^{5} c^{2} x^{3} + 2 b^{4} c^{3} x^{4}} + \frac {\left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right ) \log {\left (x + \frac {A b^{3} e^{2} - 6 A b^{2} c d e + 6 A b c^{2} d^{2} + 2 B b^{3} d e - 3 B b^{2} c d^{2} - b \left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right )}{2 A b^{2} c e^{2} - 12 A b c^{2} d e + 12 A c^{3} d^{2} + 4 B b^{2} c d e - 6 B b c^{2} d^{2}} \right )}}{b^{5}} - \frac {\left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right ) \log {\left (x + \frac {A b^{3} e^{2} - 6 A b^{2} c d e + 6 A b c^{2} d^{2} + 2 B b^{3} d e - 3 B b^{2} c d^{2} + b \left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right )}{2 A b^{2} c e^{2} - 12 A b c^{2} d e + 12 A c^{3} d^{2} + 4 B b^{2} c d e - 6 B b c^{2} d^{2}} \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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