Optimal. Leaf size=168 \[ -\frac {A b e-3 A c d+b B d}{b^4 x}-\frac {(b B-A c) (c d-b e)}{2 b^3 (b+c x)^2}-\frac {A d}{2 b^3 x^2}+\frac {\log (x) \left (-3 b c (A e+B d)+6 A c^2 d+b^2 B e\right )}{b^5}-\frac {\log (b+c x) \left (-3 b c (A e+B d)+6 A c^2 d+b^2 B e\right )}{b^5}+\frac {-2 b c (A e+B d)+3 A c^2 d+b^2 B e}{b^4 (b+c x)} \]
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Rubi [A] time = 0.21, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {771} \begin {gather*} \frac {-2 b c (A e+B d)+3 A c^2 d+b^2 B e}{b^4 (b+c x)}+\frac {\log (x) \left (-3 b c (A e+B d)+6 A c^2 d+b^2 B e\right )}{b^5}-\frac {\log (b+c x) \left (-3 b c (A e+B d)+6 A c^2 d+b^2 B e\right )}{b^5}-\frac {A b e-3 A c d+b B d}{b^4 x}-\frac {(b B-A c) (c d-b e)}{2 b^3 (b+c x)^2}-\frac {A d}{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)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {A d}{b^3 x^3}+\frac {b B d-3 A c d+A b e}{b^4 x^2}+\frac {6 A c^2 d+b^2 B e-3 b c (B d+A e)}{b^5 x}-\frac {c (b B-A c) (-c d+b e)}{b^3 (b+c x)^3}+\frac {c \left (-3 A c^2 d-b^2 B e+2 b c (B d+A e)\right )}{b^4 (b+c x)^2}+\frac {c \left (-6 A c^2 d-b^2 B e+3 b c (B d+A e)\right )}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {A d}{2 b^3 x^2}-\frac {b B d-3 A c d+A b e}{b^4 x}-\frac {(b B-A c) (c d-b e)}{2 b^3 (b+c x)^2}+\frac {3 A c^2 d+b^2 B e-2 b c (B d+A e)}{b^4 (b+c x)}+\frac {\left (6 A c^2 d+b^2 B e-3 b c (B d+A e)\right ) \log (x)}{b^5}-\frac {\left (6 A c^2 d+b^2 B e-3 b c (B d+A e)\right ) \log (b+c x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 162, normalized size = 0.96 \begin {gather*} \frac {\frac {2 b \left (-2 b c (A e+B d)+3 A c^2 d+b^2 B e\right )}{b+c x}+2 \log (x) \left (-3 b c (A e+B d)+6 A c^2 d+b^2 B e\right )-2 \log (b+c x) \left (-3 b c (A e+B d)+6 A c^2 d+b^2 B e\right )+\frac {b^2 (b B-A c) (b e-c d)}{(b+c x)^2}-\frac {A b^2 d}{x^2}-\frac {2 b (A b e-3 A c d+b B d)}{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)}{\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 = 410, normalized size = 2.44 \begin {gather*} -\frac {A b^{4} d + 2 \, {\left (3 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d - {\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} e\right )} x^{3} + 3 \, {\left (3 \, {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d - {\left (B b^{4} - 3 \, A b^{3} c\right )} e\right )} x^{2} + 2 \, {\left (A b^{4} e + {\left (B b^{4} - 2 \, A b^{3} c\right )} d\right )} x - 2 \, {\left ({\left (3 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d - {\left (B b^{2} c^{2} - 3 \, A b c^{3}\right )} e\right )} x^{4} + 2 \, {\left (3 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d - {\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} e\right )} x^{3} + {\left (3 \, {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d - {\left (B b^{4} - 3 \, A b^{3} c\right )} e\right )} x^{2}\right )} \log \left (c x + b\right ) + 2 \, {\left ({\left (3 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d - {\left (B b^{2} c^{2} - 3 \, A b c^{3}\right )} e\right )} x^{4} + 2 \, {\left (3 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d - {\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} e\right )} x^{3} + {\left (3 \, {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d - {\left (B b^{4} - 3 \, A b^{3} c\right )} e\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 225, normalized size = 1.34 \begin {gather*} -\frac {{\left (3 \, B b c d - 6 \, A c^{2} d - B b^{2} e + 3 \, A b c e\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {{\left (3 \, B b c^{2} d - 6 \, A c^{3} d - B b^{2} c e + 3 \, A b c^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {6 \, B b c^{2} d x^{3} - 12 \, A c^{3} d x^{3} - 2 \, B b^{2} c x^{3} e + 6 \, A b c^{2} x^{3} e + 9 \, B b^{2} c d x^{2} - 18 \, A b c^{2} d x^{2} - 3 \, B b^{3} x^{2} e + 9 \, A b^{2} c x^{2} e + 2 \, B b^{3} d x - 4 \, A b^{2} c d x + 2 \, A b^{3} x e + A b^{3} d}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 261, normalized size = 1.55 \begin {gather*} -\frac {A c e}{2 \left (c x +b \right )^{2} b^{2}}+\frac {A \,c^{2} d}{2 \left (c x +b \right )^{2} b^{3}}+\frac {B e}{2 \left (c x +b \right )^{2} b}-\frac {B c d}{2 \left (c x +b \right )^{2} b^{2}}-\frac {2 A c e}{\left (c x +b \right ) b^{3}}+\frac {3 A \,c^{2} d}{\left (c x +b \right ) b^{4}}-\frac {3 A c e \ln \relax (x )}{b^{4}}+\frac {3 A c e \ln \left (c x +b \right )}{b^{4}}+\frac {6 A \,c^{2} d \ln \relax (x )}{b^{5}}-\frac {6 A \,c^{2} d \ln \left (c x +b \right )}{b^{5}}+\frac {B e}{\left (c x +b \right ) b^{2}}-\frac {2 B c d}{\left (c x +b \right ) b^{3}}+\frac {B e \ln \relax (x )}{b^{3}}-\frac {B e \ln \left (c x +b \right )}{b^{3}}-\frac {3 B c d \ln \relax (x )}{b^{4}}+\frac {3 B c d \ln \left (c x +b \right )}{b^{4}}-\frac {A e}{b^{3} x}+\frac {3 A c d}{b^{4} x}-\frac {B d}{b^{3} x}-\frac {A d}{2 b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 217, normalized size = 1.29 \begin {gather*} -\frac {A b^{3} d + 2 \, {\left (3 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c - 3 \, A b c^{2}\right )} e\right )} x^{3} + 3 \, {\left (3 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d - {\left (B b^{3} - 3 \, A b^{2} c\right )} e\right )} x^{2} + 2 \, {\left (A b^{3} e + {\left (B b^{3} - 2 \, A b^{2} c\right )} d\right )} x}{2 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} + \frac {{\left (3 \, {\left (B b c - 2 \, A c^{2}\right )} d - {\left (B b^{2} - 3 \, A b c\right )} e\right )} \log \left (c x + b\right )}{b^{5}} - \frac {{\left (3 \, {\left (B b c - 2 \, A c^{2}\right )} d - {\left (B b^{2} - 3 \, A b c\right )} 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 = 1.53, size = 223, normalized size = 1.33 \begin {gather*} -\frac {\frac {x\,\left (A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^2}-\frac {3\,x^2\,\left (6\,A\,c^2\,d+B\,b^2\,e-3\,A\,b\,c\,e-3\,B\,b\,c\,d\right )}{2\,b^3}+\frac {A\,d}{2\,b}-\frac {c\,x^3\,\left (6\,A\,c^2\,d+B\,b^2\,e-3\,A\,b\,c\,e-3\,B\,b\,c\,d\right )}{b^4}}{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 (6\,A\,c^2\,d-b\,\left (3\,A\,c\,e+3\,B\,c\,d\right )+B\,b^2\,e\right )}{b\,\left (6\,A\,c^2\,d+B\,b^2\,e-3\,A\,b\,c\,e-3\,B\,b\,c\,d\right )}\right )\,\left (6\,A\,c^2\,d-b\,\left (3\,A\,c\,e+3\,B\,c\,d\right )+B\,b^2\,e\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.87, size = 449, normalized size = 2.67 \begin {gather*} \frac {- A b^{3} d + x^{3} \left (- 6 A b c^{2} e + 12 A c^{3} d + 2 B b^{2} c e - 6 B b c^{2} d\right ) + x^{2} \left (- 9 A b^{2} c e + 18 A b c^{2} d + 3 B b^{3} e - 9 B b^{2} c d\right ) + x \left (- 2 A b^{3} e + 4 A b^{2} c d - 2 B b^{3} d\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} + \frac {\left (- 3 A b c e + 6 A c^{2} d + B b^{2} e - 3 B b c d\right ) \log {\left (x + \frac {- 3 A b^{2} c e + 6 A b c^{2} d + B b^{3} e - 3 B b^{2} c d - b \left (- 3 A b c e + 6 A c^{2} d + B b^{2} e - 3 B b c d\right )}{- 6 A b c^{2} e + 12 A c^{3} d + 2 B b^{2} c e - 6 B b c^{2} d} \right )}}{b^{5}} - \frac {\left (- 3 A b c e + 6 A c^{2} d + B b^{2} e - 3 B b c d\right ) \log {\left (x + \frac {- 3 A b^{2} c e + 6 A b c^{2} d + B b^{3} e - 3 B b^{2} c d + b \left (- 3 A b c e + 6 A c^{2} d + B b^{2} e - 3 B b c d\right )}{- 6 A b c^{2} e + 12 A c^{3} d + 2 B b^{2} c e - 6 B b c^{2} d} \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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