Optimal. Leaf size=104 \[ \frac {B c x}{e^3}+\frac {d (B d-A e) (c d-b e)}{2 e^4 (d+e x)^2}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^4 (d+e x)}-\frac {(3 B c d-b B e-A c e) \log (d+e x)}{e^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 104, 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 = {785}
\begin {gather*} \frac {d (B d-A e) (c d-b e)}{2 e^4 (d+e x)^2}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^4 (d+e x)}-\frac {\log (d+e x) (-A c e-b B e+3 B c d)}{e^4}+\frac {B c x}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^3} \, dx &=\int \left (\frac {B c}{e^3}-\frac {d (B d-A e) (c d-b e)}{e^3 (d+e x)^3}+\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 (d+e x)^2}+\frac {-3 B c d+b B e+A c e}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {B c x}{e^3}+\frac {d (B d-A e) (c d-b e)}{2 e^4 (d+e x)^2}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^4 (d+e x)}-\frac {(3 B c d-b B e-A c e) \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 96, normalized size = 0.92 \begin {gather*} \frac {2 B c e x+\frac {d (B d-A e) (c d-b e)}{(d+e x)^2}+\frac {-6 B c d^2+4 b B d e+4 A c d e-2 A b e^2}{d+e x}+2 (-3 B c d+b B e+A c e) \log (d+e x)}{2 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 109, normalized size = 1.05
method | result | size |
norman | \(\frac {\frac {B c \,x^{3}}{e}-\frac {d \left (A b \,e^{2}-3 A c d e -3 B b d e +9 B c \,d^{2}\right )}{2 e^{4}}-\frac {\left (A b \,e^{2}-2 A c d e -2 B b d e +6 B c \,d^{2}\right ) x}{e^{3}}}{\left (e x +d \right )^{2}}+\frac {\left (A c e +b B e -3 B c d \right ) \ln \left (e x +d \right )}{e^{4}}\) | \(108\) |
default | \(\frac {B c x}{e^{3}}-\frac {A b \,e^{2}-2 A c d e -2 B b d e +3 B c \,d^{2}}{e^{4} \left (e x +d \right )}+\frac {d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right )}{2 e^{4} \left (e x +d \right )^{2}}+\frac {\left (A c e +b B e -3 B c d \right ) \ln \left (e x +d \right )}{e^{4}}\) | \(109\) |
risch | \(\frac {B c x}{e^{3}}+\frac {\left (-A b \,e^{2}+2 A c d e +2 B b d e -3 B c \,d^{2}\right ) x -\frac {d \left (A b \,e^{2}-3 A c d e -3 B b d e +5 B c \,d^{2}\right )}{2 e}}{e^{3} \left (e x +d \right )^{2}}+\frac {\ln \left (e x +d \right ) A c}{e^{3}}+\frac {\ln \left (e x +d \right ) b B}{e^{3}}-\frac {3 \ln \left (e x +d \right ) B c d}{e^{4}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 121, normalized size = 1.16 \begin {gather*} B c x e^{\left (-3\right )} - {\left (3 \, B c d - B b e - A c e\right )} e^{\left (-4\right )} \log \left (x e + d\right ) - \frac {5 \, B c d^{3} + A b d e^{2} - 3 \, {\left (B b e + A c e\right )} d^{2} + 2 \, {\left (3 \, B c d^{2} e + A b e^{3} - 2 \, {\left (B b e^{2} + A c e^{2}\right )} d\right )} x}{2 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.66, size = 178, normalized size = 1.71 \begin {gather*} -\frac {5 \, B c d^{3} - 2 \, {\left (B c x^{3} - A b x\right )} e^{3} - {\left (4 \, B c d x^{2} - A b d + 4 \, {\left (B b + A c\right )} d x\right )} e^{2} + {\left (4 \, B c d^{2} x - 3 \, {\left (B b + A c\right )} d^{2}\right )} e + 2 \, {\left (3 \, B c d^{3} - {\left (B b + A c\right )} x^{2} e^{3} + {\left (3 \, B c d x^{2} - 2 \, {\left (B b + A c\right )} d x\right )} e^{2} + {\left (6 \, B c d^{2} x - {\left (B b + A c\right )} d^{2}\right )} e\right )} \log \left (x e + d\right )}{2 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.77, size = 138, normalized size = 1.33 \begin {gather*} \frac {B c x}{e^{3}} + \frac {- A b d e^{2} + 3 A c d^{2} e + 3 B b d^{2} e - 5 B c d^{3} + x \left (- 2 A b e^{3} + 4 A c d e^{2} + 4 B b d e^{2} - 6 B c d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} + \frac {\left (A c e + B b e - 3 B c d\right ) \log {\left (d + e x \right )}}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.41, size = 113, normalized size = 1.09 \begin {gather*} B c x e^{\left (-3\right )} - {\left (3 \, B c d - B b e - A c e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (5 \, B c d^{3} - 3 \, B b d^{2} e - 3 \, A c d^{2} e + A b d e^{2} + 2 \, {\left (3 \, B c d^{2} e - 2 \, B b d e^{2} - 2 \, A c d e^{2} + A b e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 123, normalized size = 1.18 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,c\,e+B\,b\,e-3\,B\,c\,d\right )}{e^4}-\frac {x\,\left (A\,b\,e^2+3\,B\,c\,d^2-2\,A\,c\,d\,e-2\,B\,b\,d\,e\right )+\frac {5\,B\,c\,d^3+A\,b\,d\,e^2-3\,A\,c\,d^2\,e-3\,B\,b\,d^2\,e}{2\,e}}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2}+\frac {B\,c\,x}{e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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