Optimal. Leaf size=104 \[ \frac {d (B d-A e) (c d-b e)}{2 e^4 (d+e x)^2}-\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^4 (d+e x)}-\frac {\log (d+e x) (-A c e-b B e+3 B c d)}{e^4}+\frac {B c x}{e^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, 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 = {771} \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.
[In]
[Out]
Rule 771
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 96, normalized size = 0.92 \begin {gather*} \frac {\frac {-2 A b e^2+4 A c d e+4 b B d e-6 B c d^2}{d+e x}+\frac {d (B d-A e) (c d-b e)}{(d+e x)^2}+2 \log (d+e x) (A c e+b B e-3 B c d)+2 B c e x}{2 e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 186, normalized size = 1.79 \begin {gather*} \frac {2 \, B c e^{3} x^{3} + 4 \, B c d e^{2} x^{2} - 5 \, B c d^{3} - A b d e^{2} + 3 \, {\left (B b + A c\right )} d^{2} e - 2 \, {\left (2 \, B c d^{2} e + A b e^{3} - 2 \, {\left (B b + A c\right )} d e^{2}\right )} x - 2 \, {\left (3 \, B c d^{3} - {\left (B b + A c\right )} d^{2} e + {\left (3 \, B c d e^{2} - {\left (B b + A c\right )} e^{3}\right )} x^{2} + 2 \, {\left (3 \, B c d^{2} e - {\left (B b + A c\right )} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, 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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 174, normalized size = 1.67 \begin {gather*} \frac {A b d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {A c \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}-\frac {B b \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {B c \,d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {A b}{\left (e x +d \right ) e^{2}}+\frac {2 A c d}{\left (e x +d \right ) e^{3}}+\frac {A c \ln \left (e x +d \right )}{e^{3}}+\frac {2 B b d}{\left (e x +d \right ) e^{3}}+\frac {B b \ln \left (e x +d \right )}{e^{3}}-\frac {3 B c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 B c d \ln \left (e x +d \right )}{e^{4}}+\frac {B c x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 120, normalized size = 1.15 \begin {gather*} -\frac {5 \, B c d^{3} + A b d e^{2} - 3 \, {\left (B b + A c\right )} d^{2} e + 2 \, {\left (3 \, B c d^{2} e + A b e^{3} - 2 \, {\left (B b + A c\right )} d e^{2}\right )} x}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac {B c x}{e^{3}} - \frac {{\left (3 \, B c d - {\left (B b + A c\right )} e\right )} \log \left (e x + d\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.49, 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.
[In]
[Out]
________________________________________________________________________________________