Optimal. Leaf size=60 \[ -\frac {(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac {\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac {B e x}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ -\frac {(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac {\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac {B e x}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{(a+b x)^2} \, dx &=\int \left (\frac {B e}{b^2}+\frac {(A b-a B) (b d-a e)}{b^2 (a+b x)^2}+\frac {b B d+A b e-2 a B e}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {B e x}{b^2}-\frac {(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac {(b B d+A b e-2 a B e) \log (a+b x)}{b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 56, normalized size = 0.93 \[ \frac {-\frac {(A b-a B) (b d-a e)}{a+b x}+\log (a+b x) (-2 a B e+A b e+b B d)+b B e x}{b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 109, normalized size = 1.82 \[ \frac {B b^{2} e x^{2} + B a b e x + {\left (B a b - A b^{2}\right )} d - {\left (B a^{2} - A a b\right )} e + {\left (B a b d - {\left (2 \, B a^{2} - A a b\right )} e + {\left (B b^{2} d - {\left (2 \, B a b - A b^{2}\right )} e\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.20, size = 117, normalized size = 1.95 \[ \frac {{\left (b x + a\right )} B e}{b^{3}} - \frac {{\left (B b d - 2 \, B a e + A b e\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} + \frac {\frac {B a b^{2} d}{b x + a} - \frac {A b^{3} d}{b x + a} - \frac {B a^{2} b e}{b x + a} + \frac {A a b^{2} e}{b x + a}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 106, normalized size = 1.77 \[ \frac {A a e}{\left (b x +a \right ) b^{2}}-\frac {A d}{\left (b x +a \right ) b}+\frac {A e \ln \left (b x +a \right )}{b^{2}}-\frac {B \,a^{2} e}{\left (b x +a \right ) b^{3}}+\frac {B a d}{\left (b x +a \right ) b^{2}}-\frac {2 B a e \ln \left (b x +a \right )}{b^{3}}+\frac {B d \ln \left (b x +a \right )}{b^{2}}+\frac {B e x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.64, size = 77, normalized size = 1.28 \[ \frac {B e x}{b^{2}} + \frac {{\left (B a b - A b^{2}\right )} d - {\left (B a^{2} - A a b\right )} e}{b^{4} x + a b^{3}} + \frac {{\left (B b d - {\left (2 \, B a - A b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.12, size = 75, normalized size = 1.25 \[ \frac {\ln \left (a+b\,x\right )\,\left (A\,b\,e-2\,B\,a\,e+B\,b\,d\right )}{b^3}-\frac {A\,b^2\,d+B\,a^2\,e-A\,a\,b\,e-B\,a\,b\,d}{b\,\left (x\,b^3+a\,b^2\right )}+\frac {B\,e\,x}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.46, size = 71, normalized size = 1.18 \[ \frac {B e x}{b^{2}} + \frac {A a b e - A b^{2} d - B a^{2} e + B a b d}{a b^{3} + b^{4} x} - \frac {\left (- A b e + 2 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________