Integrand size = 18, antiderivative size = 59 \[ \int \frac {(A+B x) (d+e x)}{a+b x} \, dx=\frac {B (b d-a e) x}{b^2}+\frac {e (A+B x)^2}{2 b B}+\frac {(A b-a B) (b d-a e) \log (a+b x)}{b^3} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int \frac {(A+B x) (d+e x)}{a+b x} \, dx=\frac {(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac {B x (b d-a e)}{b^2}+\frac {e (A+B x)^2}{2 b B} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {B (b d-a e)}{b^2}+\frac {(A b-a B) (b d-a e)}{b^2 (a+b x)}+\frac {e (A+B x)}{b}\right ) \, dx \\ & = \frac {B (b d-a e) x}{b^2}+\frac {e (A+B x)^2}{2 b B}+\frac {(A b-a B) (b d-a e) \log (a+b x)}{b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) (d+e x)}{a+b x} \, dx=\frac {b x (-2 a B e+b (2 B d+2 A e+B e x))+2 (A b-a B) (b d-a e) \log (a+b x)}{2 b^3} \]
[In]
[Out]
Time = 0.70 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {\frac {1}{2} B b e \,x^{2}+A b e x -B a e x +B b d x}{b^{2}}+\frac {\left (-A a b e +A \,b^{2} d +B \,a^{2} e -B a b d \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(66\) |
norman | \(\frac {\left (A b e -B a e +B b d \right ) x}{b^{2}}+\frac {B e \,x^{2}}{2 b}-\frac {\left (A a b e -A \,b^{2} d -B \,a^{2} e +B a b d \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(67\) |
risch | \(\frac {B e \,x^{2}}{2 b}+\frac {A e x}{b}-\frac {B a e x}{b^{2}}+\frac {B d x}{b}-\frac {\ln \left (b x +a \right ) A a e}{b^{2}}+\frac {\ln \left (b x +a \right ) A d}{b}+\frac {\ln \left (b x +a \right ) B \,a^{2} e}{b^{3}}-\frac {\ln \left (b x +a \right ) B a d}{b^{2}}\) | \(90\) |
parallelrisch | \(-\frac {-B e \,x^{2} b^{2}+2 A \ln \left (b x +a \right ) a b e -2 A \ln \left (b x +a \right ) b^{2} d -2 A x \,b^{2} e -2 B \ln \left (b x +a \right ) a^{2} e +2 B \ln \left (b x +a \right ) a b d +2 B x a b e -2 B x \,b^{2} d}{2 b^{3}}\) | \(90\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) (d+e x)}{a+b x} \, dx=\frac {B b^{2} e x^{2} + 2 \, {\left (B b^{2} d - {\left (B a b - A b^{2}\right )} e\right )} x - 2 \, {\left ({\left (B a b - A b^{2}\right )} d - {\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) (d+e x)}{a+b x} \, dx=\frac {B e x^{2}}{2 b} + x \left (\frac {A e}{b} - \frac {B a e}{b^{2}} + \frac {B d}{b}\right ) + \frac {\left (- A b + B a\right ) \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{3}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.22 \[ \int \frac {(A+B x) (d+e x)}{a+b x} \, dx=\frac {B b e x^{2} + 2 \, {\left (B b d - {\left (B a - A b\right )} e\right )} x}{2 \, b^{2}} - \frac {{\left ({\left (B a b - A b^{2}\right )} d - {\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.17 \[ \int \frac {(A+B x) (d+e x)}{a+b x} \, dx=\frac {B b e x^{2} + 2 \, B b d x - 2 \, B a e x + 2 \, A b e x}{2 \, b^{2}} - \frac {{\left (B a b d - A b^{2} d - B a^{2} e + A a b e\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \]
[In]
[Out]
Time = 1.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) (d+e x)}{a+b x} \, dx=x\,\left (\frac {A\,e+B\,d}{b}-\frac {B\,a\,e}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (A\,b^2\,d+B\,a^2\,e-A\,a\,b\,e-B\,a\,b\,d\right )}{b^3}+\frac {B\,e\,x^2}{2\,b} \]
[In]
[Out]