Integrand size = 20, antiderivative size = 62 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^2} \, dx=\frac {b (b B-A c) (b+c x)^4}{4 c^3}-\frac {(2 b B-A c) (b+c x)^5}{5 c^3}+\frac {B (b+c x)^6}{6 c^3} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {779} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^2} \, dx=-\frac {(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac {b (b+c x)^4 (b B-A c)}{4 c^3}+\frac {B (b+c x)^6}{6 c^3} \]
[In]
[Out]
Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b (b B-A c) (b+c x)^3}{c^2}+\frac {(-2 b B+A c) (b+c x)^4}{c^2}+\frac {B (b+c x)^5}{c^2}\right ) \, dx \\ & = \frac {b (b B-A c) (b+c x)^4}{4 c^3}-\frac {(2 b B-A c) (b+c x)^5}{5 c^3}+\frac {B (b+c x)^6}{6 c^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^2} \, dx=\frac {1}{60} x^2 \left (30 A b^3+20 b^2 (b B+3 A c) x+45 b c (b B+A c) x^2+12 c^2 (3 b B+A c) x^3+10 B c^3 x^4\right ) \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23
method | result | size |
gosper | \(\frac {x^{2} \left (10 B \,c^{3} x^{4}+12 A \,c^{3} x^{3}+36 B b \,c^{2} x^{3}+45 A b \,c^{2} x^{2}+45 B \,b^{2} c \,x^{2}+60 A \,b^{2} c x +20 B \,b^{3} x +30 A \,b^{3}\right )}{60}\) | \(76\) |
default | \(\frac {B \,c^{3} x^{6}}{6}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{5}}{5}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{4}}{4}+\frac {\left (3 A \,b^{2} c +B \,b^{3}\right ) x^{3}}{3}+\frac {A \,b^{3} x^{2}}{2}\) | \(76\) |
risch | \(\frac {1}{6} B \,c^{3} x^{6}+\frac {1}{5} A \,c^{3} x^{5}+\frac {3}{5} x^{5} B b \,c^{2}+\frac {3}{4} x^{4} A b \,c^{2}+\frac {3}{4} x^{4} B \,b^{2} c +x^{3} A \,b^{2} c +\frac {1}{3} x^{3} B \,b^{3}+\frac {1}{2} A \,b^{3} x^{2}\) | \(77\) |
parallelrisch | \(\frac {1}{6} B \,c^{3} x^{6}+\frac {1}{5} A \,c^{3} x^{5}+\frac {3}{5} x^{5} B b \,c^{2}+\frac {3}{4} x^{4} A b \,c^{2}+\frac {3}{4} x^{4} B \,b^{2} c +x^{3} A \,b^{2} c +\frac {1}{3} x^{3} B \,b^{3}+\frac {1}{2} A \,b^{3} x^{2}\) | \(77\) |
norman | \(\frac {\left (\frac {1}{5} A \,c^{3}+\frac {3}{5} B b \,c^{2}\right ) x^{6}+\left (\frac {3}{4} A b \,c^{2}+\frac {3}{4} B \,b^{2} c \right ) x^{5}+\left (A \,b^{2} c +\frac {1}{3} B \,b^{3}\right ) x^{4}+\frac {A \,b^{3} x^{3}}{2}+\frac {B \,c^{3} x^{7}}{6}}{x}\) | \(78\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^2} \, dx=\frac {1}{6} \, B c^{3} x^{6} + \frac {1}{2} \, A b^{3} x^{2} + \frac {1}{5} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac {3}{4} \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^2} \, dx=\frac {A b^{3} x^{2}}{2} + \frac {B c^{3} x^{6}}{6} + x^{5} \left (\frac {A c^{3}}{5} + \frac {3 B b c^{2}}{5}\right ) + x^{4} \cdot \left (\frac {3 A b c^{2}}{4} + \frac {3 B b^{2} c}{4}\right ) + x^{3} \left (A b^{2} c + \frac {B b^{3}}{3}\right ) \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^2} \, dx=\frac {1}{6} \, B c^{3} x^{6} + \frac {1}{2} \, A b^{3} x^{2} + \frac {1}{5} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac {3}{4} \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^2} \, dx=\frac {1}{6} \, B c^{3} x^{6} + \frac {3}{5} \, B b c^{2} x^{5} + \frac {1}{5} \, A c^{3} x^{5} + \frac {3}{4} \, B b^{2} c x^{4} + \frac {3}{4} \, A b c^{2} x^{4} + \frac {1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + \frac {1}{2} \, A b^{3} x^{2} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^2} \, dx=x^3\,\left (\frac {B\,b^3}{3}+A\,c\,b^2\right )+x^5\,\left (\frac {A\,c^3}{5}+\frac {3\,B\,b\,c^2}{5}\right )+\frac {A\,b^3\,x^2}{2}+\frac {B\,c^3\,x^6}{6}+\frac {3\,b\,c\,x^4\,\left (A\,c+B\,b\right )}{4} \]
[In]
[Out]