Integrand size = 16, antiderivative size = 66 \[ \int \frac {x^2 (c+d x)}{a+b x} \, dx=-\frac {a (b c-a d) x}{b^3}+\frac {(b c-a d) x^2}{2 b^2}+\frac {d x^3}{3 b}+\frac {a^2 (b c-a d) \log (a+b x)}{b^4} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {x^2 (c+d x)}{a+b x} \, dx=\frac {a^2 (b c-a d) \log (a+b x)}{b^4}-\frac {a x (b c-a d)}{b^3}+\frac {x^2 (b c-a d)}{2 b^2}+\frac {d x^3}{3 b} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-b c+a d)}{b^3}+\frac {(b c-a d) x}{b^2}+\frac {d x^2}{b}-\frac {a^2 (-b c+a d)}{b^3 (a+b x)}\right ) \, dx \\ & = -\frac {a (b c-a d) x}{b^3}+\frac {(b c-a d) x^2}{2 b^2}+\frac {d x^3}{3 b}+\frac {a^2 (b c-a d) \log (a+b x)}{b^4} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (c+d x)}{a+b x} \, dx=\frac {b x \left (6 a^2 d-3 a b (2 c+d x)+b^2 x (3 c+2 d x)\right )+6 a^2 (b c-a d) \log (a+b x)}{6 b^4} \]
[In]
[Out]
Time = 0.97 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95
method | result | size |
norman | \(\frac {a \left (a d -b c \right ) x}{b^{3}}+\frac {d \,x^{3}}{3 b}-\frac {\left (a d -b c \right ) x^{2}}{2 b^{2}}-\frac {a^{2} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{4}}\) | \(63\) |
default | \(\frac {\frac {1}{3} d \,x^{3} b^{2}-\frac {1}{2} x^{2} a b d +\frac {1}{2} b^{2} c \,x^{2}+a^{2} d x -a b c x}{b^{3}}-\frac {a^{2} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{4}}\) | \(67\) |
risch | \(\frac {d \,x^{3}}{3 b}-\frac {x^{2} a d}{2 b^{2}}+\frac {c \,x^{2}}{2 b}+\frac {a^{2} d x}{b^{3}}-\frac {a c x}{b^{2}}-\frac {a^{3} \ln \left (b x +a \right ) d}{b^{4}}+\frac {a^{2} \ln \left (b x +a \right ) c}{b^{3}}\) | \(76\) |
parallelrisch | \(-\frac {-2 d \,x^{3} b^{3}+3 x^{2} a \,b^{2} d -3 x^{2} b^{3} c +6 \ln \left (b x +a \right ) a^{3} d -6 \ln \left (b x +a \right ) a^{2} b c -6 x \,a^{2} b d +6 x a \,b^{2} c}{6 b^{4}}\) | \(76\) |
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 (c+d x)}{a+b x} \, dx=\frac {2 \, b^{3} d x^{3} + 3 \, {\left (b^{3} c - a b^{2} d\right )} x^{2} - 6 \, {\left (a b^{2} c - a^{2} b d\right )} x + 6 \, {\left (a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (c+d x)}{a+b x} \, dx=- \frac {a^{2} \left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{4}} + x^{2} \left (- \frac {a d}{2 b^{2}} + \frac {c}{2 b}\right ) + x \left (\frac {a^{2} d}{b^{3}} - \frac {a c}{b^{2}}\right ) + \frac {d x^{3}}{3 b} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 (c+d x)}{a+b x} \, dx=\frac {2 \, b^{2} d x^{3} + 3 \, {\left (b^{2} c - a b d\right )} x^{2} - 6 \, {\left (a b c - a^{2} d\right )} x}{6 \, b^{3}} + \frac {{\left (a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{b^{4}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06 \[ \int \frac {x^2 (c+d x)}{a+b x} \, dx=\frac {2 \, b^{2} d x^{3} + 3 \, b^{2} c x^{2} - 3 \, a b d x^{2} - 6 \, a b c x + 6 \, a^{2} d x}{6 \, b^{3}} + \frac {{\left (a^{2} b c - a^{3} d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 (c+d x)}{a+b x} \, dx=x^2\,\left (\frac {c}{2\,b}-\frac {a\,d}{2\,b^2}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d-a^2\,b\,c\right )}{b^4}+\frac {d\,x^3}{3\,b}-\frac {a\,x\,\left (\frac {c}{b}-\frac {a\,d}{b^2}\right )}{b} \]
[In]
[Out]