Integrand size = 18, antiderivative size = 129 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=-\frac {a (b c-a d)^3 x}{b^5}+\frac {(b c-a d)^3 x^2}{2 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^2 (3 b c-a d) x^4}{4 b^2}+\frac {d^3 x^5}{5 b}+\frac {a^2 (b c-a d)^3 \log (a+b x)}{b^6} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 129, 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 = {90} \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {a^2 (b c-a d)^3 \log (a+b x)}{b^6}+\frac {d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac {a x (b c-a d)^3}{b^5}+\frac {x^2 (b c-a d)^3}{2 b^4}+\frac {d^2 x^4 (3 b c-a d)}{4 b^2}+\frac {d^3 x^5}{5 b} \]
[In]
[Out]
Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-b c+a d)^3}{b^5}+\frac {(b c-a d)^3 x}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^2}{b^3}+\frac {d^2 (3 b c-a d) x^3}{b^2}+\frac {d^3 x^4}{b}-\frac {a^2 (-b c+a d)^3}{b^5 (a+b x)}\right ) \, dx \\ & = -\frac {a (b c-a d)^3 x}{b^5}+\frac {(b c-a d)^3 x^2}{2 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^2 (3 b c-a d) x^4}{4 b^2}+\frac {d^3 x^5}{5 b}+\frac {a^2 (b c-a d)^3 \log (a+b x)}{b^6} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {60 a b (-b c+a d)^3 x+30 b^2 (b c-a d)^3 x^2+20 b^3 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3+15 b^4 d^2 (3 b c-a d) x^4+12 b^5 d^3 x^5+60 a^2 (b c-a d)^3 \log (a+b x)}{60 b^6} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.54
method | result | size |
norman | \(\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{5}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 b^{4}}+\frac {d^{3} x^{5}}{5 b}-\frac {d^{2} \left (a d -3 b c \right ) x^{4}}{4 b^{2}}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{3}}{3 b^{3}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(199\) |
default | \(\frac {\frac {1}{5} d^{3} x^{5} b^{4}-\frac {1}{4} a \,b^{3} d^{3} x^{4}+\frac {3}{4} b^{4} c \,d^{2} x^{4}+\frac {1}{3} a^{2} b^{2} d^{3} x^{3}-a \,b^{3} c \,d^{2} x^{3}+b^{4} c^{2} d \,x^{3}-\frac {1}{2} a^{3} b \,d^{3} x^{2}+\frac {3}{2} a^{2} b^{2} c \,d^{2} x^{2}-\frac {3}{2} a \,b^{3} c^{2} d \,x^{2}+\frac {1}{2} b^{4} c^{3} x^{2}+a^{4} d^{3} x -3 a^{3} b c \,d^{2} x +3 a^{2} b^{2} c^{2} d x -b^{3} c^{3} a x}{b^{5}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(223\) |
risch | \(\frac {d^{3} x^{5}}{5 b}-\frac {a \,d^{3} x^{4}}{4 b^{2}}+\frac {3 c \,d^{2} x^{4}}{4 b}+\frac {a^{2} d^{3} x^{3}}{3 b^{3}}-\frac {a c \,d^{2} x^{3}}{b^{2}}+\frac {c^{2} d \,x^{3}}{b}-\frac {a^{3} d^{3} x^{2}}{2 b^{4}}+\frac {3 a^{2} c \,d^{2} x^{2}}{2 b^{3}}-\frac {3 a \,c^{2} d \,x^{2}}{2 b^{2}}+\frac {c^{3} x^{2}}{2 b}+\frac {a^{4} d^{3} x}{b^{5}}-\frac {3 a^{3} c \,d^{2} x}{b^{4}}+\frac {3 a^{2} c^{2} d x}{b^{3}}-\frac {c^{3} a x}{b^{2}}-\frac {a^{5} \ln \left (b x +a \right ) d^{3}}{b^{6}}+\frac {3 a^{4} \ln \left (b x +a \right ) c \,d^{2}}{b^{5}}-\frac {3 a^{3} \ln \left (b x +a \right ) c^{2} d}{b^{4}}+\frac {a^{2} \ln \left (b x +a \right ) c^{3}}{b^{3}}\) | \(244\) |
parallelrisch | \(-\frac {-12 d^{3} x^{5} b^{5}+15 x^{4} a \,b^{4} d^{3}-45 x^{4} b^{5} c \,d^{2}-20 x^{3} a^{2} b^{3} d^{3}+60 x^{3} a \,b^{4} c \,d^{2}-60 x^{3} b^{5} c^{2} d +30 x^{2} a^{3} b^{2} d^{3}-90 x^{2} a^{2} b^{3} c \,d^{2}+90 x^{2} a \,b^{4} c^{2} d -30 x^{2} b^{5} c^{3}+60 \ln \left (b x +a \right ) a^{5} d^{3}-180 \ln \left (b x +a \right ) a^{4} b c \,d^{2}+180 \ln \left (b x +a \right ) a^{3} b^{2} c^{2} d -60 \ln \left (b x +a \right ) a^{2} b^{3} c^{3}-60 x \,a^{4} b \,d^{3}+180 x \,a^{3} b^{2} c \,d^{2}-180 x \,a^{2} b^{3} c^{2} d +60 x a \,b^{4} c^{3}}{60 b^{6}}\) | \(245\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.67 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {12 \, b^{5} d^{3} x^{5} + 15 \, {\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4} + 20 \, {\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{3} + 30 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} - 60 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x + 60 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.43 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=- \frac {a^{2} \left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{6}} + x^{4} \left (- \frac {a d^{3}}{4 b^{2}} + \frac {3 c d^{2}}{4 b}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{3 b^{3}} - \frac {a c d^{2}}{b^{2}} + \frac {c^{2} d}{b}\right ) + x^{2} \left (- \frac {a^{3} d^{3}}{2 b^{4}} + \frac {3 a^{2} c d^{2}}{2 b^{3}} - \frac {3 a c^{2} d}{2 b^{2}} + \frac {c^{3}}{2 b}\right ) + x \left (\frac {a^{4} d^{3}}{b^{5}} - \frac {3 a^{3} c d^{2}}{b^{4}} + \frac {3 a^{2} c^{2} d}{b^{3}} - \frac {a c^{3}}{b^{2}}\right ) + \frac {d^{3} x^{5}}{5 b} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.66 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {12 \, b^{4} d^{3} x^{5} + 15 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{4} + 20 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{3} + 30 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2} - 60 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{60 \, b^{5}} + \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.76 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {12 \, b^{4} d^{3} x^{5} + 45 \, b^{4} c d^{2} x^{4} - 15 \, a b^{3} d^{3} x^{4} + 60 \, b^{4} c^{2} d x^{3} - 60 \, a b^{3} c d^{2} x^{3} + 20 \, a^{2} b^{2} d^{3} x^{3} + 30 \, b^{4} c^{3} x^{2} - 90 \, a b^{3} c^{2} d x^{2} + 90 \, a^{2} b^{2} c d^{2} x^{2} - 30 \, a^{3} b d^{3} x^{2} - 60 \, a b^{3} c^{3} x + 180 \, a^{2} b^{2} c^{2} d x - 180 \, a^{3} b c d^{2} x + 60 \, a^{4} d^{3} x}{60 \, b^{5}} + \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.79 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=x^2\,\left (\frac {c^3}{2\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{2\,b}\right )-x^4\,\left (\frac {a\,d^3}{4\,b^2}-\frac {3\,c\,d^2}{4\,b}\right )+x^3\,\left (\frac {c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{3\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3\right )}{b^6}+\frac {d^3\,x^5}{5\,b}-\frac {a\,x\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{b} \]
[In]
[Out]