Integrand size = 18, antiderivative size = 136 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac {3 d (b c-a d)^2 x^2}{2 b^4}+\frac {d^2 (3 b c-2 a d) x^3}{3 b^3}+\frac {d^3 x^4}{4 b^2}-\frac {a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac {a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 136, 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} \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=-\frac {a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac {a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}+\frac {x (b c-4 a d) (b c-a d)^2}{b^5}+\frac {3 d x^2 (b c-a d)^2}{2 b^4}+\frac {d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac {d^3 x^4}{4 b^2} \]
[In]
[Out]
Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-4 a d) (b c-a d)^2}{b^5}+\frac {3 d (b c-a d)^2 x}{b^4}+\frac {d^2 (3 b c-2 a d) x^2}{b^3}+\frac {d^3 x^3}{b^2}-\frac {a^2 (-b c+a d)^3}{b^5 (a+b x)^2}+\frac {a (-b c+a d)^2 (-2 b c+5 a d)}{b^5 (a+b x)}\right ) \, dx \\ & = \frac {(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac {3 d (b c-a d)^2 x^2}{2 b^4}+\frac {d^2 (3 b c-2 a d) x^3}{3 b^3}+\frac {d^3 x^4}{4 b^2}-\frac {a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac {a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {12 b (b c-4 a d) (b c-a d)^2 x+18 b^2 d (b c-a d)^2 x^2+4 b^3 d^2 (3 b c-2 a d) x^3+3 b^4 d^3 x^4+\frac {12 a^2 (-b c+a d)^3}{a+b x}+12 a (b c-a d)^2 (-2 b c+5 a d) \log (a+b x)}{12 b^6} \]
[In]
[Out]
Time = 0.45 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.60
method | result | size |
default | \(-\frac {-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {2}{3} x^{3} a \,b^{2} d^{3}-x^{3} b^{3} c \,d^{2}-\frac {3}{2} x^{2} a^{2} b \,d^{3}+3 x^{2} a \,b^{2} c \,d^{2}-\frac {3}{2} x^{2} b^{3} c^{2} d +4 a^{3} d^{3} x -9 a^{2} b c \,d^{2} x +6 a \,b^{2} c^{2} d x -b^{3} c^{3} x}{b^{5}}+\frac {a \left (5 a^{3} d^{3}-12 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}+\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{6} \left (b x +a \right )}\) | \(218\) |
norman | \(\frac {-\frac {\left (5 a^{3} d^{3}-12 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{2 b^{4}}+\frac {d^{3} x^{5}}{4 b}+\frac {d \left (5 a^{2} d^{2}-12 a b c d +9 b^{2} c^{2}\right ) x^{3}}{6 b^{3}}-\frac {d^{2} \left (5 a d -12 b c \right ) x^{4}}{12 b^{2}}-\frac {\left (5 a^{5} d^{3}-12 a^{4} b c \,d^{2}+9 a^{3} b^{2} c^{2} d -2 a^{2} b^{3} c^{3}\right ) x}{b^{5} a}}{b x +a}+\frac {a \left (5 a^{3} d^{3}-12 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(218\) |
risch | \(\frac {d^{3} x^{4}}{4 b^{2}}-\frac {2 x^{3} a \,d^{3}}{3 b^{3}}+\frac {x^{3} c \,d^{2}}{b^{2}}+\frac {3 x^{2} a^{2} d^{3}}{2 b^{4}}-\frac {3 x^{2} a c \,d^{2}}{b^{3}}+\frac {3 x^{2} c^{2} d}{2 b^{2}}-\frac {4 a^{3} d^{3} x}{b^{5}}+\frac {9 a^{2} c \,d^{2} x}{b^{4}}-\frac {6 a \,c^{2} d x}{b^{3}}+\frac {c^{3} x}{b^{2}}+\frac {a^{5} d^{3}}{b^{6} \left (b x +a \right )}-\frac {3 a^{4} c \,d^{2}}{b^{5} \left (b x +a \right )}+\frac {3 a^{3} c^{2} d}{b^{4} \left (b x +a \right )}-\frac {a^{2} c^{3}}{b^{3} \left (b x +a \right )}+\frac {5 a^{4} \ln \left (b x +a \right ) d^{3}}{b^{6}}-\frac {12 a^{3} \ln \left (b x +a \right ) c \,d^{2}}{b^{5}}+\frac {9 a^{2} \ln \left (b x +a \right ) c^{2} d}{b^{4}}-\frac {2 a \ln \left (b x +a \right ) c^{3}}{b^{3}}\) | \(260\) |
parallelrisch | \(\frac {3 d^{3} x^{5} b^{5}-5 x^{4} a \,b^{4} d^{3}+12 x^{4} b^{5} c \,d^{2}+10 x^{3} a^{2} b^{3} d^{3}-24 x^{3} a \,b^{4} c \,d^{2}+18 x^{3} b^{5} c^{2} d +60 \ln \left (b x +a \right ) x \,a^{4} b \,d^{3}-144 \ln \left (b x +a \right ) x \,a^{3} b^{2} c \,d^{2}+108 \ln \left (b x +a \right ) x \,a^{2} b^{3} c^{2} d -24 \ln \left (b x +a \right ) x a \,b^{4} c^{3}-30 x^{2} a^{3} b^{2} d^{3}+72 x^{2} a^{2} b^{3} c \,d^{2}-54 x^{2} a \,b^{4} c^{2} d +12 x^{2} b^{5} c^{3}+60 \ln \left (b x +a \right ) a^{5} d^{3}-144 \ln \left (b x +a \right ) a^{4} b c \,d^{2}+108 \ln \left (b x +a \right ) a^{3} b^{2} c^{2} d -24 \ln \left (b x +a \right ) a^{2} b^{3} c^{3}+60 a^{5} d^{3}-144 a^{4} b c \,d^{2}+108 a^{3} b^{2} c^{2} d -24 a^{2} b^{3} c^{3}}{12 b^{6} \left (b x +a \right )}\) | \(317\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (130) = 260\).
Time = 0.22 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.31 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {3 \, b^{5} d^{3} x^{5} - 12 \, a^{2} b^{3} c^{3} + 36 \, a^{3} b^{2} c^{2} d - 36 \, a^{4} b c d^{2} + 12 \, a^{5} d^{3} + {\left (12 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \, {\left (9 \, b^{5} c^{2} d - 12 \, a b^{4} c d^{2} + 5 \, a^{2} b^{3} d^{3}\right )} x^{3} + 6 \, {\left (2 \, b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 12 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{2} + 12 \, {\left (a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 4 \, a^{4} b d^{3}\right )} x - 12 \, {\left (2 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 12 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.50 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {a \left (a d - b c\right )^{2} \cdot \left (5 a d - 2 b c\right ) \log {\left (a + b x \right )}}{b^{6}} + x^{3} \left (- \frac {2 a d^{3}}{3 b^{3}} + \frac {c d^{2}}{b^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{3}}{2 b^{4}} - \frac {3 a c d^{2}}{b^{3}} + \frac {3 c^{2} d}{2 b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{3}}{b^{5}} + \frac {9 a^{2} c d^{2}}{b^{4}} - \frac {6 a c^{2} d}{b^{3}} + \frac {c^{3}}{b^{2}}\right ) + \frac {a^{5} d^{3} - 3 a^{4} b c d^{2} + 3 a^{3} b^{2} c^{2} d - a^{2} b^{3} c^{3}}{a b^{6} + b^{7} x} + \frac {d^{3} x^{4}}{4 b^{2}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.62 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=-\frac {a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} d^{3} x^{4} + 4 \, {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{3} + 18 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 12 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} x}{12 \, b^{5}} - \frac {{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (130) = 260\).
Time = 0.28 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.10 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {{\left (3 \, d^{3} + \frac {4 \, {\left (3 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {6 \, {\left (3 \, b^{4} c^{2} d - 12 \, a b^{3} c d^{2} + 10 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {12 \, {\left (b^{6} c^{3} - 9 \, a b^{5} c^{2} d + 18 \, a^{2} b^{4} c d^{2} - 10 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{4}}{12 \, b^{6}} + \frac {{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{6}} - \frac {\frac {a^{2} b^{7} c^{3}}{b x + a} - \frac {3 \, a^{3} b^{6} c^{2} d}{b x + a} + \frac {3 \, a^{4} b^{5} c d^{2}}{b x + a} - \frac {a^{5} b^{4} d^{3}}{b x + a}}{b^{10}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.07 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=x\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )-x^3\,\left (\frac {2\,a\,d^3}{3\,b^3}-\frac {c\,d^2}{b^2}\right )+x^2\,\left (\frac {3\,c^2\,d}{2\,b^2}+\frac {a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{2\,b^4}\right )+\frac {a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {d^3\,x^4}{4\,b^2}+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,d^3-12\,a^3\,b\,c\,d^2+9\,a^2\,b^2\,c^2\,d-2\,a\,b^3\,c^3\right )}{b^6} \]
[In]
[Out]