Integrand size = 16, antiderivative size = 94 \[ \int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx=\frac {(A b-3 a B) x}{b^4}+\frac {B x^2}{2 b^3}+\frac {a^3 (A b-a B)}{2 b^5 (a+b x)^2}-\frac {a^2 (3 A b-4 a B)}{b^5 (a+b x)}-\frac {3 a (A b-2 a B) \log (a+b x)}{b^5} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 94, 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} \[ \int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx=\frac {a^3 (A b-a B)}{2 b^5 (a+b x)^2}-\frac {a^2 (3 A b-4 a B)}{b^5 (a+b x)}-\frac {3 a (A b-2 a B) \log (a+b x)}{b^5}+\frac {x (A b-3 a B)}{b^4}+\frac {B x^2}{2 b^3} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A b-3 a B}{b^4}+\frac {B x}{b^3}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)^3}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)^2}+\frac {3 a (-A b+2 a B)}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {(A b-3 a B) x}{b^4}+\frac {B x^2}{2 b^3}+\frac {a^3 (A b-a B)}{2 b^5 (a+b x)^2}-\frac {a^2 (3 A b-4 a B)}{b^5 (a+b x)}-\frac {3 a (A b-2 a B) \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx=\frac {2 b (A b-3 a B) x+b^2 B x^2+\frac {a^3 (A b-a B)}{(a+b x)^2}+\frac {2 a^2 (-3 A b+4 a B)}{a+b x}+6 a (-A b+2 a B) \log (a+b x)}{2 b^5} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\frac {1}{2} b B \,x^{2}+A b x -3 B a x}{b^{4}}-\frac {3 a \left (A b -2 B a \right ) \ln \left (b x +a \right )}{b^{5}}+\frac {a^{3} \left (A b -B a \right )}{2 b^{5} \left (b x +a \right )^{2}}-\frac {a^{2} \left (3 A b -4 B a \right )}{b^{5} \left (b x +a \right )}\) | \(90\) |
norman | \(\frac {\frac {\left (A b -2 B a \right ) x^{3}}{b^{2}}+\frac {B \,x^{4}}{2 b}-\frac {a^{2} \left (9 a b A -18 a^{2} B \right )}{2 b^{5}}-\frac {2 a \left (3 a b A -6 a^{2} B \right ) x}{b^{4}}}{\left (b x +a \right )^{2}}-\frac {3 a \left (A b -2 B a \right ) \ln \left (b x +a \right )}{b^{5}}\) | \(94\) |
risch | \(\frac {B \,x^{2}}{2 b^{3}}+\frac {A x}{b^{3}}-\frac {3 B a x}{b^{4}}+\frac {\left (-3 a^{2} b A +4 a^{3} B \right ) x -\frac {a^{3} \left (5 A b -7 B a \right )}{2 b}}{b^{4} \left (b x +a \right )^{2}}-\frac {3 a \ln \left (b x +a \right ) A}{b^{4}}+\frac {6 a^{2} \ln \left (b x +a \right ) B}{b^{5}}\) | \(98\) |
parallelrisch | \(-\frac {-B \,x^{4} b^{4}+6 A \ln \left (b x +a \right ) x^{2} a \,b^{3}-2 A \,x^{3} b^{4}-12 B \ln \left (b x +a \right ) x^{2} a^{2} b^{2}+4 B \,x^{3} a \,b^{3}+12 A \ln \left (b x +a \right ) x \,a^{2} b^{2}-24 B \ln \left (b x +a \right ) x \,a^{3} b +6 A \ln \left (b x +a \right ) a^{3} b +12 A x \,a^{2} b^{2}-12 B \ln \left (b x +a \right ) a^{4}-24 B x \,a^{3} b +9 A \,a^{3} b -18 B \,a^{4}}{2 b^{5} \left (b x +a \right )^{2}}\) | \(162\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.82 \[ \int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx=\frac {B b^{4} x^{4} + 7 \, B a^{4} - 5 \, A a^{3} b - 2 \, {\left (2 \, B a b^{3} - A b^{4}\right )} x^{3} - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x + 6 \, {\left (2 \, B a^{4} - A a^{3} b + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 2 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14 \[ \int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx=\frac {B x^{2}}{2 b^{3}} + \frac {3 a \left (- A b + 2 B a\right ) \log {\left (a + b x \right )}}{b^{5}} + x \left (\frac {A}{b^{3}} - \frac {3 B a}{b^{4}}\right ) + \frac {- 5 A a^{3} b + 7 B a^{4} + x \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15 \[ \int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx=\frac {7 \, B a^{4} - 5 \, A a^{3} b + 2 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {B b x^{2} - 2 \, {\left (3 \, B a - A b\right )} x}{2 \, b^{4}} + \frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \log \left (b x + a\right )}{b^{5}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx=\frac {3 \, {\left (2 \, B a^{2} - A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {B b^{3} x^{2} - 6 \, B a b^{2} x + 2 \, A b^{3} x}{2 \, b^{6}} + \frac {7 \, B a^{4} - 5 \, A a^{3} b + 2 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15 \[ \int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx=\frac {x\,\left (4\,B\,a^3-3\,A\,a^2\,b\right )+\frac {7\,B\,a^4-5\,A\,a^3\,b}{2\,b}}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}+x\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )+\frac {B\,x^2}{2\,b^3}+\frac {\ln \left (a+b\,x\right )\,\left (6\,B\,a^2-3\,A\,a\,b\right )}{b^5} \]
[In]
[Out]