Integrand size = 17, antiderivative size = 44 \[ \int \frac {x^5}{a x^2+b x^3} \, dx=\frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b}-\frac {a^3 \log (a+b x)}{b^4} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45} \[ \int \frac {x^5}{a x^2+b x^3} \, dx=-\frac {a^3 \log (a+b x)}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b} \]
[In]
[Out]
Rule 45
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{a+b x} \, dx \\ & = \int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx \\ & = \frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b}-\frac {a^3 \log (a+b x)}{b^4} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{a x^2+b x^3} \, dx=\frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b}-\frac {a^3 \log (a+b x)}{b^4} \]
[In]
[Out]
Time = 2.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\frac {1}{3} b^{2} x^{3}-\frac {1}{2} a b \,x^{2}+a^{2} x}{b^{3}}-\frac {a^{3} \ln \left (b x +a \right )}{b^{4}}\) | \(41\) |
risch | \(\frac {a^{2} x}{b^{3}}-\frac {a \,x^{2}}{2 b^{2}}+\frac {x^{3}}{3 b}-\frac {a^{3} \ln \left (b x +a \right )}{b^{4}}\) | \(41\) |
parallelrisch | \(-\frac {-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x}{6 b^{4}}\) | \(42\) |
norman | \(\frac {\frac {a^{2} x^{2}}{b^{3}}+\frac {x^{4}}{3 b}-\frac {a \,x^{3}}{2 b^{2}}}{x}-\frac {a^{3} \ln \left (b x +a \right )}{b^{4}}\) | \(48\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93 \[ \int \frac {x^5}{a x^2+b x^3} \, dx=\frac {2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )}{6 \, b^{4}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{a x^2+b x^3} \, dx=- \frac {a^{3} \log {\left (a + b x \right )}}{b^{4}} + \frac {a^{2} x}{b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{3}}{3 b} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {x^5}{a x^2+b x^3} \, dx=-\frac {a^{3} \log \left (b x + a\right )}{b^{4}} + \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{6 \, b^{3}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {x^5}{a x^2+b x^3} \, dx=-\frac {a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{6 \, b^{3}} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.91 \[ \int \frac {x^5}{a x^2+b x^3} \, dx=\frac {x^3}{3\,b}-\frac {a^3\,\ln \left (a+b\,x\right )}{b^4}-\frac {a\,x^2}{2\,b^2}+\frac {a^2\,x}{b^3} \]
[In]
[Out]