Integrand size = 17, antiderivative size = 41 \[ \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx=a c x+\frac {a d x^n}{n}+\frac {b d x^{2 n}}{2 n}+\frac {b c x^{1+n}}{1+n} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1905, 14} \[ \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx=a c x+\frac {a d x^n}{n}+\frac {b c x^{n+1}}{n+1}+\frac {b d x^{2 n}}{2 n} \]
[In]
[Out]
Rule 14
Rule 1905
Rubi steps \begin{align*} \text {integral}& = c \int \left (a+b x^n\right ) \, dx+d \int x^{-1+n} \left (a+b x^n\right ) \, dx \\ & = a c x+\frac {b c x^{1+n}}{1+n}+d \int \left (a x^{-1+n}+b x^{-1+2 n}\right ) \, dx \\ & = a c x+\frac {a d x^n}{n}+\frac {b d x^{2 n}}{2 n}+\frac {b c x^{1+n}}{1+n} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx=\frac {2 a \left (c n x+d x^n\right )+b x^n \left (\frac {2 c n x}{1+n}+d x^n\right )}{2 n} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.05
method | result | size |
risch | \(a c x +\frac {b d \,x^{2 n}}{2 n}+\frac {\left (n b c x +a d n +a d \right ) x^{n}}{n \left (1+n \right )}\) | \(43\) |
norman | \(a c x +\frac {a d \,{\mathrm e}^{n \ln \left (x \right )}}{n}+\frac {b c x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {b d \,{\mathrm e}^{2 n \ln \left (x \right )}}{2 n}\) | \(45\) |
parallelrisch | \(\frac {x \,x^{n} x^{-1+n} b d n +x \,x^{n} x^{-1+n} b d +2 x^{n} b c n x +2 x \,x^{-1+n} a d n +2 a c x \,n^{2}+2 x \,x^{-1+n} a d +2 a c x n}{2 n \left (1+n \right )}\) | \(81\) |
[In]
[Out]
none
Time = 0.53 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.37 \[ \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx=\frac {2 \, {\left (a c n^{2} + a c n\right )} x + {\left (b d n + b d\right )} x^{2 \, n} + 2 \, {\left (b c n x + a d n + a d\right )} x^{n}}{2 \, {\left (n^{2} + n\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (36) = 72\).
Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 4.39 \[ \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx=\begin {cases} a c x - \frac {a d}{x} + b c \log {\left (x \right )} - \frac {b d}{2 x^{2}} & \text {for}\: n = -1 \\\left (a + b\right ) \left (c x + d \log {\left (x \right )}\right ) & \text {for}\: n = 0 \\\frac {2 a c n^{2} x}{2 n^{2} + 2 n} + \frac {2 a c n x}{2 n^{2} + 2 n} + \frac {2 a d n x x^{n - 1}}{2 n^{2} + 2 n} + \frac {2 a d x x^{n - 1}}{2 n^{2} + 2 n} + \frac {2 b c n x x^{n}}{2 n^{2} + 2 n} + \frac {b d n x x^{n} x^{n - 1}}{2 n^{2} + 2 n} + \frac {b d x x^{n} x^{n - 1}}{2 n^{2} + 2 n} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx=a c x + \frac {b d x^{2 \, n}}{2 \, n} + \frac {b c x^{n + 1}}{n + 1} + \frac {a d x^{n}}{n} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.59 \[ \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx=\frac {2 \, a c n^{2} x + 2 \, b c n x x^{n} + 2 \, a c n x + b d n x^{2 \, n} + 2 \, a d n x^{n} + b d x^{2 \, n} + 2 \, a d x^{n}}{2 \, {\left (n^{2} + n\right )}} \]
[In]
[Out]
Time = 9.75 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right ) \, dx=a\,c\,x+\frac {a\,d\,x^n}{n}+\frac {b\,d\,x^{2\,n}}{2\,n}+\frac {b\,c\,x\,x^n}{n+1} \]
[In]
[Out]