Integrand size = 17, antiderivative size = 83 \[ \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=-\frac {a x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b^2 (1+p)}+\frac {x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{2+p}}{b^2 (2+p)} \]
-a*x^2*(a+b*(c*x^n)^(1/n))^(p+1)/b^2/(p+1)/((c*x^n)^(2/n))+x^2*(a+b*(c*x^n )^(1/n))^(2+p)/b^2/(2+p)/((c*x^n)^(2/n))
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.76 \[ \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {x^2 \left (c x^n\right )^{-2/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p} \left (-a+b (1+p) \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2 (1+p) (2+p)} \]
(x^2*(a + b*(c*x^n)^n^(-1))^(1 + p)*(-a + b*(1 + p)*(c*x^n)^n^(-1)))/(b^2* (1 + p)*(2 + p)*(c*x^n)^(2/n))
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {892, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle x^2 \left (c x^n\right )^{-2/n} \int \left (c x^n\right )^{\frac {1}{n}} \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^pd\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle x^2 \left (c x^n\right )^{-2/n} \int \left (\frac {\left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^{p+1}}{b}-\frac {a \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^p}{b}\right )d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2 \left (c x^n\right )^{-2/n} \left (\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac {a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1}}{b^2 (p+1)}\right )\) |
(x^2*(-((a*(a + b*(c*x^n)^n^(-1))^(1 + p))/(b^2*(1 + p))) + (a + b*(c*x^n) ^n^(-1))^(2 + p)/(b^2*(2 + p))))/(c*x^n)^(2/n)
3.31.24.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> Simp[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1)) Subst[Int[x^m*(a + b *x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x ] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.71 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.66
method | result | size |
risch | \(\frac {{\left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}^{1+p} c^{-\frac {1}{n}} x^{2} \left (x^{n}\right )^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{b \left (1+p \right )}-\frac {\left (x^{n}\right )^{-\frac {2}{n}} c^{-\frac {2}{n}} x^{2} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}} {\left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}^{2+p}}{b^{2} \left (1+p \right ) \left (2+p \right )}\) | \(304\) |
(b*(x^n)^(1/n)*c^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x ^n))*(csgn(I*c)-csgn(I*c*x^n))/n)+a)^(1+p)/(c^(1/n))*x^2/((x^n)^(1/n))*exp (-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c* x^n))/n)/b/(1+p)-1/b^2/(1+p)/((x^n)^(1/n))^2/(c^(1/n))^2*x^2*exp(-I*Pi*csg n(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)*(b*(x ^n)^(1/n)*c^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))* (csgn(I*c)-csgn(I*c*x^n))/n)+a)^(2+p)/(2+p)
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (a b c^{\left (\frac {1}{n}\right )} p x + {\left (b^{2} p + b^{2}\right )} c^{\frac {2}{n}} x^{2} - a^{2}\right )} {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p}}{{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )} c^{\frac {2}{n}}} \]
(a*b*c^(1/n)*p*x + (b^2*p + b^2)*c^(2/n)*x^2 - a^2)*(b*c^(1/n)*x + a)^p/(( b^2*p^2 + 3*b^2*p + 2*b^2)*c^(2/n))
\[ \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int x \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \]
\[ \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int { {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p} x \,d x } \]
Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.64 \[ \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{2} c^{\frac {2}{n}} p x^{2} + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b c^{\left (\frac {1}{n}\right )} p x + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{2} c^{\frac {2}{n}} x^{2} - {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{2}}{b^{2} c^{\frac {2}{n}} p^{2} + 3 \, b^{2} c^{\frac {2}{n}} p + 2 \, b^{2} c^{\frac {2}{n}}} \]
((b*c^(1/n)*x + a)^p*b^2*c^(2/n)*p*x^2 + (b*c^(1/n)*x + a)^p*a*b*c^(1/n)*p *x + (b*c^(1/n)*x + a)^p*b^2*c^(2/n)*x^2 - (b*c^(1/n)*x + a)^p*a^2)/(b^2*c ^(2/n)*p^2 + 3*b^2*c^(2/n)*p + 2*b^2*c^(2/n))
Timed out. \[ \int x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int x\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \]