Integrand size = 19, antiderivative size = 126 \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {a^2 x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b^3 (1+p)}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{2+p}}{b^3 (2+p)}+\frac {x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{3+p}}{b^3 (3+p)} \]
a^2*x^3*(a+b*(c*x^n)^(1/n))^(p+1)/b^3/(p+1)/((c*x^n)^(3/n))-2*a*x^3*(a+b*( c*x^n)^(1/n))^(2+p)/b^3/(2+p)/((c*x^n)^(3/n))+x^3*(a+b*(c*x^n)^(1/n))^(3+p )/b^3/(3+p)/((c*x^n)^(3/n))
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.75 \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p} \left (2 a^2-2 a b (1+p) \left (c x^n\right )^{\frac {1}{n}}+b^2 \left (2+3 p+p^2\right ) \left (c x^n\right )^{2/n}\right )}{b^3 (1+p) (2+p) (3+p)} \]
(x^3*(a + b*(c*x^n)^n^(-1))^(1 + p)*(2*a^2 - 2*a*b*(1 + p)*(c*x^n)^n^(-1) + b^2*(2 + 3*p + p^2)*(c*x^n)^(2/n)))/(b^3*(1 + p)*(2 + p)*(3 + p)*(c*x^n) ^(3/n))
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.79, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, 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.
\(\displaystyle \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle x^3 \left (c x^n\right )^{-3/n} \int \left (c x^n\right )^{2/n} \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^pd\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle x^3 \left (c x^n\right )^{-3/n} \int \left (\frac {a^2 \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^p}{b^2}-\frac {2 a \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^{p+1}}{b^2}+\frac {\left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^{p+2}}{b^2}\right )d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^3 \left (c x^n\right )^{-3/n} \left (\frac {a^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1}}{b^3 (p+1)}-\frac {2 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+2}}{b^3 (p+2)}+\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+3}}{b^3 (p+3)}\right )\) |
(x^3*((a^2*(a + b*(c*x^n)^n^(-1))^(1 + p))/(b^3*(1 + p)) - (2*a*(a + b*(c* x^n)^n^(-1))^(2 + p))/(b^3*(2 + p)) + (a + b*(c*x^n)^n^(-1))^(3 + p)/(b^3* (3 + p))))/(c*x^n)^(3/n)
3.31.23.3.1 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 = 14.90 (sec) , antiderivative size = 787, normalized size of antiderivative = 6.25
(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^3/((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)-2/b/(1+p)*(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)/(3+p )*x^3/((x^n)^(1/n))/(c^(1/n))*exp(-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+cs gn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)-2/b^2/(1+p)*(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)/(3+p)/(2+p)*a*x^3/((x^n)^(1/n))^2/(c^(1/n))^2*exp(- 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)-2/b^2/(1+p)*(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)/(3+p)/(2+p)*a*x^ 3/((x^n)^(1/n))^2/(c^(1/n))^2*exp(-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)*p+2/b^3/(1+p)*(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)/(3+p)*a^2*x^3/((x^n)^(1/n))^3/(c^(1/n))^3/(2+p)*exp(- 3/2*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)
Time = 0.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02 \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=-\frac {{\left (2 \, a^{2} b c^{\left (\frac {1}{n}\right )} p x - {\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} c^{\frac {3}{n}} x^{3} - {\left (a b^{2} p^{2} + a b^{2} p\right )} c^{\frac {2}{n}} x^{2} - 2 \, a^{3}\right )} {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p}}{{\left (b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}\right )} c^{\frac {3}{n}}} \]
-(2*a^2*b*c^(1/n)*p*x - (b^3*p^2 + 3*b^3*p + 2*b^3)*c^(3/n)*x^3 - (a*b^2*p ^2 + a*b^2*p)*c^(2/n)*x^2 - 2*a^3)*(b*c^(1/n)*x + a)^p/((b^3*p^3 + 6*b^3*p ^2 + 11*b^3*p + 6*b^3)*c^(3/n))
\[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int x^{2} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \]
\[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int { {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p} x^{2} \,d x } \]
Time = 0.35 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.93 \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{3} c^{\frac {3}{n}} p^{2} x^{3} + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b^{2} c^{\frac {2}{n}} p^{2} x^{2} + 3 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{3} c^{\frac {3}{n}} p x^{3} + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b^{2} c^{\frac {2}{n}} p x^{2} + 2 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{3} c^{\frac {3}{n}} x^{3} - 2 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{2} b c^{\left (\frac {1}{n}\right )} p x + 2 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{3}}{b^{3} c^{\frac {3}{n}} p^{3} + 6 \, b^{3} c^{\frac {3}{n}} p^{2} + 11 \, b^{3} c^{\frac {3}{n}} p + 6 \, b^{3} c^{\frac {3}{n}}} \]
((b*c^(1/n)*x + a)^p*b^3*c^(3/n)*p^2*x^3 + (b*c^(1/n)*x + a)^p*a*b^2*c^(2/ n)*p^2*x^2 + 3*(b*c^(1/n)*x + a)^p*b^3*c^(3/n)*p*x^3 + (b*c^(1/n)*x + a)^p *a*b^2*c^(2/n)*p*x^2 + 2*(b*c^(1/n)*x + a)^p*b^3*c^(3/n)*x^3 - 2*(b*c^(1/n )*x + a)^p*a^2*b*c^(1/n)*p*x + 2*(b*c^(1/n)*x + a)^p*a^3)/(b^3*c^(3/n)*p^3 + 6*b^3*c^(3/n)*p^2 + 11*b^3*c^(3/n)*p + 6*b^3*c^(3/n))
Timed out. \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int x^2\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \]