Integrand size = 19, antiderivative size = 90 \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x^3 \left (c x^n\right )^{-2/n}}{b^2}-\frac {a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3} \]
x^3/b^2/((c*x^n)^(2/n))-a^2*x^3/b^3/((c*x^n)^(3/n))/(a+b*(c*x^n)^(1/n))-2* a*x^3*ln(a+b*(c*x^n)^(1/n))/b^3/((c*x^n)^(3/n))
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x^3 \left (c x^n\right )^{-3/n} \left (b \left (c x^n\right )^{\frac {1}{n}}-\frac {a^2}{a+b \left (c x^n\right )^{\frac {1}{n}}}-2 a \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{b^3} \]
(x^3*(b*(c*x^n)^n^(-1) - a^2/(a + b*(c*x^n)^n^(-1)) - 2*a*Log[a + b*(c*x^n )^n^(-1)]))/(b^3*(c*x^n)^(3/n))
Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {892, 49, 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.
\(\displaystyle \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle x^3 \left (c x^n\right )^{-3/n} \int \frac {\left (c x^n\right )^{2/n}}{\left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^2}d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle x^3 \left (c x^n\right )^{-3/n} \int \left (\frac {a^2}{b^2 \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^2}-\frac {2 a}{b^2 \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )}+\frac {1}{b^2}\right )d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^3 \left (c x^n\right )^{-3/n} \left (-\frac {a^2}{b^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 a \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{b^2}\right )\) |
(x^3*((c*x^n)^n^(-1)/b^2 - a^2/(b^3*(a + b*(c*x^n)^n^(-1))) - (2*a*Log[a + b*(c*x^n)^n^(-1)])/b^3))/(c*x^n)^(3/n)
3.31.14.3.1 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}, x] && IGtQ[m, 0] && IGtQ[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 = 4.13 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.29
method | result | size |
risch | \(\frac {x^{3}}{a \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}-\frac {\left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} x^{3} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{a b}+\frac {2 \left (x^{n}\right )^{-\frac {2}{n}} c^{-\frac {2}{n}} x^{3} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{b^{2}}-\frac {2 a \ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} x^{3} {\mathrm e}^{-\frac {3 i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{b^{3}}\) | \(386\) |
x^3/a/(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/a/((x^n)^(1/n))/(c^(1/n))*x^3 *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+2/((x^n)^(1/n))^2/(c^(1/n))^2*x^3*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)/b^2-2*a*ln(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))*(c sgn(I*c)-csgn(I*c*x^n))/n)+a)/((x^n)^(1/n))/(c^(1/n))*c^(-2/n)*(x^n)^(-2/n )*x^3/b^3*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 = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {b^{2} c^{\frac {2}{n}} x^{2} + a b c^{\left (\frac {1}{n}\right )} x - a^{2} - 2 \, {\left (a b c^{\left (\frac {1}{n}\right )} x + a^{2}\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{b^{4} c^{\frac {4}{n}} x + a b^{3} c^{\frac {3}{n}}} \]
(b^2*c^(2/n)*x^2 + a*b*c^(1/n)*x - a^2 - 2*(a*b*c^(1/n)*x + a^2)*log(b*c^( 1/n)*x + a))/(b^4*c^(4/n)*x + a*b^3*c^(3/n))
\[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {x^{2}}{\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \]
\[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {x^{2}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}} \,d x } \]
\[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {x^{2}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {x^2}{{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \]