Integrand size = 17, antiderivative size = 67 \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \] Output:
a*x^2/b^2/((c*x^n)^(2/n))/(a+b*(c*x^n)^(1/n))+x^2*ln(a+b*(c*x^n)^(1/n))/b^ 2/((c*x^n)^(2/n))
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {a}{a+b \left (c x^n\right )^{\frac {1}{n}}}+\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{b^2} \] Input:
Integrate[x/(a + b*(c*x^n)^n^(-1))^2,x]
Output:
(x^2*(a/(a + b*(c*x^n)^n^(-1)) + Log[a + b*(c*x^n)^n^(-1)]))/(b^2*(c*x^n)^ (2/n))
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, 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.
\(\displaystyle \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle x^2 \left (c x^n\right )^{-2/n} \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^2}d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle x^2 \left (c x^n\right )^{-2/n} \int \left (\frac {1}{b \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )}-\frac {a}{b \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^2}\right )d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2 \left (c x^n\right )^{-2/n} \left (\frac {a}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2}\right )\) |
Input:
Int[x/(a + b*(c*x^n)^n^(-1))^2,x]
Output:
(x^2*(a/(b^2*(a + b*(c*x^n)^n^(-1))) + Log[a + b*(c*x^n)^n^(-1)]/b^2))/(c* x^n)^(2/n)
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 = 0.14 (sec) , antiderivative size = 310, normalized size of antiderivative = 4.63
method | result | size |
risch | \(\frac {x^{2}}{a \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}-\frac {\left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} x^{2} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{a b}+\frac {\ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (c^{-\frac {1}{n}}\right )^{2} {\left (\left (x^{n}\right )^{-\frac {1}{n}}\right )}^{2} x^{2} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{b^{2}}\) | \(310\) |
Input:
int(x/(a+b*(c*x^n)^(1/n))^2,x,method=_RETURNVERBOSE)
Output:
1/a*x^2/(b*c^(1/n)*(x^n)^(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)+a)-1/a/((x^n)^(1/n))/(c^(1/n))*x ^2*exp(-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csg n(I*c*x^n))/n)/b+ln(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(-csg n(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)+a)/(c^(1/n))/((x^n)^( 1/n))*(x^n)^(-1/n)*c^(-1/n)*x^2/b^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)
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) + a}{b^{3} c^{\frac {3}{n}} x + a b^{2} c^{\frac {2}{n}}} \] Input:
integrate(x/(a+b*(c*x^n)^(1/n))^2,x, algorithm="fricas")
Output:
((b*c^(1/n)*x + a)*log(b*c^(1/n)*x + a) + a)/(b^3*c^(3/n)*x + a*b^2*c^(2/n ))
\[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {x}{\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \] Input:
integrate(x/(a+b*(c*x**n)**(1/n))**2,x)
Output:
Integral(x/(a + b*(c*x**n)**(1/n))**2, x)
\[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {x}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}} \,d x } \] Input:
integrate(x/(a+b*(c*x^n)^(1/n))^2,x, algorithm="maxima")
Output:
x^2/(a*b*c^(1/n)*(x^n)^(1/n) + a^2) - integrate(x/(a*b*c^(1/n)*(x^n)^(1/n) + a^2), x)
\[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {x}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}} \,d x } \] Input:
integrate(x/(a+b*(c*x^n)^(1/n))^2,x, algorithm="giac")
Output:
integrate(x/((c*x^n)^(1/n)*b + a)^2, x)
Timed out. \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {x}{{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \] Input:
int(x/(a + b*(c*x^n)^(1/n))^2,x)
Output:
int(x/(a + b*(c*x^n)^(1/n))^2, x)
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {c^{\frac {1}{n}} \mathrm {log}\left (c^{\frac {1}{n}} b x +a \right ) b x -c^{\frac {1}{n}} b x +\mathrm {log}\left (c^{\frac {1}{n}} b x +a \right ) a}{c^{\frac {2}{n}} b^{2} \left (c^{\frac {1}{n}} b x +a \right )} \] Input:
int(x/(a+b*(c*x^n)^(1/n))^2,x)
Output:
(c**(1/n)*log(c**(1/n)*b*x + a)*b*x - c**(1/n)*b*x + log(c**(1/n)*b*x + a) *a)/(c**(2/n)*b**2*(c**(1/n)*b*x + a))