Integrand size = 19, antiderivative size = 60 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=-\frac {1}{a x}-\frac {b \left (c x^n\right )^{\frac {1}{n}} \log (x)}{a^2 x}+\frac {b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2 x} \] Output:
-1/a/x-b*(c*x^n)^(1/n)*ln(x)/a^2/x+b*(c*x^n)^(1/n)*ln(a+b*(c*x^n)^(1/n))/a ^2/x
Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=-\frac {a+b \left (c x^n\right )^{\frac {1}{n}} \log (x)-b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2 x} \] Input:
Integrate[1/(x^2*(a + b*(c*x^n)^n^(-1))),x]
Output:
-((a + b*(c*x^n)^n^(-1)*Log[x] - b*(c*x^n)^n^(-1)*Log[a + b*(c*x^n)^n^(-1) ])/(a^2*x))
Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {892, 54, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \int \frac {\left (c x^n\right )^{-2/n}}{b \left (c x^n\right )^{\frac {1}{n}}+a}d\left (c x^n\right )^{\frac {1}{n}}}{x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \int \left (\frac {\left (c x^n\right )^{-2/n}}{a}-\frac {b \left (c x^n\right )^{-1/n}}{a^2}+\frac {b^2}{a^2 \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )}\right )d\left (c x^n\right )^{\frac {1}{n}}}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \left (-\frac {b \log \left (\left (c x^n\right )^{\frac {1}{n}}\right )}{a^2}+\frac {b \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2}-\frac {\left (c x^n\right )^{-1/n}}{a}\right )}{x}\) |
Input:
Int[1/(x^2*(a + b*(c*x^n)^n^(-1))),x]
Output:
((c*x^n)^n^(-1)*(-(1/(a*(c*x^n)^n^(-1))) - (b*Log[(c*x^n)^n^(-1)])/a^2 + ( b*Log[a + b*(c*x^n)^n^(-1)])/a^2))/x
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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 = 0.18 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.67
method | result | size |
risch | \(-\frac {1}{a x}-\frac {b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}} \ln \left (x \right )}{a^{2} x}+\frac {\ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{x \,a^{2}}\) | \(220\) |
Input:
int(1/x^2/(a+b*(c*x^n)^(1/n)),x,method=_RETURNVERBOSE)
Output:
-1/a/x-1/a^2*b/x*(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)*ln(x)+ln(b*c^(1/n)*(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)-cs gn(I*c*x^n))/n)+a)*c^(1/n)*(x^n)^(1/n)/x*b/a^2*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)
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\frac {b c^{\left (\frac {1}{n}\right )} x \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) - b c^{\left (\frac {1}{n}\right )} x \log \left (x\right ) - a}{a^{2} x} \] Input:
integrate(1/x^2/(a+b*(c*x^n)^(1/n)),x, algorithm="fricas")
Output:
(b*c^(1/n)*x*log(b*c^(1/n)*x + a) - b*c^(1/n)*x*log(x) - a)/(a^2*x)
\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int \frac {1}{x^{2} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )}\, dx \] Input:
integrate(1/x**2/(a+b*(c*x**n)**(1/n)),x)
Output:
Integral(1/(x**2*(a + b*(c*x**n)**(1/n))), x)
\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*(c*x^n)^(1/n)),x, algorithm="maxima")
Output:
integrate(1/(((c*x^n)^(1/n)*b + a)*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a+b*(c*x^n)^(1/n)),x, algorithm="giac")
Output:
integrate(1/(((c*x^n)^(1/n)*b + a)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int \frac {1}{x^2\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )} \,d x \] Input:
int(1/(x^2*(a + b*(c*x^n)^(1/n))),x)
Output:
int(1/(x^2*(a + b*(c*x^n)^(1/n))), x)
Time = 0.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\frac {c^{\frac {1}{n}} \mathrm {log}\left (c^{\frac {1}{n}} b x +a \right ) b x -c^{\frac {1}{n}} \mathrm {log}\left (x \right ) b x -a}{a^{2} x} \] Input:
int(1/x^2/(a+b*(c*x^n)^(1/n)),x)
Output:
(c**(1/n)*log(c**(1/n)*b*x + a)*b*x - c**(1/n)*log(x)*b*x - a)/(a**2*x)