\(\int \frac {1}{x^2 (a+b (c x^n)^{\frac {1}{n}})} \, dx\) [3011]

Optimal result

Integrand size = 19, antiderivative size = 60 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=-\frac {1}{a x}-\frac {b \left (c x^n\right )^{\frac {1}{n}} \log (x)}{a^2 x}+\frac {b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2 x} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {375, 46} \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=-\frac {b \log (x) \left (c x^n\right )^{\frac {1}{n}}}{a^2 x}+\frac {b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2 x}-\frac {1}{a x} \]

[In]

Int[1/(x^2*(a + b*(c*x^n)^n^(-1))),x]

[Out]

-(1/(a*x)) - (b*(c*x^n)^n^(-1)*Log[x])/(a^2*x) + (b*(c*x^n)^n^(-1)*Log[a + b*(c*x^n)^n^(-1)])/(a^2*x)

Rule 46

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] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 375

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(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)]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x} \\ & = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{x} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=-\frac {a+b \left (c x^n\right )^{\frac {1}{n}} \log (x)-b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2 x} \]

[In]

Integrate[1/(x^2*(a + b*(c*x^n)^n^(-1))),x]

[Out]

-((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))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.30 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.67

method	result	size
risch	\(\frac {\ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{x \,a^{2}}-\frac {1}{a x}-\frac {b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}} \ln \left (x \right )}{a^{2} x}\)	\(220\)

[In]

int(1/x^2/(a+b*(c*x^n)^(1/n)),x,method=_RETURNVERBOSE)

[Out]

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))*(csgn(I*c)-csgn(I*c*x^n))/n)+
a)*b/x*c^(1/n)*(x^n)^(1/n)/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)-1/a/x-1/a^2*b/x*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)-c
sgn(I*c*x^n))/n)*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\frac {b c^{\left (\frac {1}{n}\right )} x \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) - b c^{\left (\frac {1}{n}\right )} x \log \left (x\right ) - a}{a^{2} x} \]

[In]

integrate(1/x^2/(a+b*(c*x^n)^(1/n)),x, algorithm="fricas")

[Out]

(b*c^(1/n)*x*log(b*c^(1/n)*x + a) - b*c^(1/n)*x*log(x) - a)/(a^2*x)

Sympy [F]

\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int \frac {1}{x^{2} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )}\, dx \]

[In]

integrate(1/x**2/(a+b*(c*x**n)**(1/n)),x)

[Out]

Integral(1/(x**2*(a + b*(c*x**n)**(1/n))), x)

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*(c*x^n)^(1/n)),x, algorithm="maxima")

[Out]

integrate(1/(((c*x^n)^(1/n)*b + a)*x^2), x)

Giac [F]

\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*(c*x^n)^(1/n)),x, algorithm="giac")

[Out]

integrate(1/(((c*x^n)^(1/n)*b + a)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \, dx=\int \frac {1}{x^2\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )} \,d x \]

[In]

int(1/(x^2*(a + b*(c*x^n)^(1/n))),x)

[Out]

int(1/(x^2*(a + b*(c*x^n)^(1/n))), x)