\(\int \frac {x}{(a+b (c x^n)^{\frac {1}{n}})^2} \, dx\) [3015]

Optimal result

Integrand size = 17, antiderivative size = 67 \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]

[Out]

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))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {375, 45} \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2} \]

[In]

Int[x/(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

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

Rule 45

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] && NeQ[b*c - a*d, 0] && 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])

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x^2 \left (c x^n\right )^{-2/n} \left (\frac {a}{a+b \left (c x^n\right )^{\frac {1}{n}}}+\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{b^2} \]

[In]

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

[Out]

(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))

Maple [C] (warning: unable to verify)

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

Time = 4.23 (sec) , antiderivative size = 310, normalized size of antiderivative = 4.63

method	result	size
risch	\(\frac {x^{2}}{a \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}-\frac {\left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} x^{2} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{a b}+\frac {\ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (c^{-\frac {1}{n}}\right )^{2} {\left (\left (x^{n}\right )^{-\frac {1}{n}}\right )}^{2} x^{2} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{b^{2}}\)	\(310\)

[In]

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

[Out]

x^2/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^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*(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)-c
sgn(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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) + a}{b^{3} c^{\frac {3}{n}} x + a b^{2} c^{\frac {2}{n}}} \]

[In]

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

[Out]

((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))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x/(a + b*(c*x^n)^(1/n))^2,x)

[Out]

int(x/(a + b*(c*x^n)^(1/n))^2, x)