\(\int \frac {1}{x (a+b (c x^n)^{\frac {1}{n}})^2} \, dx\) [3017]

Optimal result

Integrand size = 19, antiderivative size = 45 \[ \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {1}{a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, 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} \[ \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2}+\frac {\log (x)}{a^2}+\frac {1}{a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )} \]

[In]

Int[1/(x*(a + b*(c*x^n)^n^(-1))^2),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {\frac {a}{a+b \left (c x^n\right )^{\frac {1}{n}}}+\log \left (\left (c x^n\right )^{\frac {1}{n}}\right )-\log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 4.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(-\frac {\ln \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}{a^{2}}+\frac {1}{a \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}+\frac {\ln \left (\left (c \,x^{n}\right )^{\frac {1}{n}}\right )}{a^{2}}\)	\(54\)
default	\(-\frac {\ln \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}{a^{2}}+\frac {1}{a \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}+\frac {\ln \left (\left (c \,x^{n}\right )^{\frac {1}{n}}\right )}{a^{2}}\)	\(54\)
parallelrisch	\(\frac {\ln \left (x \right ) x^{2} \left (c \,x^{n}\right )^{\frac {1}{n}} b^{2}-\ln \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right ) x^{2} \left (c \,x^{n}\right )^{\frac {1}{n}} b^{2}+\ln \left (x \right ) x^{2} a b -\ln \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right ) x^{2} a b +a b \,x^{2}}{a^{2} b \,x^{2} \left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}\)	\(111\)
risch	\(\frac {\ln \left (c \right )}{a^{2} n}+\frac {\ln \left (x^{n}\right )}{a^{2} n}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2 a^{2} n}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2 a^{2} n}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i x^{n}\right )}{2 a^{2} n}-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )}{2 a^{2} n}+\frac {1}{a \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}-\frac {\ln \left (c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+\frac {a}{b}\right )}{a^{2}}\)	\(271\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {b c^{\left (\frac {1}{n}\right )} x \log \left (x\right ) - {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right ) + a \log \left (x\right ) + a}{a^{2} b c^{\left (\frac {1}{n}\right )} x + a^{3}} \]

[In]

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

[Out]

(b*c^(1/n)*x*log(x) - (b*c^(1/n)*x + a)*log(b*c^(1/n)*x + a) + a*log(x) + a)/(a^2*b*c^(1/n)*x + a^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (37) = 74\).

Time = 2.22 (sec) , antiderivative size = 189, normalized size of antiderivative = 4.20 \[ \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (c x^{n}\right )^{- \frac {2}{n}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\left (x \right )}}{a^{2}} & \text {for}\: b = 0 \\- \frac {\left (c x^{n}\right )^{- \frac {2}{n}}}{2 b^{2}} & \text {for}\: a = 0 \\\tilde {\infty } \log {\left (x \right )} & \text {for}\: b = - a \left (c x^{n}\right )^{- \frac {1}{n}} \\\frac {a \log {\left (x \right )}}{a^{3} + a^{2} b \left (c x^{n}\right )^{\frac {1}{n}}} - \frac {a \log {\left (\frac {a}{b} + \left (c x^{n}\right )^{\frac {1}{n}} \right )}}{a^{3} + a^{2} b \left (c x^{n}\right )^{\frac {1}{n}}} + \frac {a}{a^{3} + a^{2} b \left (c x^{n}\right )^{\frac {1}{n}}} + \frac {b \left (c x^{n}\right )^{\frac {1}{n}} \log {\left (x \right )}}{a^{3} + a^{2} b \left (c x^{n}\right )^{\frac {1}{n}}} - \frac {b \left (c x^{n}\right )^{\frac {1}{n}} \log {\left (\frac {a}{b} + \left (c x^{n}\right )^{\frac {1}{n}} \right )}}{a^{3} + a^{2} b \left (c x^{n}\right )^{\frac {1}{n}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo/(c*x**n)**(2/n), Eq(a, 0) & Eq(b, 0)), (log(x)/a**2, Eq(b, 0)), (-1/(2*b**2*(c*x**n)**(2/n)), E
q(a, 0)), (zoo*log(x), Eq(b, -a/(c*x**n)**(1/n))), (a*log(x)/(a**3 + a**2*b*(c*x**n)**(1/n)) - a*log(a/b + (c*
x**n)**(1/n))/(a**3 + a**2*b*(c*x**n)**(1/n)) + a/(a**3 + a**2*b*(c*x**n)**(1/n)) + b*(c*x**n)**(1/n)*log(x)/(
a**3 + a**2*b*(c*x**n)**(1/n)) - b*(c*x**n)**(1/n)*log(a/b + (c*x**n)**(1/n))/(a**3 + a**2*b*(c*x**n)**(1/n)),
True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {1}{a b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{2}} + \frac {\log \left (x\right )}{a^{2}} - \frac {\log \left (\frac {b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a}{b c^{\left (\frac {1}{n}\right )}}\right )}{a^{2}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.50 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {\ln \left (x\right )}{a^2}-\frac {\ln \left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}{a^2}+\frac {1}{a^2+a\,b\,{\left (c\,x^n\right )}^{1/n}} \]

[In]

int(1/(x*(a + b*(c*x^n)^(1/n))^2),x)

[Out]

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