3.3017 \(\int \frac{1}{x (a+b (c x^n)^{\frac{1}{n}})^2} \, dx\)

Optimal. Leaf size=45 \[ -\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \]

[Out]

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

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Rubi [A]  time = 0.0202457, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {368, 44} \[ -\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \]

Antiderivative was successfully verified.

[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 368

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)]

Rule 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\operatorname{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]  time = 0.0335881, size = 40, normalized size = 0.89 \[ \frac{\frac{a}{a+b \left (c x^n\right )^{\frac{1}{n}}}-\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+\log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 54, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( \sqrt [n]{c{x}^{n}} \right ) }{{a}^{2}}}-{\frac{\ln \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }{{a}^{2}}}+{\frac{1}{a \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.985928, size = 82, normalized size = 1.82 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.54628, size = 138, normalized size = 3.07 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)