3.3011 \(\int \frac{1}{x^2 (a+b (c x^n)^{\frac{1}{n}})} \, dx\)

Optimal. Leaf size=60 \[ -\frac{b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^2 x}+\frac{b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2 x}-\frac{1}{a x} \]

[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)

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Rubi [A]  time = 0.0232409, antiderivative size = 60, 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} \[ -\frac{b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^2 x}+\frac{b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2 x}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[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 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^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{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}} \operatorname{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]  time = 0.0238341, size = 49, normalized size = 0.82 \[ -\frac{-b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+a+b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^2 x} \]

Antiderivative was successfully verified.

[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))

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Maple [C]  time = 0.097, size = 331, normalized size = 5.5 \begin{align*} -{\frac{1}{ax}}+{\frac{b\sqrt [n]{c}}{{a}^{2}}\ln \left ( b{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}x+a \right ){{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}}-{\frac{b\ln \left ( x \right ) \sqrt [n]{c}}{{a}^{2}}{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/x+ln(b*exp(-1/2*(I*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*x^n)-I*Pi*csgn(I*c*
x^n)^2*csgn(I*c)+I*Pi*csgn(I*c*x^n)^3+2*n*ln(x)-2*ln(c)-2*ln(x^n))/n)*x+a)*b*c^(1/n)/a^2*exp(-1/2*(I*Pi*csgn(I
*c*x^n)*csgn(I*c)*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*csgn(I*c*x^
n)^3+2*n*ln(x)-2*ln(x^n))/n)-ln(x)*b*c^(1/n)/a^2*exp(-1/2*(I*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-I*Pi*csgn(
I*c*x^n)^2*csgn(I*x^n)-I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*csgn(I*c*x^n)^3+2*n*ln(x)-2*ln(x^n))/n)

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Maxima [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 )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.53864, size = 93, normalized size = 1.55 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)