3.3019 \(\int \frac{1}{x^3 (a+b (c x^n)^{\frac{1}{n}})^2} \, dx\)

Optimal. Leaf size=125 \[ \frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}-\frac{1}{2 a^2 x^2} \]

[Out]

-1/(2*a^2*x^2) + (2*b*(c*x^n)^n^(-1))/(a^3*x^2) + (b^2*(c*x^n)^(2/n))/(a^3*x^2*(a + b*(c*x^n)^n^(-1))) + (3*b^
2*(c*x^n)^(2/n)*Log[x])/(a^4*x^2) - (3*b^2*(c*x^n)^(2/n)*Log[a + b*(c*x^n)^n^(-1)])/(a^4*x^2)

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Rubi [A]  time = 0.0501029, antiderivative size = 125, 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} \[ \frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}-\frac{1}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*(c*x^n)^n^(-1))^2),x]

[Out]

-1/(2*a^2*x^2) + (2*b*(c*x^n)^n^(-1))/(a^3*x^2) + (b^2*(c*x^n)^(2/n))/(a^3*x^2*(a + b*(c*x^n)^n^(-1))) + (3*b^
2*(c*x^n)^(2/n)*Log[x])/(a^4*x^2) - (3*b^2*(c*x^n)^(2/n)*Log[a + b*(c*x^n)^n^(-1)])/(a^4*x^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^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x^2}\\ &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x^2}\\ &=-\frac{1}{2 a^2 x^2}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}+\frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 b^2 \left (c x^n\right )^{2/n} \log (x)}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}\\ \end{align*}

Mathematica [A]  time = 0.187761, size = 99, normalized size = 0.79 \[ \frac{\left (c x^n\right )^{2/n} \left (a \left (\frac{2 b^2}{a+b \left (c x^n\right )^{\frac{1}{n}}}-a \left (c x^n\right )^{-2/n}+4 b \left (c x^n\right )^{-1/n}\right )-6 b^2 \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+6 b^2 \log (x)\right )}{2 a^4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*(c*x^n)^n^(-1))^2),x]

[Out]

((c*x^n)^(2/n)*(a*(-(a/(c*x^n)^(2/n)) + (4*b)/(c*x^n)^n^(-1) + (2*b^2)/(a + b*(c*x^n)^n^(-1))) + 6*b^2*Log[x]
- 6*b^2*Log[a + b*(c*x^n)^n^(-1)]))/(2*a^4*x^2)

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Maple [C]  time = 0.107, size = 558, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a+b*(c*x^n)^(1/n))^2,x)

[Out]

1/a/x^2/(a+b*exp(-1/2*(I*Pi*csgn(I*c*x^n)^3-I*Pi*csgn(I*c*x^n)^2*csgn(I*c)-I*Pi*csgn(I*c*x^n)^2*csgn(I*x^n)+I*
Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-2*ln(c)-2*ln(x^n))/n))+3/a^3/x*b*c^(1/n)*exp(-1/2*(I*Pi*csgn(I*c*x^n)*c
sgn(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)-3/2/a^2/x^2+3/a^4*ln(x)*b^2*(c^(1/n))^2*exp(-(I*Pi*csgn(I*c*x^n)*csgn(I*c)*csgn(I*x^n)-I*P
i*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)-3/a^
4*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^2*(c^(1/n))^2*exp(-(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*} \frac{1}{a b c^{\left (\frac{1}{n}\right )} x^{2}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x^{2}} + 3 \, \int \frac{1}{a b c^{\left (\frac{1}{n}\right )} x^{3}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(c*x^n)^(1/n))^2,x, algorithm="maxima")

[Out]

1/(a*b*c^(1/n)*x^2*(x^n)^(1/n) + a^2*x^2) + 3*integrate(1/(a*b*c^(1/n)*x^3*(x^n)^(1/n) + a^2*x^3), x)

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Fricas [A]  time = 1.57348, size = 262, normalized size = 2.1 \begin{align*} \frac{6 \, b^{3} c^{\frac{3}{n}} x^{3} \log \left (x\right ) + 3 \, a^{2} b c^{\left (\frac{1}{n}\right )} x - a^{3} + 6 \,{\left (a b^{2} x^{2} \log \left (x\right ) + a b^{2} x^{2}\right )} c^{\frac{2}{n}} - 6 \,{\left (b^{3} c^{\frac{3}{n}} x^{3} + a b^{2} c^{\frac{2}{n}} x^{2}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \,{\left (a^{4} b c^{\left (\frac{1}{n}\right )} x^{3} + a^{5} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(c*x^n)^(1/n))^2,x, algorithm="fricas")

[Out]

1/2*(6*b^3*c^(3/n)*x^3*log(x) + 3*a^2*b*c^(1/n)*x - a^3 + 6*(a*b^2*x^2*log(x) + a*b^2*x^2)*c^(2/n) - 6*(b^3*c^
(3/n)*x^3 + a*b^2*c^(2/n)*x^2)*log(b*c^(1/n)*x + a))/(a^4*b*c^(1/n)*x^3 + a^5*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b*(c*x**n)**(1/n))**2,x)

[Out]

Integral(1/(x**3*(a + b*(c*x**n)**(1/n))**2), 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 )}^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(c*x^n)^(1/n))^2,x, algorithm="giac")

[Out]

integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^3), x)