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

Optimal. Leaf size=20 \[ -\frac{1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2} \]

[Out]

-1/(2*(a*x + b*Log[c*x^n]^2)^2)

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Rubi [A]  time = 0.159982, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {2561, 2544} \[ -\frac{1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*(a*x + b*Log[c*x^n]^2)^2)

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 2544

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x
_)^(m_.)))/(x_), x_Symbol] :> Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] /; FreeQ[{a, b, c,
 d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]

Rubi steps

\begin{align*} \int \frac{a x^3+2 b n x^2 \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^3} \, dx &=\int \frac{x^2 \left (a x+2 b n \log \left (c x^n\right )\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^3} \, dx\\ &=\int \frac{a x+2 b n \log \left (c x^n\right )}{x \left (a x+b \log ^2\left (c x^n\right )\right )^3} \, dx\\ &=-\frac{1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0141217, size = 20, normalized size = 1. \[ -\frac{1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*(a*x + b*Log[c*x^n]^2)^2)

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Maple [C]  time = 0.085, size = 451, normalized size = 22.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/(-b*Pi^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2+2*b*Pi^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-b*Pi^2*
csgn(I*c)^2*csgn(I*c*x^n)^4+2*b*Pi^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-4*b*Pi^2*csgn(I*c)*csgn(I*x^n)*cs
gn(I*c*x^n)^4+2*b*Pi^2*csgn(I*c)*csgn(I*c*x^n)^5-b*Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+2*b*Pi^2*csgn(I*x^n)*csg
n(I*c*x^n)^5-b*Pi^2*csgn(I*c*x^n)^6+4*I*b*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*b*ln(x^n)*Pi*csgn(I*x^n)*cs
gn(I*c*x^n)^2-4*I*b*ln(c)*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+4*I*b*ln(c)*Pi*csgn(I*c)*csgn(I*c*x^n)^2+4*I*
b*ln(x^n)*Pi*csgn(I*c)*csgn(I*c*x^n)^2-4*I*b*ln(x^n)*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-4*I*b*ln(c)*Pi*csg
n(I*c*x^n)^3-4*I*b*ln(x^n)*Pi*csgn(I*c*x^n)^3+4*b*ln(c)^2+8*b*ln(c)*ln(x^n)+4*b*ln(x^n)^2+4*a*x)^2

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Maxima [B]  time = 1.42503, size = 128, normalized size = 6.4 \begin{align*} -\frac{1}{2 \,{\left (b^{2} \log \left (c\right )^{4} + 4 \, b^{2} \log \left (c\right ) \log \left (x^{n}\right )^{3} + b^{2} \log \left (x^{n}\right )^{4} + 2 \, a b x \log \left (c\right )^{2} + a^{2} x^{2} + 2 \,{\left (3 \, b^{2} \log \left (c\right )^{2} + a b x\right )} \log \left (x^{n}\right )^{2} + 4 \,{\left (b^{2} \log \left (c\right )^{3} + a b x \log \left (c\right )\right )} \log \left (x^{n}\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.02759, size = 254, normalized size = 12.7 \begin{align*} -\frac{1}{2 \,{\left (b^{2} n^{4} \log \left (x\right )^{4} + 4 \, b^{2} n^{3} \log \left (c\right ) \log \left (x\right )^{3} + b^{2} \log \left (c\right )^{4} + 2 \, a b x \log \left (c\right )^{2} + a^{2} x^{2} + 2 \,{\left (3 \, b^{2} n^{2} \log \left (c\right )^{2} + a b n^{2} x\right )} \log \left (x\right )^{2} + 4 \,{\left (b^{2} n \log \left (c\right )^{3} + a b n x \log \left (c\right )\right )} \log \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [B]  time = 1.27848, size = 413, normalized size = 20.65 \begin{align*} -\frac{4 \, a b n^{2} x + a^{2} x^{2}}{2 \,{\left (4 \, a b^{3} n^{6} x \log \left (x\right )^{4} + 16 \, a b^{3} n^{5} x \log \left (c\right ) \log \left (x\right )^{3} + a^{2} b^{2} n^{4} x^{2} \log \left (x\right )^{4} + 24 \, a b^{3} n^{4} x \log \left (c\right )^{2} \log \left (x\right )^{2} + 4 \, a^{2} b^{2} n^{3} x^{2} \log \left (c\right ) \log \left (x\right )^{3} + 16 \, a b^{3} n^{3} x \log \left (c\right )^{3} \log \left (x\right ) + 8 \, a^{2} b^{2} n^{4} x^{2} \log \left (x\right )^{2} + 6 \, a^{2} b^{2} n^{2} x^{2} \log \left (c\right )^{2} \log \left (x\right )^{2} + 4 \, a b^{3} n^{2} x \log \left (c\right )^{4} + 16 \, a^{2} b^{2} n^{3} x^{2} \log \left (c\right ) \log \left (x\right ) + 4 \, a^{2} b^{2} n x^{2} \log \left (c\right )^{3} \log \left (x\right ) + 2 \, a^{3} b n^{2} x^{3} \log \left (x\right )^{2} + 8 \, a^{2} b^{2} n^{2} x^{2} \log \left (c\right )^{2} + a^{2} b^{2} x^{2} \log \left (c\right )^{4} + 4 \, a^{3} b n x^{3} \log \left (c\right ) \log \left (x\right ) + 4 \, a^{3} b n^{2} x^{3} + 2 \, a^{3} b x^{3} \log \left (c\right )^{2} + a^{4} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*a*b*n^2*x + a^2*x^2)/(4*a*b^3*n^6*x*log(x)^4 + 16*a*b^3*n^5*x*log(c)*log(x)^3 + a^2*b^2*n^4*x^2*log(x)
^4 + 24*a*b^3*n^4*x*log(c)^2*log(x)^2 + 4*a^2*b^2*n^3*x^2*log(c)*log(x)^3 + 16*a*b^3*n^3*x*log(c)^3*log(x) + 8
*a^2*b^2*n^4*x^2*log(x)^2 + 6*a^2*b^2*n^2*x^2*log(c)^2*log(x)^2 + 4*a*b^3*n^2*x*log(c)^4 + 16*a^2*b^2*n^3*x^2*
log(c)*log(x) + 4*a^2*b^2*n*x^2*log(c)^3*log(x) + 2*a^3*b*n^2*x^3*log(x)^2 + 8*a^2*b^2*n^2*x^2*log(c)^2 + a^2*
b^2*x^2*log(c)^4 + 4*a^3*b*n*x^3*log(c)*log(x) + 4*a^3*b*n^2*x^3 + 2*a^3*b*x^3*log(c)^2 + a^4*x^4)