### 3.27 $$\int \frac{a x^2+2 b n x \log (c x^n)}{(a x^2+b x \log ^2(c x^n))^2} \, dx$$

Optimal. Leaf size=18 $-\frac{1}{a x+b \log ^2\left (c x^n\right )}$

[Out]

-(a*x + b*Log[c*x^n]^2)^(-1)

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Rubi [A]  time = 0.133133, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.054, Rules used = {2561, 2544} $-\frac{1}{a x+b \log ^2\left (c x^n\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a*x + b*Log[c*x^n]^2)^(-1)

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

Mathematica [A]  time = 0.0133318, size = 18, normalized size = 1. $-\frac{1}{a x+b \log ^2\left (c x^n\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a*x + b*Log[c*x^n]^2)^(-1)

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Maple [C]  time = 0.072, size = 451, normalized size = 25.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(-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)

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Maxima [A]  time = 1.21717, size = 42, normalized size = 2.33 \begin{align*} -\frac{1}{b \log \left (c\right )^{2} + 2 \, b \log \left (c\right ) \log \left (x^{n}\right ) + b \log \left (x^{n}\right )^{2} + a x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.86091, size = 84, normalized size = 4.67 \begin{align*} -\frac{1}{b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.45101, size = 42, normalized size = 2.33 \begin{align*} -\frac{1}{b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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