\(\int \frac {a x^2+2 b n x \log (c x^n)}{(a x^2+b x \log ^2(c x^n))^2} \, dx\) [27]

Optimal result

Integrand size = 37, antiderivative size = 18 \[ \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx=-\frac {1}{a x+b \log ^2\left (c x^n\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2641, 2624} \[ \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx=-\frac {1}{a x+b \log ^2\left (c x^n\right )} \]

[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 2624

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]

Rule 2641

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx=-\frac {1}{a x+b \log ^2\left (c x^n\right )} \]

[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)

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
parallelrisch	\(-\frac {1}{a x +b \ln \left (c \,x^{n}\right )^{2}}\)	\(19\)
risch	\(-\frac {4}{-b \,\pi ^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 b \,\pi ^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-b \,\pi ^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 b \,\pi ^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 b \,\pi ^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 b \,\pi ^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-b \,\pi ^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 b \,\pi ^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-b \,\pi ^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+4 i \pi \ln \left (c \right ) b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-4 i \pi \ln \left (c \right ) b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \pi \ln \left (c \right ) b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 i b \ln \left (x^{n}\right ) \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi \ln \left (c \right ) b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-4 i b \ln \left (x^{n}\right ) \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4 i b \ln \left (x^{n}\right ) \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-4 i b \ln \left (x^{n}\right ) \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+4 b \ln \left (c \right )^{2}+8 b \ln \left (c \right ) \ln \left (x^{n}\right )+4 b \ln \left (x^{n}\right )^{2}+4 a x}\)	\(451\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx=-\frac {1}{b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a x} \]

[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)

Sympy [A] (verification not implemented)

Time = 11.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx=- \frac {1}{a x + b \log {\left (c x^{n} \right )}^{2}} \]

[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]

-1/(a*x + b*log(c*x**n)**2)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx=-\frac {1}{b \log \left (c\right )^{2} + 2 \, b \log \left (c\right ) \log \left (x^{n}\right ) + b \log \left (x^{n}\right )^{2} + a x} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx=-\frac {1}{b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a x} \]

[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)

Mupad [B] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx=-\frac {1}{b\,{\ln \left (c\,x^n\right )}^2+a\,x} \]

[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]

-1/(a*x + b*log(c*x^n)^2)