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]

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

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Rubi [A]  time = 0.13, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {1}{a x+b \log ^2\left (c x^n\right )} \]

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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fricas [A]  time = 0.44, size = 31, normalized size = 1.72 \[ -\frac {1}{b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + b \log \relax (c)^{2} + a x} \]

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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giac [A]  time = 0.18, size = 31, normalized size = 1.72 \[ -\frac {1}{b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + b \log \relax (c)^{2} + a x} \]

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)

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maple [C]  time = 0.31, size = 451, normalized size = 25.06 \[ -\frac {4}{-\pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 \pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-\pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (c )-4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (x^{n}\right )+4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{n}\right )+4 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{n}\right )-4 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-4 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (x^{n}\right )+4 b \ln \relax (c )^{2}+8 b \ln \relax (c ) \ln \left (x^{n}\right )+4 b \ln \left (x^{n}\right )^{2}+4 a x} \]

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/(-Pi^2*b*csgn(I*x^n)^2*csgn(I*c*x^n)^4+2*Pi^2*b*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-Pi^2*b*csgn(I*c)^2*
csgn(I*x^n)^2*csgn(I*c*x^n)^2+2*Pi^2*b*csgn(I*x^n)*csgn(I*c*x^n)^5-4*Pi^2*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n
)^4+2*Pi^2*b*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-Pi^2*b*csgn(I*c*x^n)^6+2*Pi^2*b*csgn(I*c)*csgn(I*c*x^n)^5
-Pi^2*b*csgn(I*c)^2*csgn(I*c*x^n)^4+4*I*b*ln(x^n)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*b*ln(x^n)*Pi*csgn(I*c*x^n
)^2*csgn(I*c)-4*I*Pi*b*csgn(I*c*x^n)^3*ln(c)-4*I*b*ln(x^n)*Pi*csgn(I*c*x^n)^3-4*I*b*ln(x^n)*Pi*csgn(I*x^n)*csg
n(I*c*x^n)*csgn(I*c)+4*I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2*ln(c)+4*I*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2*ln(c)-4*I*P
i*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*ln(c)+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 = 0.65, size = 31, normalized size = 1.72 \[ -\frac {1}{b \log \relax (c)^{2} + 2 \, b \log \relax (c) \log \left (x^{n}\right ) + b \log \left (x^{n}\right )^{2} + a x} \]

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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mupad [B]  time = 0.26, size = 18, normalized size = 1.00 \[ -\frac {1}{b\,{\ln \left (c\,x^n\right )}^2+a\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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