3.161 \(\int \frac {(a+b \log (c x^n)) (d+e \log (f x^r))}{x^3} \, dx\)

Optimal. Leaf size=83 \[ -\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {e r \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{8 x^2}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b e n r}{8 x^2} \]

[Out]

-1/8*b*e*n*r/x^2-1/8*e*r*(2*a+b*n+2*b*ln(c*x^n))/x^2-1/4*b*n*(d+e*ln(f*x^r))/x^2-1/2*(a+b*ln(c*x^n))*(d+e*ln(f
*x^r))/x^2

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Rubi [A]  time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2304, 2366, 12} \[ -\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {e r \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{8 x^2}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b e n r}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x^3,x]

[Out]

-(b*e*n*r)/(8*x^2) - (e*r*(2*a + b*n + 2*b*Log[c*x^n]))/(8*x^2) - (b*n*(d + e*Log[f*x^r]))/(4*x^2) - ((a + b*L
og[c*x^n])*(d + e*Log[f*x^r]))/(2*x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx &=-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-(e r) \int \frac {-2 a \left (1+\frac {b n}{2 a}\right )-2 b \log \left (c x^n\right )}{4 x^3} \, dx\\ &=-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {1}{4} (e r) \int \frac {-2 a \left (1+\frac {b n}{2 a}\right )-2 b \log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac {b e n r}{8 x^2}-\frac {e r \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{8 x^2}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 64, normalized size = 0.77 \[ -\frac {e (2 a+b n) \log \left (f x^r\right )+2 a d+a e r+b \log \left (c x^n\right ) \left (2 d+2 e \log \left (f x^r\right )+e r\right )+b d n+b e n r}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x^3,x]

[Out]

-1/4*(2*a*d + b*d*n + a*e*r + b*e*n*r + e*(2*a + b*n)*Log[f*x^r] + b*Log[c*x^n]*(2*d + e*r + 2*e*Log[f*x^r]))/
x^2

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fricas [A]  time = 0.85, size = 95, normalized size = 1.14 \[ -\frac {2 \, b e n r \log \relax (x)^{2} + b d n + 2 \, a d + {\left (b e n + a e\right )} r + {\left (b e r + 2 \, b d\right )} \log \relax (c) + {\left (b e n + 2 \, b e \log \relax (c) + 2 \, a e\right )} \log \relax (f) + 2 \, {\left (b e r \log \relax (c) + b e n \log \relax (f) + b d n + {\left (b e n + a e\right )} r\right )} \log \relax (x)}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^3,x, algorithm="fricas")

[Out]

-1/4*(2*b*e*n*r*log(x)^2 + b*d*n + 2*a*d + (b*e*n + a*e)*r + (b*e*r + 2*b*d)*log(c) + (b*e*n + 2*b*e*log(c) +
2*a*e)*log(f) + 2*(b*e*r*log(c) + b*e*n*log(f) + b*d*n + (b*e*n + a*e)*r)*log(x))/x^2

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giac [A]  time = 0.24, size = 116, normalized size = 1.40 \[ -\frac {2 \, b n r e \log \relax (x)^{2} + 2 \, b n r e \log \relax (x) + 2 \, b r e \log \relax (c) \log \relax (x) + 2 \, b n e \log \relax (f) \log \relax (x) + b n r e + b r e \log \relax (c) + b n e \log \relax (f) + 2 \, b e \log \relax (c) \log \relax (f) + 2 \, b d n \log \relax (x) + 2 \, a r e \log \relax (x) + b d n + a r e + 2 \, b d \log \relax (c) + 2 \, a e \log \relax (f) + 2 \, a d}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^3,x, algorithm="giac")

[Out]

-1/4*(2*b*n*r*e*log(x)^2 + 2*b*n*r*e*log(x) + 2*b*r*e*log(c)*log(x) + 2*b*n*e*log(f)*log(x) + b*n*r*e + b*r*e*
log(c) + b*n*e*log(f) + 2*b*e*log(c)*log(f) + 2*b*d*n*log(x) + 2*a*r*e*log(x) + b*d*n + a*r*e + 2*b*d*log(c) +
2*a*e*log(f) + 2*a*d)/x^2

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maple [C]  time = 0.34, size = 1442, normalized size = 17.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*x^n)+a)*(d+e*ln(f*x^r))/x^3,x)

[Out]

-1/4*e*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-I*Pi*b*csgn(I*c*x^n)^3+I
*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2+2*b*ln(c)+b*n+2*b*ln(x^n)+2*a)/x^2*ln(x^r)-1/8*(2*b*d*n+2*a*e*r+2*b*e*r*ln(x^n
)+4*b*e*ln(f)*ln(x^n)+2*b*e*n*ln(f)+2*b*e*r*ln(c)+4*b*e*ln(c)*ln(f)+4*b*d*ln(x^n)+2*I*Pi*b*d*csgn(I*x^n)*csgn(
I*c*x^n)^2+4*a*e*ln(f)+4*a*d+2*b*e*n*r+4*b*d*ln(c)-Pi^2*b*e*csgn(I*c)*csgn(I*f)*csgn(I*x^n)*csgn(I*x^r)*csgn(I
*c*x^n)*csgn(I*f*x^r)-Pi^2*b*e*csgn(I*c)*csgn(I*f)*csgn(I*c*x^n)^2*csgn(I*f*x^r)^2-Pi^2*b*e*csgn(I*c)*csgn(I*x
^r)*csgn(I*c*x^n)^2*csgn(I*f*x^r)^2+2*I*Pi*a*e*csgn(I*f)*csgn(I*f*x^r)^2+2*I*Pi*a*e*csgn(I*x^r)*csgn(I*f*x^r)^
2-2*I*Pi*b*e*csgn(I*f*x^r)^3*ln(x^n)+I*Pi*b*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi*b*e*r*csgn(I*c*x^n)^2*csgn(I*
c)+I*n*Pi*b*e*csgn(I*f)*csgn(I*f*x^r)^2+I*n*Pi*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2-I*Pi*b*e*r*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)+2*I*Pi*b*d*csgn(I*c)*csgn(I*c*x^n)^2+Pi^2*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*f*x^r)^3-2*I*
Pi*b*e*csgn(I*f)*csgn(I*x^r)*csgn(I*f*x^r)*ln(c)-2*I*Pi*b*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*ln(f)-2*I*Pi*b
*d*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+Pi^2*b*e*csgn(I*f)*csgn(I*c*x^n)^3*csgn(I*f*x^r)^2+Pi^2*b*e*csgn(I*x^r)
*csgn(I*c*x^n)^3*csgn(I*f*x^r)^2+Pi^2*b*e*csgn(I*c)*csgn(I*c*x^n)^2*csgn(I*f*x^r)^3-2*I*Pi*b*e*csgn(I*f*x^r)^3
*ln(c)-2*I*Pi*b*e*csgn(I*c*x^n)^3*ln(f)-I*Pi*b*e*r*csgn(I*c*x^n)^3+2*I*Pi*b*e*csgn(I*f)*csgn(I*f*x^r)^2*ln(c)-
Pi^2*b*e*csgn(I*f)*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*f*x^r)^2-2*I*Pi*b*d*csgn(I*c*x^n)^3-2*I*Pi*b*e*csgn(I*f)
*csgn(I*x^r)*csgn(I*f*x^r)*ln(x^n)+Pi^2*b*e*csgn(I*f)*csgn(I*x^n)*csgn(I*x^r)*csgn(I*c*x^n)^2*csgn(I*f*x^r)+Pi
^2*b*e*csgn(I*c)*csgn(I*f)*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*f*x^r)^2+2*I*Pi*b*e*csgn(I*f)*csgn(I*f*x^r)^2*ln(x
^n)+2*I*Pi*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2*ln(x^n)-Pi^2*b*e*csgn(I*c*x^n)^3*csgn(I*f*x^r)^3-2*I*Pi*a*e*csgn(I*
f*x^r)^3+Pi^2*b*e*csgn(I*c)*csgn(I*x^n)*csgn(I*x^r)*csgn(I*c*x^n)*csgn(I*f*x^r)^2+Pi^2*b*e*csgn(I*c)*csgn(I*f)
*csgn(I*x^r)*csgn(I*c*x^n)^2*csgn(I*f*x^r)-I*n*Pi*b*e*csgn(I*f)*csgn(I*x^r)*csgn(I*f*x^r)-Pi^2*b*e*csgn(I*f)*c
sgn(I*x^r)*csgn(I*c*x^n)^3*csgn(I*f*x^r)-I*n*Pi*b*e*csgn(I*f*x^r)^3+2*I*Pi*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2*ln(
c)+2*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*ln(f)+2*I*Pi*b*e*csgn(I*c)*csgn(I*c*x^n)^2*ln(f)-2*I*Pi*a*e*csgn(I*f
)*csgn(I*x^r)*csgn(I*f*x^r)-Pi^2*b*e*csgn(I*x^n)*csgn(I*x^r)*csgn(I*c*x^n)^2*csgn(I*f*x^r)^2-Pi^2*b*e*csgn(I*c
)*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*f*x^r)^3)/x^2

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maxima [A]  time = 0.75, size = 93, normalized size = 1.12 \[ -\frac {1}{4} \, b e {\left (\frac {r}{x^{2}} + \frac {2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} \log \left (c x^{n}\right ) - \frac {b e n {\left (r + \log \relax (f) + \log \left (x^{r}\right )\right )}}{4 \, x^{2}} - \frac {b d n}{4 \, x^{2}} - \frac {a e r}{4 \, x^{2}} - \frac {b d \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a e \log \left (f x^{r}\right )}{2 \, x^{2}} - \frac {a d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^3,x, algorithm="maxima")

[Out]

-1/4*b*e*(r/x^2 + 2*log(f*x^r)/x^2)*log(c*x^n) - 1/4*b*e*n*(r + log(f) + log(x^r))/x^2 - 1/4*b*d*n/x^2 - 1/4*a
*e*r/x^2 - 1/2*b*d*log(c*x^n)/x^2 - 1/2*a*e*log(f*x^r)/x^2 - 1/2*a*d/x^2

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mupad [B]  time = 3.94, size = 83, normalized size = 1.00 \[ -\ln \left (f\,x^r\right )\,\left (\frac {a\,e}{2\,x^2}+\frac {b\,e\,n}{4\,x^2}+\frac {b\,e\,\ln \left (c\,x^n\right )}{2\,x^2}\right )-\frac {\frac {a\,d}{2}+\frac {b\,d\,n}{4}+\frac {a\,e\,r}{4}+\frac {b\,e\,n\,r}{4}}{x^2}-\frac {b\,\ln \left (c\,x^n\right )\,\left (2\,d+e\,r\right )}{4\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*log(f*x^r))*(a + b*log(c*x^n)))/x^3,x)

[Out]

- log(f*x^r)*((a*e)/(2*x^2) + (b*e*n)/(4*x^2) + (b*e*log(c*x^n))/(2*x^2)) - ((a*d)/2 + (b*d*n)/4 + (a*e*r)/4 +
(b*e*n*r)/4)/x^2 - (b*log(c*x^n)*(2*d + e*r))/(4*x^2)

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sympy [B]  time = 7.58, size = 201, normalized size = 2.42 \[ - \frac {a d}{2 x^{2}} - \frac {a e r \log {\relax (x )}}{2 x^{2}} - \frac {a e r}{4 x^{2}} - \frac {a e \log {\relax (f )}}{2 x^{2}} - \frac {b d n \log {\relax (x )}}{2 x^{2}} - \frac {b d n}{4 x^{2}} - \frac {b d \log {\relax (c )}}{2 x^{2}} - \frac {b e n r \log {\relax (x )}^{2}}{2 x^{2}} - \frac {b e n r \log {\relax (x )}}{2 x^{2}} - \frac {b e n r}{4 x^{2}} - \frac {b e n \log {\relax (f )} \log {\relax (x )}}{2 x^{2}} - \frac {b e n \log {\relax (f )}}{4 x^{2}} - \frac {b e r \log {\relax (c )} \log {\relax (x )}}{2 x^{2}} - \frac {b e r \log {\relax (c )}}{4 x^{2}} - \frac {b e \log {\relax (c )} \log {\relax (f )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*(d+e*ln(f*x**r))/x**3,x)

[Out]

-a*d/(2*x**2) - a*e*r*log(x)/(2*x**2) - a*e*r/(4*x**2) - a*e*log(f)/(2*x**2) - b*d*n*log(x)/(2*x**2) - b*d*n/(
4*x**2) - b*d*log(c)/(2*x**2) - b*e*n*r*log(x)**2/(2*x**2) - b*e*n*r*log(x)/(2*x**2) - b*e*n*r/(4*x**2) - b*e*
n*log(f)*log(x)/(2*x**2) - b*e*n*log(f)/(4*x**2) - b*e*r*log(c)*log(x)/(2*x**2) - b*e*r*log(c)/(4*x**2) - b*e*
log(c)*log(f)/(2*x**2)

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