3.2.61 \(\int \frac {(a+b \log (c x^n)) (d+e \log (f x^r))}{x^3} \, dx\) [161]

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

[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 = 0.05, 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 = {2341, 2413, 12} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(b*e*n*r)/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 2341

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 2413

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 = 0.04, size = 64, normalized size = 0.77 \begin {gather*} -\frac {2 a d+b d n+a e r+b e n r+e (2 a+b n) \log \left (f x^r\right )+b \log \left (c x^n\right ) \left (2 d+e r+2 e \log \left (f x^r\right )\right )}{4 x^2} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 1442, normalized size = 17.37

method	result	size
risch	\(\text {Expression too large to display}\)	\(1442\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 97, normalized size = 1.17 \begin {gather*} -\frac {1}{4} \, b {\left (\frac {r}{x^{2}} + \frac {2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} e \log \left (c x^{n}\right ) - \frac {b n {\left (r + \log \left (f\right ) + \log \left (x^{r}\right )\right )} e}{4 \, x^{2}} - \frac {b d n}{4 \, x^{2}} - \frac {a r e}{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}} \end {gather*}

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*(r/x^2 + 2*log(f*x^r)/x^2)*e*log(c*x^n) - 1/4*b*n*(r + log(f) + log(x^r))*e/x^2 - 1/4*b*d*n/x^2 - 1/4*a
*r*e/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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Fricas [A]
time = 0.36, size = 100, normalized size = 1.20 \begin {gather*} -\frac {2 \, b n r e \log \left (x\right )^{2} + b d n + {\left (b n + a\right )} r e + 2 \, a d + {\left (b r e + 2 \, b d\right )} \log \left (c\right ) + {\left (2 \, b e \log \left (c\right ) + {\left (b n + 2 \, a\right )} e\right )} \log \left (f\right ) + 2 \, {\left (b r e \log \left (c\right ) + b n e \log \left (f\right ) + b d n + {\left (b n + a\right )} r e\right )} \log \left (x\right )}{4 \, x^{2}} \end {gather*}

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

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Sympy [A]
time = 1.23, size = 128, normalized size = 1.54 \begin {gather*} - \frac {a d}{2 x^{2}} - \frac {a e r}{4 x^{2}} - \frac {a e \log {\left (f x^{r} \right )}}{2 x^{2}} - \frac {b d n}{4 x^{2}} - \frac {b d \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b e n r}{4 x^{2}} - \frac {b e n \log {\left (f x^{r} \right )}}{4 x^{2}} - \frac {b e r \log {\left (c x^{n} \right )}}{4 x^{2}} - \frac {b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{2 x^{2}} \end {gather*}

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

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Giac [A]
time = 3.86, size = 116, normalized size = 1.40 \begin {gather*} -\frac {2 \, b n r e \log \left (x\right )^{2} + 2 \, b n r e \log \left (x\right ) + 2 \, b r e \log \left (c\right ) \log \left (x\right ) + 2 \, b n e \log \left (f\right ) \log \left (x\right ) + b n r e + b r e \log \left (c\right ) + b n e \log \left (f\right ) + 2 \, b e \log \left (c\right ) \log \left (f\right ) + 2 \, b d n \log \left (x\right ) + 2 \, a r e \log \left (x\right ) + b d n + a r e + 2 \, b d \log \left (c\right ) + 2 \, a e \log \left (f\right ) + 2 \, a d}{4 \, x^{2}} \end {gather*}

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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Mupad [B]
time = 3.94, size = 83, normalized size = 1.00 \begin {gather*} -\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} \end {gather*}

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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