3.2.62 \(\int \frac {(a+b \log (c x^n)) (d+e \log (f x^r))}{x^4} \, dx\) [162]

Optimal. Leaf size=83 \[ -\frac {b e n r}{27 x^3}-\frac {e r \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{27 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3} \]

[Out]

-1/27*b*e*n*r/x^3-1/27*e*r*(3*a+b*n+3*b*ln(c*x^n))/x^3-1/9*b*n*(d+e*ln(f*x^r))/x^3-1/3*(a+b*ln(c*x^n))*(d+e*ln
(f*x^r))/x^3

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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 )}{3 x^3}-\frac {e r \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{27 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {b e n r}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x^4,x]

[Out]

-1/27*(b*e*n*r)/x^3 - (e*r*(3*a + b*n + 3*b*Log[c*x^n]))/(27*x^3) - (b*n*(d + e*Log[f*x^r]))/(9*x^3) - ((a + b
*Log[c*x^n])*(d + e*Log[f*x^r]))/(3*x^3)

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^4} \, dx &=-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-(e r) \int \frac {-3 a \left (1+\frac {b n}{3 a}\right )-3 b \log \left (c x^n\right )}{9 x^4} \, dx\\ &=-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {1}{9} (e r) \int \frac {-3 a \left (1+\frac {b n}{3 a}\right )-3 b \log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac {b e n r}{27 x^3}-\frac {e r \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{27 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 69, normalized size = 0.83 \begin {gather*} -\frac {9 a d+3 b d n+3 a e r+2 b e n r+3 e (3 a+b n) \log \left (f x^r\right )+3 b \log \left (c x^n\right ) \left (3 d+e r+3 e \log \left (f x^r\right )\right )}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x^4,x]

[Out]

-1/27*(9*a*d + 3*b*d*n + 3*a*e*r + 2*b*e*n*r + 3*e*(3*a + b*n)*Log[f*x^r] + 3*b*Log[c*x^n]*(3*d + e*r + 3*e*Lo
g[f*x^r]))/x^3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 1451, normalized size = 17.48

method	result	size
risch	\(\text {Expression too large to display}\)	\(1451\)

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^4,x,method=_RETURNVERBOSE)

[Out]

-1/18*e*(-3*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+3*I*b*Pi*csgn(I*x^n)
*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*c*x^n)^3+6*b*ln(c)+2*b*n+6*ln(x^n)*b+6*a)/x^3*ln(x^r)-1/108*(12*a*e*r+12*b*d*
n+36*a*d-6*I*Pi*b*e*r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-6*I*n*Pi*b*e*csgn(I*f)*csgn(I*x^r)*csgn(I*f*x^r)-18*
I*Pi*b*d*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+36*d*b*ln(c)+9*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)^2-18*I*Pi*b*d*csgn(I*c*x^n)^3+6*I*n*Pi*b*e*csgn(I*f)*csgn(I*f*x^r)^2+18*I*Pi*b*e*csgn(I*x
^r)*csgn(I*f*x^r)^2*ln(x^n)+36*a*ln(f)*e+36*b*ln(c)*ln(f)*e+12*b*ln(c)*e*r+9*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)+9*Pi^2*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^
n)+9*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+12*n*ln(f)*b*e+36*ln(f)*b*e*ln(x^n
)+12*b*e*r*ln(x^n)+18*I*Pi*ln(f)*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2+18*I*Pi*ln(c)*b*e*csgn(I*f)*csgn(I*f*x^r)^2+6
*I*n*Pi*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2+8*b*e*n*r+36*b*d*ln(x^n)-18*I*Pi*ln(f)*b*e*csgn(I*c)*csgn(I*x^n)*csgn(
I*c*x^n)-9*Pi^2*b*e*csgn(I*f*x^r)^3*csgn(I*c*x^n)^3-9*Pi^2*b*e*csgn(I*f)*csgn(I*x^r)*csgn(I*f*x^r)*csgn(I*c)*c
sgn(I*x^n)*csgn(I*c*x^n)-18*I*Pi*a*e*csgn(I*f*x^r)^3+9*Pi^2*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2*csgn(I*c*x^n)^3-18
*I*Pi*ln(c)*b*e*csgn(I*f*x^r)^3+18*I*Pi*a*e*csgn(I*f)*csgn(I*f*x^r)^2+18*I*Pi*a*e*csgn(I*x^r)*csgn(I*f*x^r)^2-
18*I*Pi*ln(c)*b*e*csgn(I*f)*csgn(I*x^r)*csgn(I*f*x^r)-18*I*Pi*a*e*csgn(I*f)*csgn(I*x^r)*csgn(I*f*x^r)+18*I*Pi*
ln(c)*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2-9*Pi^2*b*e*csgn(I*f)*csgn(I*f*x^r)^2*csgn(I*c)*csgn(I*c*x^n)^2-9*Pi^2*b*
e*csgn(I*f)*csgn(I*f*x^r)^2*csgn(I*x^n)*csgn(I*c*x^n)^2-9*Pi^2*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2*csgn(I*c)*csgn(
I*c*x^n)^2-9*Pi^2*b*e*csgn(I*x^r)*csgn(I*f*x^r)^2*csgn(I*x^n)*csgn(I*c*x^n)^2-18*I*Pi*b*e*csgn(I*f)*csgn(I*x^r
)*csgn(I*f*x^r)*ln(x^n)-6*I*Pi*b*e*r*csgn(I*c*x^n)^3-18*I*Pi*b*e*csgn(I*f*x^r)^3*ln(x^n)+9*Pi^2*b*e*csgn(I*f*x
^r)^3*csgn(I*c)*csgn(I*c*x^n)^2+9*Pi^2*b*e*csgn(I*f*x^r)^3*csgn(I*x^n)*csgn(I*c*x^n)^2+9*Pi^2*b*e*csgn(I*f)*cs
gn(I*f*x^r)^2*csgn(I*c*x^n)^3-9*Pi^2*b*e*csgn(I*f*x^r)^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-6*I*n*Pi*b*e*csgn
(I*f*x^r)^3+18*I*Pi*b*d*csgn(I*c)*csgn(I*c*x^n)^2+18*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2-18*I*Pi*ln(f)*b*e*cs
gn(I*c*x^n)^3-9*Pi^2*b*e*csgn(I*f)*csgn(I*x^r)*csgn(I*f*x^r)*csgn(I*c*x^n)^3+18*I*Pi*b*e*csgn(I*f)*csgn(I*f*x^
r)^2*ln(x^n)+6*I*Pi*b*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2+6*I*Pi*b*e*r*csgn(I*c)*csgn(I*c*x^n)^2+18*I*Pi*ln(f)*b*e
*csgn(I*c)*csgn(I*c*x^n)^2)/x^3

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Maxima [A]
time = 0.30, size = 103, normalized size = 1.24 \begin {gather*} -\frac {1}{9} \, b {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} e \log \left (c x^{n}\right ) - \frac {b n {\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )} e}{27 \, x^{3}} - \frac {b d n}{9 \, x^{3}} - \frac {a r e}{9 \, x^{3}} - \frac {b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a e \log \left (f x^{r}\right )}{3 \, x^{3}} - \frac {a d}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

-1/9*b*(r/x^3 + 3*log(f*x^r)/x^3)*e*log(c*x^n) - 1/27*b*n*(2*r + 3*log(f) + 3*log(x^r))*e/x^3 - 1/9*b*d*n/x^3
- 1/9*a*r*e/x^3 - 1/3*b*d*log(c*x^n)/x^3 - 1/3*a*e*log(f*x^r)/x^3 - 1/3*a*d/x^3

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Fricas [A]
time = 0.38, size = 112, normalized size = 1.35 \begin {gather*} -\frac {9 \, b n r e \log \left (x\right )^{2} + 3 \, b d n + {\left (2 \, b n + 3 \, a\right )} r e + 9 \, a d + 3 \, {\left (b r e + 3 \, b d\right )} \log \left (c\right ) + 3 \, {\left (3 \, b e \log \left (c\right ) + {\left (b n + 3 \, a\right )} e\right )} \log \left (f\right ) + 3 \, {\left (3 \, b r e \log \left (c\right ) + 3 \, b n e \log \left (f\right ) + 3 \, b d n + {\left (2 \, b n + 3 \, a\right )} r e\right )} \log \left (x\right )}{27 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

-1/27*(9*b*n*r*e*log(x)^2 + 3*b*d*n + (2*b*n + 3*a)*r*e + 9*a*d + 3*(b*r*e + 3*b*d)*log(c) + 3*(3*b*e*log(c) +
(b*n + 3*a)*e)*log(f) + 3*(3*b*r*e*log(c) + 3*b*n*e*log(f) + 3*b*d*n + (2*b*n + 3*a)*r*e)*log(x))/x^3

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Sympy [A]
time = 2.66, size = 129, normalized size = 1.55 \begin {gather*} - \frac {a d}{3 x^{3}} - \frac {a e r}{9 x^{3}} - \frac {a e \log {\left (f x^{r} \right )}}{3 x^{3}} - \frac {b d n}{9 x^{3}} - \frac {b d \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {2 b e n r}{27 x^{3}} - \frac {b e n \log {\left (f x^{r} \right )}}{9 x^{3}} - \frac {b e r \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{3 x^{3}} \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**4,x)

[Out]

-a*d/(3*x**3) - a*e*r/(9*x**3) - a*e*log(f*x**r)/(3*x**3) - b*d*n/(9*x**3) - b*d*log(c*x**n)/(3*x**3) - 2*b*e*
n*r/(27*x**3) - b*e*n*log(f*x**r)/(9*x**3) - b*e*r*log(c*x**n)/(9*x**3) - b*e*log(c*x**n)*log(f*x**r)/(3*x**3)

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Giac [A]
time = 6.28, size = 121, normalized size = 1.46 \begin {gather*} -\frac {9 \, b n r e \log \left (x\right )^{2} + 6 \, b n r e \log \left (x\right ) + 9 \, b r e \log \left (c\right ) \log \left (x\right ) + 9 \, b n e \log \left (f\right ) \log \left (x\right ) + 2 \, b n r e + 3 \, b r e \log \left (c\right ) + 3 \, b n e \log \left (f\right ) + 9 \, b e \log \left (c\right ) \log \left (f\right ) + 9 \, b d n \log \left (x\right ) + 9 \, a r e \log \left (x\right ) + 3 \, b d n + 3 \, a r e + 9 \, b d \log \left (c\right ) + 9 \, a e \log \left (f\right ) + 9 \, a d}{27 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

-1/27*(9*b*n*r*e*log(x)^2 + 6*b*n*r*e*log(x) + 9*b*r*e*log(c)*log(x) + 9*b*n*e*log(f)*log(x) + 2*b*n*r*e + 3*b
*r*e*log(c) + 3*b*n*e*log(f) + 9*b*e*log(c)*log(f) + 9*b*d*n*log(x) + 9*a*r*e*log(x) + 3*b*d*n + 3*a*r*e + 9*b
*d*log(c) + 9*a*e*log(f) + 9*a*d)/x^3

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Mupad [B]
time = 3.93, size = 83, normalized size = 1.00 \begin {gather*} -\ln \left (f\,x^r\right )\,\left (\frac {a\,e}{3\,x^3}+\frac {b\,e\,n}{9\,x^3}+\frac {b\,e\,\ln \left (c\,x^n\right )}{3\,x^3}\right )-\frac {\frac {a\,d}{3}+\frac {b\,d\,n}{9}+\frac {a\,e\,r}{9}+\frac {2\,b\,e\,n\,r}{27}}{x^3}-\frac {b\,\ln \left (c\,x^n\right )\,\left (3\,d+e\,r\right )}{9\,x^3} \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^4,x)

[Out]

- log(f*x^r)*((a*e)/(3*x^3) + (b*e*n)/(9*x^3) + (b*e*log(c*x^n))/(3*x^3)) - ((a*d)/3 + (b*d*n)/9 + (a*e*r)/9 +
(2*b*e*n*r)/27)/x^3 - (b*log(c*x^n)*(3*d + e*r))/(9*x^3)

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