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

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

[Out]

-(b*e*n*r)/(27*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)

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Rubi [A]  time = 0.0744238, antiderivative size = 83, normalized size of antiderivative = 1., 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} \[ -\frac{\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac{e r \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{27 x^3}-\frac{b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac{b e n r}{27 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*n*r)/(27*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 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])

Rule 12

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

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*}

Mathematica [A]  time = 0.0741346, size = 69, normalized size = 0.83 \[ -\frac{3 e (3 a+b n) \log \left (f x^r\right )+9 a d+3 a e r+3 b \log \left (c x^n\right ) \left (3 d+3 e \log \left (f x^r\right )+e r\right )+3 b d n+2 b e n r}{27 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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*Log[f*x
^r]))/(27*x^3)

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Maple [C]  time = 0.204, size = 1451, normalized size = 17.5 \begin{align*} \text{result too large to display} \end{align*}

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)

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

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Maxima [A]  time = 1.1751, size = 134, normalized size = 1.61 \begin{align*} -\frac{1}{9} \, b e{\left (\frac{r}{x^{3}} + \frac{3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} \log \left (c x^{n}\right ) - \frac{b e n{\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )}}{27 \, x^{3}} - \frac{b d n}{9 \, x^{3}} - \frac{a e r}{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{align*}

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*e*(r/x^3 + 3*log(f*x^r)/x^3)*log(c*x^n) - 1/27*b*e*n*(2*r + 3*log(f) + 3*log(x^r))/x^3 - 1/9*b*d*n/x^3
- 1/9*a*e*r/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 = 0.974325, size = 294, normalized size = 3.54 \begin{align*} -\frac{9 \, b e n r \log \left (x\right )^{2} + 3 \, b d n + 9 \, a d +{\left (2 \, b e n + 3 \, a e\right )} r + 3 \,{\left (b e r + 3 \, b d\right )} \log \left (c\right ) + 3 \,{\left (b e n + 3 \, b e \log \left (c\right ) + 3 \, a e\right )} \log \left (f\right ) + 3 \,{\left (3 \, b e r \log \left (c\right ) + 3 \, b e n \log \left (f\right ) + 3 \, b d n +{\left (2 \, b e n + 3 \, a e\right )} r\right )} \log \left (x\right )}{27 \, x^{3}} \end{align*}

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

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Sympy [B]  time = 32.1929, size = 204, normalized size = 2.46 \begin{align*} - \frac{a d}{3 x^{3}} - \frac{a e r \log{\left (x \right )}}{3 x^{3}} - \frac{a e r}{9 x^{3}} - \frac{a e \log{\left (f \right )}}{3 x^{3}} - \frac{b d n \log{\left (x \right )}}{3 x^{3}} - \frac{b d n}{9 x^{3}} - \frac{b d \log{\left (c \right )}}{3 x^{3}} - \frac{b e n r \log{\left (x \right )}^{2}}{3 x^{3}} - \frac{2 b e n r \log{\left (x \right )}}{9 x^{3}} - \frac{2 b e n r}{27 x^{3}} - \frac{b e n \log{\left (f \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{b e n \log{\left (f \right )}}{9 x^{3}} - \frac{b e r \log{\left (c \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{b e r \log{\left (c \right )}}{9 x^{3}} - \frac{b e \log{\left (c \right )} \log{\left (f \right )}}{3 x^{3}} \end{align*}

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

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Giac [A]  time = 1.18625, size = 163, normalized size = 1.96 \begin{align*} -\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{align*}

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