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

Optimal. Leaf size=57 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{b d n}{9 x^3}-\frac{b e n}{4 x^2} \]

[Out]

-(b*d*n)/(9*x^3) - (b*e*n)/(4*x^2) - (d*(a + b*Log[c*x^n]))/(3*x^3) - (e*(a + b*Log[c*x^n]))/(2*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0447902, antiderivative size = 48, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {43, 2334, 12} \[ -\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{9 x^3}-\frac{b e n}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*n)/(9*x^3) - (b*e*n)/(4*x^2) - (((2*d)/x^3 + (3*e)/x^2)*(a + b*Log[c*x^n]))/6

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

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{(d+e x) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-2 d-3 e x}{6 x^4} \, dx\\ &=-\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int \frac{-2 d-3 e x}{x^4} \, dx\\ &=-\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int \left (-\frac{2 d}{x^4}-\frac{3 e}{x^3}\right ) \, dx\\ &=-\frac{b d n}{9 x^3}-\frac{b e n}{4 x^2}-\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0229789, size = 47, normalized size = 0.82 \[ -\frac{6 a (2 d+3 e x)+6 b (2 d+3 e x) \log \left (c x^n\right )+b n (4 d+9 e x)}{36 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*a*(2*d + 3*e*x) + b*n*(4*d + 9*e*x) + 6*b*(2*d + 3*e*x)*Log[c*x^n])/(36*x^3)

________________________________________________________________________________________

Maple [C]  time = 0.11, size = 235, normalized size = 4.1 \begin{align*} -{\frac{b \left ( 3\,ex+2\,d \right ) \ln \left ({x}^{n} \right ) }{6\,{x}^{3}}}-{\frac{9\,i\pi \,bex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-9\,i\pi \,bex{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -9\,i\pi \,bex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+9\,i\pi \,bex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +18\,\ln \left ( c \right ) bex+9\,benx+18\,aex+6\,i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-6\,i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -6\,i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+6\,i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +12\,\ln \left ( c \right ) bd+4\,bdn+12\,ad}{36\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(a+b*ln(c*x^n))/x^4,x)

[Out]

-1/6*b*(3*e*x+2*d)/x^3*ln(x^n)-1/36*(9*I*Pi*b*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-9*I*Pi*b*e*x*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)-9*I*Pi*b*e*x*csgn(I*c*x^n)^3+9*I*Pi*b*e*x*csgn(I*c*x^n)^2*csgn(I*c)+18*ln(c)*b*e*x+9*b*e*n*x+
18*a*e*x+6*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2-6*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-6*I*Pi*b*d*csgn
(I*c*x^n)^3+6*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)+12*ln(c)*b*d+4*b*d*n+12*a*d)/x^3

________________________________________________________________________________________

Maxima [A]  time = 1.15926, size = 77, normalized size = 1.35 \begin{align*} -\frac{b e n}{4 \, x^{2}} - \frac{b e \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{b d n}{9 \, x^{3}} - \frac{a e}{2 \, x^{2}} - \frac{b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a d}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))/x^4,x, algorithm="maxima")

[Out]

-1/4*b*e*n/x^2 - 1/2*b*e*log(c*x^n)/x^2 - 1/9*b*d*n/x^3 - 1/2*a*e/x^2 - 1/3*b*d*log(c*x^n)/x^3 - 1/3*a*d/x^3

________________________________________________________________________________________

Fricas [A]  time = 1.04731, size = 154, normalized size = 2.7 \begin{align*} -\frac{4 \, b d n + 12 \, a d + 9 \,{\left (b e n + 2 \, a e\right )} x + 6 \,{\left (3 \, b e x + 2 \, b d\right )} \log \left (c\right ) + 6 \,{\left (3 \, b e n x + 2 \, b d n\right )} \log \left (x\right )}{36 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))/x^4,x, algorithm="fricas")

[Out]

-1/36*(4*b*d*n + 12*a*d + 9*(b*e*n + 2*a*e)*x + 6*(3*b*e*x + 2*b*d)*log(c) + 6*(3*b*e*n*x + 2*b*d*n)*log(x))/x
^3

________________________________________________________________________________________

Sympy [A]  time = 2.55703, size = 88, normalized size = 1.54 \begin{align*} - \frac{a d}{3 x^{3}} - \frac{a e}{2 x^{2}} - \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 \log{\left (x \right )}}{2 x^{2}} - \frac{b e n}{4 x^{2}} - \frac{b e \log{\left (c \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*ln(c*x**n))/x**4,x)

[Out]

-a*d/(3*x**3) - a*e/(2*x**2) - b*d*n*log(x)/(3*x**3) - b*d*n/(9*x**3) - b*d*log(c)/(3*x**3) - b*e*n*log(x)/(2*
x**2) - b*e*n/(4*x**2) - b*e*log(c)/(2*x**2)

________________________________________________________________________________________

Giac [A]  time = 1.2936, size = 78, normalized size = 1.37 \begin{align*} -\frac{18 \, b n x e \log \left (x\right ) + 9 \, b n x e + 18 \, b x e \log \left (c\right ) + 12 \, b d n \log \left (x\right ) + 4 \, b d n + 18 \, a x e + 12 \, b d \log \left (c\right ) + 12 \, a d}{36 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))/x^4,x, algorithm="giac")

[Out]

-1/36*(18*b*n*x*e*log(x) + 9*b*n*x*e + 18*b*x*e*log(c) + 12*b*d*n*log(x) + 4*b*d*n + 18*a*x*e + 12*b*d*log(c)
+ 12*a*d)/x^3