3.372 \(\int \frac{(d+e x^r) (a+b \log (c x^n))}{x^5} \, dx\)

Optimal. Leaf size=71 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac{e x^{r-4} \left (a+b \log \left (c x^n\right )\right )}{4-r}-\frac{b d n}{16 x^4}-\frac{b e n x^{r-4}}{(4-r)^2} \]

[Out]

-(b*d*n)/(16*x^4) - (b*e*n*x^(-4 + r))/(4 - r)^2 - (d*(a + b*Log[c*x^n]))/(4*x^4) - (e*x^(-4 + r)*(a + b*Log[c
*x^n]))/(4 - r)

________________________________________________________________________________________

Rubi [A]  time = 0.0741268, antiderivative size = 63, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {14, 2334} \[ -\frac{1}{4} \left (\frac{d}{x^4}+\frac{4 e x^{r-4}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{16 x^4}-\frac{b e n x^{r-4}}{(4-r)^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)*(a + b*Log[c*x^n]))/x^5,x]

[Out]

-(b*d*n)/(16*x^4) - (b*e*n*x^(-4 + r))/(4 - r)^2 - ((d/x^4 + (4*e*x^(-4 + r))/(4 - r))*(a + b*Log[c*x^n]))/4

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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])

Rubi steps

\begin{align*} \int \frac{\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx &=-\frac{1}{4} \left (\frac{d}{x^4}+\frac{4 e x^{-4+r}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{d}{4 x^5}+\frac{e x^{-5+r}}{-4+r}\right ) \, dx\\ &=-\frac{b d n}{16 x^4}-\frac{b e n x^{-4+r}}{(4-r)^2}-\frac{1}{4} \left (\frac{d}{x^4}+\frac{4 e x^{-4+r}}{4-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.111571, size = 72, normalized size = 1.01 \[ -\frac{4 a (r-4) \left (d (r-4)-4 e x^r\right )+4 b (r-4) \log \left (c x^n\right ) \left (d (r-4)-4 e x^r\right )+b n \left (d (r-4)^2+16 e x^r\right )}{16 (r-4)^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)*(a + b*Log[c*x^n]))/x^5,x]

[Out]

-(4*a*(-4 + r)*(d*(-4 + r) - 4*e*x^r) + b*n*(d*(-4 + r)^2 + 16*e*x^r) + 4*b*(-4 + r)*(d*(-4 + r) - 4*e*x^r)*Lo
g[c*x^n])/(16*(-4 + r)^2*x^4)

________________________________________________________________________________________

Maple [C]  time = 0.164, size = 613, normalized size = 8.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)*(a+b*ln(c*x^n))/x^5,x)

[Out]

-1/4*b*(d*r-4*e*x^r-4*d)/(-4+r)/x^4*ln(x^n)-1/16*(64*a*d-16*x^r*a*e*r+16*x^r*b*e*n+8*I*Pi*b*e*csgn(I*x^n)*csgn
(I*c*x^n)*csgn(I*c)*x^r*r-8*b*d*n*r+16*b*d*n+64*x^r*a*e-32*ln(c)*b*d*r+4*ln(c)*b*d*r^2+32*I*Pi*b*d*csgn(I*x^n)
*csgn(I*c*x^n)^2-32*I*Pi*b*e*csgn(I*c*x^n)^3*x^r+32*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)-2*I*Pi*b*d*r^2*csgn(I*c
*x^n)^3-16*ln(c)*b*e*x^r*r+64*ln(c)*b*e*x^r-8*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r*r+4*a*d*r^2-8*I*Pi*b*e*cs
gn(I*x^n)*csgn(I*c*x^n)^2*x^r*r+16*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*r+64*ln(c)*b*d-32*a*d*r+b*d*n*
r^2+16*I*Pi*b*d*csgn(I*c*x^n)^3*r-16*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2*r-16*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I
*c)*r-2*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-32*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-3
2*I*Pi*b*d*csgn(I*c*x^n)^3+8*I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r+2*I*Pi*b*d*r^2*csgn(I*c*x^n)^2*csgn(I*c)+2*I*Pi*b*
d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+32*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+32*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x
^n)^2*x^r-32*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/(-4+r)^2/x^4

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.33431, size = 360, normalized size = 5.07 \begin{align*} -\frac{16 \, b d n +{\left (b d n + 4 \, a d\right )} r^{2} + 64 \, a d - 8 \,{\left (b d n + 4 \, a d\right )} r + 16 \,{\left (b e n - a e r + 4 \, a e -{\left (b e r - 4 \, b e\right )} \log \left (c\right ) -{\left (b e n r - 4 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 4 \,{\left (b d r^{2} - 8 \, b d r + 16 \, b d\right )} \log \left (c\right ) + 4 \,{\left (b d n r^{2} - 8 \, b d n r + 16 \, b d n\right )} \log \left (x\right )}{16 \,{\left (r^{2} - 8 \, r + 16\right )} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^5,x, algorithm="fricas")

[Out]

-1/16*(16*b*d*n + (b*d*n + 4*a*d)*r^2 + 64*a*d - 8*(b*d*n + 4*a*d)*r + 16*(b*e*n - a*e*r + 4*a*e - (b*e*r - 4*
b*e)*log(c) - (b*e*n*r - 4*b*e*n)*log(x))*x^r + 4*(b*d*r^2 - 8*b*d*r + 16*b*d)*log(c) + 4*(b*d*n*r^2 - 8*b*d*n
*r + 16*b*d*n)*log(x))/((r^2 - 8*r + 16)*x^4)

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x**r)*(a+b*ln(c*x**n))/x**5,x)

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [B]  time = 1.33008, size = 536, normalized size = 7.55 \begin{align*} -\frac{b d n r^{2} \log \left (x\right )}{4 \,{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac{b n r x^{r} e \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{b d n r^{2}}{16 \,{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{b d r^{2} \log \left (c\right )}{4 \,{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac{b r x^{r} e \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac{2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{4 \, b n x^{r} e \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac{b d n r}{2 \,{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{a d r^{2}}{4 \,{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{b n x^{r} e}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac{a r x^{r} e}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac{2 \, b d r \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{4 \, b x^{r} e \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{4 \, b d n \log \left (x\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{b d n}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} + \frac{2 \, a d r}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{4 \, a x^{r} e}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{4 \, b d \log \left (c\right )}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} - \frac{4 \, a d}{{\left (r^{2} - 8 \, r + 16\right )} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)*(a+b*log(c*x^n))/x^5,x, algorithm="giac")

[Out]

-1/4*b*d*n*r^2*log(x)/((r^2 - 8*r + 16)*x^4) + b*n*r*x^r*e*log(x)/((r^2 - 8*r + 16)*x^4) - 1/16*b*d*n*r^2/((r^
2 - 8*r + 16)*x^4) - 1/4*b*d*r^2*log(c)/((r^2 - 8*r + 16)*x^4) + b*r*x^r*e*log(c)/((r^2 - 8*r + 16)*x^4) + 2*b
*d*n*r*log(x)/((r^2 - 8*r + 16)*x^4) - 4*b*n*x^r*e*log(x)/((r^2 - 8*r + 16)*x^4) + 1/2*b*d*n*r/((r^2 - 8*r + 1
6)*x^4) - 1/4*a*d*r^2/((r^2 - 8*r + 16)*x^4) - b*n*x^r*e/((r^2 - 8*r + 16)*x^4) + a*r*x^r*e/((r^2 - 8*r + 16)*
x^4) + 2*b*d*r*log(c)/((r^2 - 8*r + 16)*x^4) - 4*b*x^r*e*log(c)/((r^2 - 8*r + 16)*x^4) - 4*b*d*n*log(x)/((r^2
- 8*r + 16)*x^4) - b*d*n/((r^2 - 8*r + 16)*x^4) + 2*a*d*r/((r^2 - 8*r + 16)*x^4) - 4*a*x^r*e/((r^2 - 8*r + 16)
*x^4) - 4*b*d*log(c)/((r^2 - 8*r + 16)*x^4) - 4*a*d/((r^2 - 8*r + 16)*x^4)