3.371 \(\int \frac{(d+e x^r) (a+b \log (c x^n))}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0743005, 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} \[ -\frac{1}{2} \left (\frac{d}{x^2}+\frac{2 e x^{r-2}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{4 x^2}-\frac{b e n x^{r-2}}{(2-r)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.108623, size = 72, normalized size = 1.01 \[ -\frac{2 a (r-2) \left (d (r-2)-2 e x^r\right )+2 b (r-2) \log \left (c x^n\right ) \left (d (r-2)-2 e x^r\right )+b n \left (d (r-2)^2+4 e x^r\right )}{4 (r-2)^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.158, 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^3,x)

[Out]

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

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.4095, size = 351, normalized size = 4.94 \begin{align*} -\frac{4 \, b d n +{\left (b d n + 2 \, a d\right )} r^{2} + 8 \, a d - 4 \,{\left (b d n + 2 \, a d\right )} r + 4 \,{\left (b e n - a e r + 2 \, a e -{\left (b e r - 2 \, b e\right )} \log \left (c\right ) -{\left (b e n r - 2 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 2 \,{\left (b d r^{2} - 4 \, b d r + 4 \, b d\right )} \log \left (c\right ) + 2 \,{\left (b d n r^{2} - 4 \, b d n r + 4 \, b d n\right )} \log \left (x\right )}{4 \,{\left (r^{2} - 4 \, r + 4\right )} x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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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**3,x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.37193, size = 535, normalized size = 7.54 \begin{align*} -\frac{b d n r^{2} \log \left (x\right )}{2 \,{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac{b n r x^{r} e \log \left (x\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{b d n r^{2}}{4 \,{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{b d r^{2} \log \left (c\right )}{2 \,{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac{b r x^{r} e \log \left (c\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac{2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{2 \, b n x^{r} e \log \left (x\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac{b d n r}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{a d r^{2}}{2 \,{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{b n x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac{a r x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac{2 \, b d r \log \left (c\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{2 \, b x^{r} e \log \left (c\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{2 \, b d n \log \left (x\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{b d n}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} + \frac{2 \, a d r}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{2 \, a x^{r} e}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{2 \, b d \log \left (c\right )}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} - \frac{2 \, a d}{{\left (r^{2} - 4 \, r + 4\right )} x^{2}} \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^3,x, algorithm="giac")

[Out]

-1/2*b*d*n*r^2*log(x)/((r^2 - 4*r + 4)*x^2) + b*n*r*x^r*e*log(x)/((r^2 - 4*r + 4)*x^2) - 1/4*b*d*n*r^2/((r^2 -
4*r + 4)*x^2) - 1/2*b*d*r^2*log(c)/((r^2 - 4*r + 4)*x^2) + b*r*x^r*e*log(c)/((r^2 - 4*r + 4)*x^2) + 2*b*d*n*r
*log(x)/((r^2 - 4*r + 4)*x^2) - 2*b*n*x^r*e*log(x)/((r^2 - 4*r + 4)*x^2) + b*d*n*r/((r^2 - 4*r + 4)*x^2) - 1/2
*a*d*r^2/((r^2 - 4*r + 4)*x^2) - b*n*x^r*e/((r^2 - 4*r + 4)*x^2) + a*r*x^r*e/((r^2 - 4*r + 4)*x^2) + 2*b*d*r*l
og(c)/((r^2 - 4*r + 4)*x^2) - 2*b*x^r*e*log(c)/((r^2 - 4*r + 4)*x^2) - 2*b*d*n*log(x)/((r^2 - 4*r + 4)*x^2) -
b*d*n/((r^2 - 4*r + 4)*x^2) + 2*a*d*r/((r^2 - 4*r + 4)*x^2) - 2*a*x^r*e/((r^2 - 4*r + 4)*x^2) - 2*b*d*log(c)/(
(r^2 - 4*r + 4)*x^2) - 2*a*d/((r^2 - 4*r + 4)*x^2)