3.378 \(\int \frac{(d+e x^r) (a+b \log (c x^n))}{x^6} \, dx\)

Optimal. Leaf size=71 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac{b d n}{25 x^5}-\frac{b e n x^{r-5}}{(5-r)^2} \]

[Out]

-(b*d*n)/(25*x^5) - (b*e*n*x^(-5 + r))/(5 - r)^2 - (d*(a + b*Log[c*x^n]))/(5*x^5) - (e*x^(-5 + r)*(a + b*Log[c
*x^n]))/(5 - r)

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Rubi [A]  time = 0.0721168, 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} \[ -\frac{1}{5} \left (\frac{d}{x^5}+\frac{5 e x^{r-5}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{25 x^5}-\frac{b e n x^{r-5}}{(5-r)^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-(b*d*n)/(25*x^5) - (b*e*n*x^(-5 + r))/(5 - r)^2 - ((d/x^5 + (5*e*x^(-5 + r))/(5 - r))*(a + b*Log[c*x^n]))/5

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^6} \, dx &=-\frac{1}{5} \left (\frac{d}{x^5}+\frac{5 e x^{-5+r}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{d}{5 x^6}+\frac{e x^{-6+r}}{-5+r}\right ) \, dx\\ &=-\frac{b d n}{25 x^5}-\frac{b e n x^{-5+r}}{(5-r)^2}-\frac{1}{5} \left (\frac{d}{x^5}+\frac{5 e x^{-5+r}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.093609, size = 72, normalized size = 1.01 \[ -\frac{5 a (r-5) \left (d (r-5)-5 e x^r\right )+5 b (r-5) \log \left (c x^n\right ) \left (d (r-5)-5 e x^r\right )+b n \left (d (r-5)^2+25 e x^r\right )}{25 (r-5)^2 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-(5*a*(-5 + r)*(d*(-5 + r) - 5*e*x^r) + b*n*(d*(-5 + r)^2 + 25*e*x^r) + 5*b*(-5 + r)*(d*(-5 + r) - 5*e*x^r)*Lo
g[c*x^n])/(25*(-5 + r)^2*x^5)

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Maple [C]  time = 0.167, size = 614, normalized size = 8.7 \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^6,x)

[Out]

-1/5*b*(d*r-5*e*x^r-5*d)/(-5+r)/x^5*ln(x^n)-1/50*(250*a*d-50*x^r*a*e*r+50*x^r*b*e*n-25*I*Pi*b*e*csgn(I*c*x^n)^
2*csgn(I*c)*x^r*r-25*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r*r+50*I*Pi*b*d*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)-20*b*d*n*r-5*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-125*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)*x^r+50*b*d*n+250*x^r*a*e-100*ln(c)*b*d*r+10*ln(c)*b*d*r^2-50*ln(c)*b*e*x^r*r+250*ln(c)*b*e*x^r+10*a*d*r^2
+250*ln(c)*b*d-100*a*d*r+2*b*d*n*r^2+50*I*Pi*b*d*csgn(I*c*x^n)^3*r+25*I*Pi*b*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(
I*c)*x^r*r+5*I*Pi*b*d*r^2*csgn(I*c*x^n)^2*csgn(I*c)+5*I*Pi*b*d*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2-125*I*Pi*b*d*cs
gn(I*c*x^n)^3-50*I*Pi*b*d*r*csgn(I*c*x^n)^2*csgn(I*c)+125*I*Pi*b*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+125*I*Pi*b*e*
csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-50*I*Pi*b*d*r*csgn(I*x^n)*csgn(I*c*x^n)^2+25*I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r+12
5*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2-125*I*Pi*b*e*csgn(I*c*x^n)^3*x^r+125*I*Pi*b*d*csgn(I*c*x^n)^2*csgn(I*c)
-5*I*Pi*b*d*r^2*csgn(I*c*x^n)^3-125*I*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/(-5+r)^2/x^5

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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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.33956, size = 367, normalized size = 5.17 \begin{align*} -\frac{25 \, b d n +{\left (b d n + 5 \, a d\right )} r^{2} + 125 \, a d - 10 \,{\left (b d n + 5 \, a d\right )} r + 25 \,{\left (b e n - a e r + 5 \, a e -{\left (b e r - 5 \, b e\right )} \log \left (c\right ) -{\left (b e n r - 5 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 5 \,{\left (b d r^{2} - 10 \, b d r + 25 \, b d\right )} \log \left (c\right ) + 5 \,{\left (b d n r^{2} - 10 \, b d n r + 25 \, b d n\right )} \log \left (x\right )}{25 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} \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^6,x, algorithm="fricas")

[Out]

-1/25*(25*b*d*n + (b*d*n + 5*a*d)*r^2 + 125*a*d - 10*(b*d*n + 5*a*d)*r + 25*(b*e*n - a*e*r + 5*a*e - (b*e*r -
5*b*e)*log(c) - (b*e*n*r - 5*b*e*n)*log(x))*x^r + 5*(b*d*r^2 - 10*b*d*r + 25*b*d)*log(c) + 5*(b*d*n*r^2 - 10*b
*d*n*r + 25*b*d*n)*log(x))/((r^2 - 10*r + 25)*x^5)

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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**6,x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.2723, size = 536, normalized size = 7.55 \begin{align*} -\frac{b d n r^{2} \log \left (x\right )}{5 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{b n r x^{r} e \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{b d n r^{2}}{25 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{b d r^{2} \log \left (c\right )}{5 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{b r x^{r} e \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, b n x^{r} e \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{2 \, b d n r}{5 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{a d r^{2}}{5 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{b n x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{a r x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{2 \, b d r \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, b x^{r} e \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, b d n \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{b d n}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{2 \, a d r}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, a x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, b d \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, a d}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} \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^6,x, algorithm="giac")

[Out]

-1/5*b*d*n*r^2*log(x)/((r^2 - 10*r + 25)*x^5) + b*n*r*x^r*e*log(x)/((r^2 - 10*r + 25)*x^5) - 1/25*b*d*n*r^2/((
r^2 - 10*r + 25)*x^5) - 1/5*b*d*r^2*log(c)/((r^2 - 10*r + 25)*x^5) + b*r*x^r*e*log(c)/((r^2 - 10*r + 25)*x^5)
+ 2*b*d*n*r*log(x)/((r^2 - 10*r + 25)*x^5) - 5*b*n*x^r*e*log(x)/((r^2 - 10*r + 25)*x^5) + 2/5*b*d*n*r/((r^2 -
10*r + 25)*x^5) - 1/5*a*d*r^2/((r^2 - 10*r + 25)*x^5) - b*n*x^r*e/((r^2 - 10*r + 25)*x^5) + a*r*x^r*e/((r^2 -
10*r + 25)*x^5) + 2*b*d*r*log(c)/((r^2 - 10*r + 25)*x^5) - 5*b*x^r*e*log(c)/((r^2 - 10*r + 25)*x^5) - 5*b*d*n*
log(x)/((r^2 - 10*r + 25)*x^5) - b*d*n/((r^2 - 10*r + 25)*x^5) + 2*a*d*r/((r^2 - 10*r + 25)*x^5) - 5*a*x^r*e/(
(r^2 - 10*r + 25)*x^5) - 5*b*d*log(c)/((r^2 - 10*r + 25)*x^5) - 5*a*d/((r^2 - 10*r + 25)*x^5)