3.18 \(\int \frac{(d+e x)^2 (a+b \log (c x^n))}{x^6} \, dx\)

Optimal. Leaf size=95 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{d e \left (a+b \log \left (c x^n\right )\right )}{2 x^4}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{b d^2 n}{25 x^5}-\frac{b d e n}{8 x^4}-\frac{b e^2 n}{9 x^3} \]

[Out]

-(b*d^2*n)/(25*x^5) - (b*d*e*n)/(8*x^4) - (b*e^2*n)/(9*x^3) - (d^2*(a + b*Log[c*x^n]))/(5*x^5) - (d*e*(a + b*L
og[c*x^n]))/(2*x^4) - (e^2*(a + b*Log[c*x^n]))/(3*x^3)

________________________________________________________________________________________

Rubi [A]  time = 0.075784, antiderivative size = 74, normalized size of antiderivative = 0.78, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {43, 2334, 12, 14} \[ -\frac{1}{30} \left (\frac{6 d^2}{x^5}+\frac{15 d e}{x^4}+\frac{10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 n}{25 x^5}-\frac{b d e n}{8 x^4}-\frac{b e^2 n}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^2*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-(b*d^2*n)/(25*x^5) - (b*d*e*n)/(8*x^4) - (b*e^2*n)/(9*x^3) - (((6*d^2)/x^5 + (15*d*e)/x^4 + (10*e^2)/x^3)*(a
+ b*Log[c*x^n]))/30

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]]

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]]

Rubi steps

\begin{align*} \int \frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac{1}{30} \left (\frac{6 d^2}{x^5}+\frac{15 d e}{x^4}+\frac{10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-6 d^2-15 d e x-10 e^2 x^2}{30 x^6} \, dx\\ &=-\frac{1}{30} \left (\frac{6 d^2}{x^5}+\frac{15 d e}{x^4}+\frac{10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{30} (b n) \int \frac{-6 d^2-15 d e x-10 e^2 x^2}{x^6} \, dx\\ &=-\frac{1}{30} \left (\frac{6 d^2}{x^5}+\frac{15 d e}{x^4}+\frac{10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{30} (b n) \int \left (-\frac{6 d^2}{x^6}-\frac{15 d e}{x^5}-\frac{10 e^2}{x^4}\right ) \, dx\\ &=-\frac{b d^2 n}{25 x^5}-\frac{b d e n}{8 x^4}-\frac{b e^2 n}{9 x^3}-\frac{1}{30} \left (\frac{6 d^2}{x^5}+\frac{15 d e}{x^4}+\frac{10 e^2}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0386351, size = 80, normalized size = 0.84 \[ -\frac{60 a \left (6 d^2+15 d e x+10 e^2 x^2\right )+60 b \left (6 d^2+15 d e x+10 e^2 x^2\right ) \log \left (c x^n\right )+b n \left (72 d^2+225 d e x+200 e^2 x^2\right )}{1800 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^2*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-(60*a*(6*d^2 + 15*d*e*x + 10*e^2*x^2) + b*n*(72*d^2 + 225*d*e*x + 200*e^2*x^2) + 60*b*(6*d^2 + 15*d*e*x + 10*
e^2*x^2)*Log[c*x^n])/(1800*x^5)

________________________________________________________________________________________

Maple [C]  time = 0.121, size = 403, normalized size = 4.2 \begin{align*} -{\frac{b \left ( 10\,{e}^{2}{x}^{2}+15\,dex+6\,{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{30\,{x}^{5}}}-{\frac{-180\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-300\,i\pi \,b{e}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -300\,i\pi \,b{e}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-180\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +600\,\ln \left ( c \right ) b{e}^{2}{x}^{2}+200\,b{e}^{2}n{x}^{2}+600\,a{e}^{2}{x}^{2}+180\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+450\,i\pi \,bdex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +300\,i\pi \,b{e}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +180\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +900\,\ln \left ( c \right ) bdex+225\,bdenx+900\,adex-450\,i\pi \,bdex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+450\,i\pi \,bdex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+300\,i\pi \,b{e}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-450\,i\pi \,bdex{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +360\,\ln \left ( c \right ) b{d}^{2}+72\,b{d}^{2}n+360\,a{d}^{2}}{1800\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a+b*ln(c*x^n))/x^6,x)

[Out]

-1/30*b*(10*e^2*x^2+15*d*e*x+6*d^2)/x^5*ln(x^n)-1/1800*(-180*I*Pi*b*d^2*csgn(I*c*x^n)^3-300*I*Pi*b*e^2*x^2*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-300*I*Pi*b*e^2*x^2*csgn(I*c*x^n)^3-180*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*c
sgn(I*c)+600*ln(c)*b*e^2*x^2+200*b*e^2*n*x^2+600*a*e^2*x^2+180*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+450*I*Pi
*b*d*e*x*csgn(I*c*x^n)^2*csgn(I*c)+300*I*Pi*b*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)+180*I*Pi*b*d^2*csgn(I*c*x^n)^2
*csgn(I*c)+900*ln(c)*b*d*e*x+225*b*d*e*n*x+900*a*d*e*x-450*I*Pi*b*d*e*x*csgn(I*c*x^n)^3+450*I*Pi*b*d*e*x*csgn(
I*x^n)*csgn(I*c*x^n)^2+300*I*Pi*b*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-450*I*Pi*b*d*e*x*csgn(I*x^n)*csgn(I*c*x^
n)*csgn(I*c)+360*ln(c)*b*d^2+72*b*d^2*n+360*a*d^2)/x^5

________________________________________________________________________________________

Maxima [A]  time = 1.4579, size = 135, normalized size = 1.42 \begin{align*} -\frac{b e^{2} n}{9 \, x^{3}} - \frac{b e^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{b d e n}{8 \, x^{4}} - \frac{a e^{2}}{3 \, x^{3}} - \frac{b d e \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac{b d^{2} n}{25 \, x^{5}} - \frac{a d e}{2 \, x^{4}} - \frac{b d^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{a d^{2}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))/x^6,x, algorithm="maxima")

[Out]

-1/9*b*e^2*n/x^3 - 1/3*b*e^2*log(c*x^n)/x^3 - 1/8*b*d*e*n/x^4 - 1/3*a*e^2/x^3 - 1/2*b*d*e*log(c*x^n)/x^4 - 1/2
5*b*d^2*n/x^5 - 1/2*a*d*e/x^4 - 1/5*b*d^2*log(c*x^n)/x^5 - 1/5*a*d^2/x^5

________________________________________________________________________________________

Fricas [A]  time = 1.01916, size = 273, normalized size = 2.87 \begin{align*} -\frac{72 \, b d^{2} n + 360 \, a d^{2} + 200 \,{\left (b e^{2} n + 3 \, a e^{2}\right )} x^{2} + 225 \,{\left (b d e n + 4 \, a d e\right )} x + 60 \,{\left (10 \, b e^{2} x^{2} + 15 \, b d e x + 6 \, b d^{2}\right )} \log \left (c\right ) + 60 \,{\left (10 \, b e^{2} n x^{2} + 15 \, b d e n x + 6 \, b d^{2} n\right )} \log \left (x\right )}{1800 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))/x^6,x, algorithm="fricas")

[Out]

-1/1800*(72*b*d^2*n + 360*a*d^2 + 200*(b*e^2*n + 3*a*e^2)*x^2 + 225*(b*d*e*n + 4*a*d*e)*x + 60*(10*b*e^2*x^2 +
15*b*d*e*x + 6*b*d^2)*log(c) + 60*(10*b*e^2*n*x^2 + 15*b*d*e*n*x + 6*b*d^2*n)*log(x))/x^5

________________________________________________________________________________________

Sympy [A]  time = 8.20365, size = 153, normalized size = 1.61 \begin{align*} - \frac{a d^{2}}{5 x^{5}} - \frac{a d e}{2 x^{4}} - \frac{a e^{2}}{3 x^{3}} - \frac{b d^{2} n \log{\left (x \right )}}{5 x^{5}} - \frac{b d^{2} n}{25 x^{5}} - \frac{b d^{2} \log{\left (c \right )}}{5 x^{5}} - \frac{b d e n \log{\left (x \right )}}{2 x^{4}} - \frac{b d e n}{8 x^{4}} - \frac{b d e \log{\left (c \right )}}{2 x^{4}} - \frac{b e^{2} n \log{\left (x \right )}}{3 x^{3}} - \frac{b e^{2} n}{9 x^{3}} - \frac{b e^{2} \log{\left (c \right )}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(a+b*ln(c*x**n))/x**6,x)

[Out]

-a*d**2/(5*x**5) - a*d*e/(2*x**4) - a*e**2/(3*x**3) - b*d**2*n*log(x)/(5*x**5) - b*d**2*n/(25*x**5) - b*d**2*l
og(c)/(5*x**5) - b*d*e*n*log(x)/(2*x**4) - b*d*e*n/(8*x**4) - b*d*e*log(c)/(2*x**4) - b*e**2*n*log(x)/(3*x**3)
- b*e**2*n/(9*x**3) - b*e**2*log(c)/(3*x**3)

________________________________________________________________________________________

Giac [A]  time = 1.28953, size = 146, normalized size = 1.54 \begin{align*} -\frac{600 \, b n x^{2} e^{2} \log \left (x\right ) + 900 \, b d n x e \log \left (x\right ) + 200 \, b n x^{2} e^{2} + 225 \, b d n x e + 600 \, b x^{2} e^{2} \log \left (c\right ) + 900 \, b d x e \log \left (c\right ) + 360 \, b d^{2} n \log \left (x\right ) + 72 \, b d^{2} n + 600 \, a x^{2} e^{2} + 900 \, a d x e + 360 \, b d^{2} \log \left (c\right ) + 360 \, a d^{2}}{1800 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*log(c*x^n))/x^6,x, algorithm="giac")

[Out]

-1/1800*(600*b*n*x^2*e^2*log(x) + 900*b*d*n*x*e*log(x) + 200*b*n*x^2*e^2 + 225*b*d*n*x*e + 600*b*x^2*e^2*log(c
) + 900*b*d*x*e*log(c) + 360*b*d^2*n*log(x) + 72*b*d^2*n + 600*a*x^2*e^2 + 900*a*d*x*e + 360*b*d^2*log(c) + 36
0*a*d^2)/x^5