3.30 \(\int \frac{(d+e x)^3 (a+b \log (c x^n))}{x^8} \, dx\)

Optimal. Leaf size=133 \[ -\frac{d^2 e \left (a+b \log \left (c x^n\right )\right )}{2 x^6}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac{3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac{b d^2 e n}{12 x^6}-\frac{b d^3 n}{49 x^7}-\frac{3 b d e^2 n}{25 x^5}-\frac{b e^3 n}{16 x^4} \]

[Out]

-(b*d^3*n)/(49*x^7) - (b*d^2*e*n)/(12*x^6) - (3*b*d*e^2*n)/(25*x^5) - (b*e^3*n)/(16*x^4) - (d^3*(a + b*Log[c*x
^n]))/(7*x^7) - (d^2*e*(a + b*Log[c*x^n]))/(2*x^6) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log[c*
x^n]))/(4*x^4)

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Rubi [A]  time = 0.106254, antiderivative size = 100, normalized size of antiderivative = 0.75, 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} \[ -\frac{1}{140} \left (\frac{70 d^2 e}{x^6}+\frac{20 d^3}{x^7}+\frac{84 d e^2}{x^5}+\frac{35 e^3}{x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d^2 e n}{12 x^6}-\frac{b d^3 n}{49 x^7}-\frac{3 b d e^2 n}{25 x^5}-\frac{b e^3 n}{16 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(a + b*Log[c*x^n]))/x^8,x]

[Out]

-(b*d^3*n)/(49*x^7) - (b*d^2*e*n)/(12*x^6) - (3*b*d*e^2*n)/(25*x^5) - (b*e^3*n)/(16*x^4) - (((20*d^3)/x^7 + (7
0*d^2*e)/x^6 + (84*d*e^2)/x^5 + (35*e^3)/x^4)*(a + b*Log[c*x^n]))/140

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)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx &=-\frac{1}{140} \left (\frac{20 d^3}{x^7}+\frac{70 d^2 e}{x^6}+\frac{84 d e^2}{x^5}+\frac{35 e^3}{x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-20 d^3-70 d^2 e x-84 d e^2 x^2-35 e^3 x^3}{140 x^8} \, dx\\ &=-\frac{1}{140} \left (\frac{20 d^3}{x^7}+\frac{70 d^2 e}{x^6}+\frac{84 d e^2}{x^5}+\frac{35 e^3}{x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{140} (b n) \int \frac{-20 d^3-70 d^2 e x-84 d e^2 x^2-35 e^3 x^3}{x^8} \, dx\\ &=-\frac{1}{140} \left (\frac{20 d^3}{x^7}+\frac{70 d^2 e}{x^6}+\frac{84 d e^2}{x^5}+\frac{35 e^3}{x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{140} (b n) \int \left (-\frac{20 d^3}{x^8}-\frac{70 d^2 e}{x^7}-\frac{84 d e^2}{x^6}-\frac{35 e^3}{x^5}\right ) \, dx\\ &=-\frac{b d^3 n}{49 x^7}-\frac{b d^2 e n}{12 x^6}-\frac{3 b d e^2 n}{25 x^5}-\frac{b e^3 n}{16 x^4}-\frac{1}{140} \left (\frac{20 d^3}{x^7}+\frac{70 d^2 e}{x^6}+\frac{84 d e^2}{x^5}+\frac{35 e^3}{x^4}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0520666, size = 113, normalized size = 0.85 \[ -\frac{420 a \left (70 d^2 e x+20 d^3+84 d e^2 x^2+35 e^3 x^3\right )+420 b \left (70 d^2 e x+20 d^3+84 d e^2 x^2+35 e^3 x^3\right ) \log \left (c x^n\right )+b n \left (4900 d^2 e x+1200 d^3+7056 d e^2 x^2+3675 e^3 x^3\right )}{58800 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(a + b*Log[c*x^n]))/x^8,x]

[Out]

-(420*a*(20*d^3 + 70*d^2*e*x + 84*d*e^2*x^2 + 35*e^3*x^3) + b*n*(1200*d^3 + 4900*d^2*e*x + 7056*d*e^2*x^2 + 36
75*e^3*x^3) + 420*b*(20*d^3 + 70*d^2*e*x + 84*d*e^2*x^2 + 35*e^3*x^3)*Log[c*x^n])/(58800*x^7)

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Maple [C]  time = 0.138, size = 571, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(a+b*ln(c*x^n))/x^8,x)

[Out]

-1/140*b*(35*e^3*x^3+84*d*e^2*x^2+70*d^2*e*x+20*d^3)/x^7*ln(x^n)-1/58800*(8400*a*d^3-7350*I*Pi*b*e^3*x^3*csgn(
I*c*x^n)^3+4200*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+14700*ln(c)*b*e^3*x^3+35280*a*d*e^2*x^2+29400*a*d^2*e*x
+8400*ln(c)*b*d^3+14700*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-17640*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)-14700*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-17640*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^
3-14700*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^3+14700*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^2*csgn(I*c)-7350*I*Pi*b*e^3*x^3*csgn
(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+7350*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-4200*I*Pi*b*d^3*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)+7350*I*Pi*b*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)+4200*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+
35280*ln(c)*b*d*e^2*x^2+29400*ln(c)*b*d^2*e*x+14700*a*e^3*x^3+17640*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)
^2+17640*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)+1200*b*d^3*n-4200*I*Pi*b*d^3*csgn(I*c*x^n)^3+3675*b*e^3*n*
x^3+4900*b*d^2*e*n*x+7056*b*d*e^2*n*x^2)/x^7

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Maxima [A]  time = 1.12394, size = 193, normalized size = 1.45 \begin{align*} -\frac{b e^{3} n}{16 \, x^{4}} - \frac{b e^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac{3 \, b d e^{2} n}{25 \, x^{5}} - \frac{a e^{3}}{4 \, x^{4}} - \frac{3 \, b d e^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{b d^{2} e n}{12 \, x^{6}} - \frac{3 \, a d e^{2}}{5 \, x^{5}} - \frac{b d^{2} e \log \left (c x^{n}\right )}{2 \, x^{6}} - \frac{b d^{3} n}{49 \, x^{7}} - \frac{a d^{2} e}{2 \, x^{6}} - \frac{b d^{3} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac{a d^{3}}{7 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^8,x, algorithm="maxima")

[Out]

-1/16*b*e^3*n/x^4 - 1/4*b*e^3*log(c*x^n)/x^4 - 3/25*b*d*e^2*n/x^5 - 1/4*a*e^3/x^4 - 3/5*b*d*e^2*log(c*x^n)/x^5
- 1/12*b*d^2*e*n/x^6 - 3/5*a*d*e^2/x^5 - 1/2*b*d^2*e*log(c*x^n)/x^6 - 1/49*b*d^3*n/x^7 - 1/2*a*d^2*e/x^6 - 1/
7*b*d^3*log(c*x^n)/x^7 - 1/7*a*d^3/x^7

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Fricas [A]  time = 1.04453, size = 393, normalized size = 2.95 \begin{align*} -\frac{1200 \, b d^{3} n + 8400 \, a d^{3} + 3675 \,{\left (b e^{3} n + 4 \, a e^{3}\right )} x^{3} + 7056 \,{\left (b d e^{2} n + 5 \, a d e^{2}\right )} x^{2} + 4900 \,{\left (b d^{2} e n + 6 \, a d^{2} e\right )} x + 420 \,{\left (35 \, b e^{3} x^{3} + 84 \, b d e^{2} x^{2} + 70 \, b d^{2} e x + 20 \, b d^{3}\right )} \log \left (c\right ) + 420 \,{\left (35 \, b e^{3} n x^{3} + 84 \, b d e^{2} n x^{2} + 70 \, b d^{2} e n x + 20 \, b d^{3} n\right )} \log \left (x\right )}{58800 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^8,x, algorithm="fricas")

[Out]

-1/58800*(1200*b*d^3*n + 8400*a*d^3 + 3675*(b*e^3*n + 4*a*e^3)*x^3 + 7056*(b*d*e^2*n + 5*a*d*e^2)*x^2 + 4900*(
b*d^2*e*n + 6*a*d^2*e)*x + 420*(35*b*e^3*x^3 + 84*b*d*e^2*x^2 + 70*b*d^2*e*x + 20*b*d^3)*log(c) + 420*(35*b*e^
3*n*x^3 + 84*b*d*e^2*n*x^2 + 70*b*d^2*e*n*x + 20*b*d^3*n)*log(x))/x^7

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Sympy [A]  time = 17.8332, size = 224, normalized size = 1.68 \begin{align*} - \frac{a d^{3}}{7 x^{7}} - \frac{a d^{2} e}{2 x^{6}} - \frac{3 a d e^{2}}{5 x^{5}} - \frac{a e^{3}}{4 x^{4}} - \frac{b d^{3} n \log{\left (x \right )}}{7 x^{7}} - \frac{b d^{3} n}{49 x^{7}} - \frac{b d^{3} \log{\left (c \right )}}{7 x^{7}} - \frac{b d^{2} e n \log{\left (x \right )}}{2 x^{6}} - \frac{b d^{2} e n}{12 x^{6}} - \frac{b d^{2} e \log{\left (c \right )}}{2 x^{6}} - \frac{3 b d e^{2} n \log{\left (x \right )}}{5 x^{5}} - \frac{3 b d e^{2} n}{25 x^{5}} - \frac{3 b d e^{2} \log{\left (c \right )}}{5 x^{5}} - \frac{b e^{3} n \log{\left (x \right )}}{4 x^{4}} - \frac{b e^{3} n}{16 x^{4}} - \frac{b e^{3} \log{\left (c \right )}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a+b*ln(c*x**n))/x**8,x)

[Out]

-a*d**3/(7*x**7) - a*d**2*e/(2*x**6) - 3*a*d*e**2/(5*x**5) - a*e**3/(4*x**4) - b*d**3*n*log(x)/(7*x**7) - b*d*
*3*n/(49*x**7) - b*d**3*log(c)/(7*x**7) - b*d**2*e*n*log(x)/(2*x**6) - b*d**2*e*n/(12*x**6) - b*d**2*e*log(c)/
(2*x**6) - 3*b*d*e**2*n*log(x)/(5*x**5) - 3*b*d*e**2*n/(25*x**5) - 3*b*d*e**2*log(c)/(5*x**5) - b*e**3*n*log(x
)/(4*x**4) - b*e**3*n/(16*x**4) - b*e**3*log(c)/(4*x**4)

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Giac [A]  time = 1.33508, size = 213, normalized size = 1.6 \begin{align*} -\frac{14700 \, b n x^{3} e^{3} \log \left (x\right ) + 35280 \, b d n x^{2} e^{2} \log \left (x\right ) + 29400 \, b d^{2} n x e \log \left (x\right ) + 3675 \, b n x^{3} e^{3} + 7056 \, b d n x^{2} e^{2} + 4900 \, b d^{2} n x e + 14700 \, b x^{3} e^{3} \log \left (c\right ) + 35280 \, b d x^{2} e^{2} \log \left (c\right ) + 29400 \, b d^{2} x e \log \left (c\right ) + 8400 \, b d^{3} n \log \left (x\right ) + 1200 \, b d^{3} n + 14700 \, a x^{3} e^{3} + 35280 \, a d x^{2} e^{2} + 29400 \, a d^{2} x e + 8400 \, b d^{3} \log \left (c\right ) + 8400 \, a d^{3}}{58800 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^8,x, algorithm="giac")

[Out]

-1/58800*(14700*b*n*x^3*e^3*log(x) + 35280*b*d*n*x^2*e^2*log(x) + 29400*b*d^2*n*x*e*log(x) + 3675*b*n*x^3*e^3
+ 7056*b*d*n*x^2*e^2 + 4900*b*d^2*n*x*e + 14700*b*x^3*e^3*log(c) + 35280*b*d*x^2*e^2*log(c) + 29400*b*d^2*x*e*
log(c) + 8400*b*d^3*n*log(x) + 1200*b*d^3*n + 14700*a*x^3*e^3 + 35280*a*d*x^2*e^2 + 29400*a*d^2*x*e + 8400*b*d
^3*log(c) + 8400*a*d^3)/x^7