3.29 \(\int \frac{(d+e x)^3 (a+b \log (c x^n))}{x^7} \, dx\)

Optimal. Leaf size=133 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac{3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{3 b d^2 e n}{25 x^5}-\frac{b d^3 n}{36 x^6}-\frac{3 b d e^2 n}{16 x^4}-\frac{b e^3 n}{9 x^3} \]

[Out]

-(b*d^3*n)/(36*x^6) - (3*b*d^2*e*n)/(25*x^5) - (3*b*d*e^2*n)/(16*x^4) - (b*e^3*n)/(9*x^3) - (d^3*(a + b*Log[c*
x^n]))/(6*x^6) - (3*d^2*e*(a + b*Log[c*x^n]))/(5*x^5) - (3*d*e^2*(a + b*Log[c*x^n]))/(4*x^4) - (e^3*(a + b*Log
[c*x^n]))/(3*x^3)

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Rubi [A]  time = 0.0959554, 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} \[ -\frac{1}{60} \left (\frac{36 d^2 e}{x^5}+\frac{10 d^3}{x^6}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{25 x^5}-\frac{b d^3 n}{36 x^6}-\frac{3 b d e^2 n}{16 x^4}-\frac{b e^3 n}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(a + b*Log[c*x^n]))/x^7,x]

[Out]

-(b*d^3*n)/(36*x^6) - (3*b*d^2*e*n)/(25*x^5) - (3*b*d*e^2*n)/(16*x^4) - (b*e^3*n)/(9*x^3) - (((10*d^3)/x^6 + (
36*d^2*e)/x^5 + (45*d*e^2)/x^4 + (20*e^3)/x^3)*(a + b*Log[c*x^n]))/60

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^7} \, dx &=-\frac{1}{60} \left (\frac{10 d^3}{x^6}+\frac{36 d^2 e}{x^5}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-10 d^3-36 d^2 e x-45 d e^2 x^2-20 e^3 x^3}{60 x^7} \, dx\\ &=-\frac{1}{60} \left (\frac{10 d^3}{x^6}+\frac{36 d^2 e}{x^5}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int \frac{-10 d^3-36 d^2 e x-45 d e^2 x^2-20 e^3 x^3}{x^7} \, dx\\ &=-\frac{1}{60} \left (\frac{10 d^3}{x^6}+\frac{36 d^2 e}{x^5}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int \left (-\frac{10 d^3}{x^7}-\frac{36 d^2 e}{x^6}-\frac{45 d e^2}{x^5}-\frac{20 e^3}{x^4}\right ) \, dx\\ &=-\frac{b d^3 n}{36 x^6}-\frac{3 b d^2 e n}{25 x^5}-\frac{3 b d e^2 n}{16 x^4}-\frac{b e^3 n}{9 x^3}-\frac{1}{60} \left (\frac{10 d^3}{x^6}+\frac{36 d^2 e}{x^5}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0515982, size = 113, normalized size = 0.85 \[ -\frac{60 a \left (36 d^2 e x+10 d^3+45 d e^2 x^2+20 e^3 x^3\right )+60 b \left (36 d^2 e x+10 d^3+45 d e^2 x^2+20 e^3 x^3\right ) \log \left (c x^n\right )+b n \left (432 d^2 e x+100 d^3+675 d e^2 x^2+400 e^3 x^3\right )}{3600 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(a + b*Log[c*x^n]))/x^7,x]

[Out]

-(60*a*(10*d^3 + 36*d^2*e*x + 45*d*e^2*x^2 + 20*e^3*x^3) + b*n*(100*d^3 + 432*d^2*e*x + 675*d*e^2*x^2 + 400*e^
3*x^3) + 60*b*(10*d^3 + 36*d^2*e*x + 45*d*e^2*x^2 + 20*e^3*x^3)*Log[c*x^n])/(3600*x^6)

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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^7,x)

[Out]

-1/60*b*(20*e^3*x^3+45*d*e^2*x^2+36*d^2*e*x+10*d^3)/x^6*ln(x^n)-1/3600*(600*a*d^3+300*I*Pi*b*d^3*csgn(I*c*x^n)
^2*csgn(I*c)-600*I*Pi*b*e^3*x^3*csgn(I*c*x^n)^3+1200*ln(c)*b*e^3*x^3+2700*a*d*e^2*x^2+2160*a*d^2*e*x+600*ln(c)
*b*d^3-300*I*Pi*b*d^3*csgn(I*c*x^n)^3-600*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1350*I*Pi*b*d*e^2
*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+1080*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^2*csgn(I*c)+1350*I*Pi*b*d*e^2*x^2*csgn(I*c*
x^n)^2*csgn(I*c)+1080*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-1080*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)-1350*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^3-1080*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^3-1350*I*Pi*b*d*e^2*x^2*csg
n(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+300*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+2700*ln(c)*b*d*e^2*x^2+2160*ln(c)*
b*d^2*e*x+1200*a*e^3*x^3+100*b*d^3*n+600*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-300*I*Pi*b*d^3*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)+600*I*Pi*b*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)+400*b*e^3*n*x^3+432*b*d^2*e*n*x+675*b*d*
e^2*n*x^2)/x^6

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Maxima [A]  time = 1.10918, size = 193, normalized size = 1.45 \begin{align*} -\frac{b e^{3} n}{9 \, x^{3}} - \frac{b e^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{3 \, b d e^{2} n}{16 \, x^{4}} - \frac{a e^{3}}{3 \, x^{3}} - \frac{3 \, b d e^{2} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac{3 \, b d^{2} e n}{25 \, x^{5}} - \frac{3 \, a d e^{2}}{4 \, x^{4}} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{b d^{3} n}{36 \, x^{6}} - \frac{3 \, a d^{2} e}{5 \, x^{5}} - \frac{b d^{3} \log \left (c x^{n}\right )}{6 \, x^{6}} - \frac{a d^{3}}{6 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

-1/9*b*e^3*n/x^3 - 1/3*b*e^3*log(c*x^n)/x^3 - 3/16*b*d*e^2*n/x^4 - 1/3*a*e^3/x^3 - 3/4*b*d*e^2*log(c*x^n)/x^4
- 3/25*b*d^2*e*n/x^5 - 3/4*a*d*e^2/x^4 - 3/5*b*d^2*e*log(c*x^n)/x^5 - 1/36*b*d^3*n/x^6 - 3/5*a*d^2*e/x^5 - 1/6
*b*d^3*log(c*x^n)/x^6 - 1/6*a*d^3/x^6

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Fricas [A]  time = 1.082, size = 382, normalized size = 2.87 \begin{align*} -\frac{100 \, b d^{3} n + 600 \, a d^{3} + 400 \,{\left (b e^{3} n + 3 \, a e^{3}\right )} x^{3} + 675 \,{\left (b d e^{2} n + 4 \, a d e^{2}\right )} x^{2} + 432 \,{\left (b d^{2} e n + 5 \, a d^{2} e\right )} x + 60 \,{\left (20 \, b e^{3} x^{3} + 45 \, b d e^{2} x^{2} + 36 \, b d^{2} e x + 10 \, b d^{3}\right )} \log \left (c\right ) + 60 \,{\left (20 \, b e^{3} n x^{3} + 45 \, b d e^{2} n x^{2} + 36 \, b d^{2} e n x + 10 \, b d^{3} n\right )} \log \left (x\right )}{3600 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

-1/3600*(100*b*d^3*n + 600*a*d^3 + 400*(b*e^3*n + 3*a*e^3)*x^3 + 675*(b*d*e^2*n + 4*a*d*e^2)*x^2 + 432*(b*d^2*
e*n + 5*a*d^2*e)*x + 60*(20*b*e^3*x^3 + 45*b*d*e^2*x^2 + 36*b*d^2*e*x + 10*b*d^3)*log(c) + 60*(20*b*e^3*n*x^3
+ 45*b*d*e^2*n*x^2 + 36*b*d^2*e*n*x + 10*b*d^3*n)*log(x))/x^6

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Sympy [A]  time = 17.4311, size = 231, normalized size = 1.74 \begin{align*} - \frac{a d^{3}}{6 x^{6}} - \frac{3 a d^{2} e}{5 x^{5}} - \frac{3 a d e^{2}}{4 x^{4}} - \frac{a e^{3}}{3 x^{3}} - \frac{b d^{3} n \log{\left (x \right )}}{6 x^{6}} - \frac{b d^{3} n}{36 x^{6}} - \frac{b d^{3} \log{\left (c \right )}}{6 x^{6}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{5 x^{5}} - \frac{3 b d^{2} e n}{25 x^{5}} - \frac{3 b d^{2} e \log{\left (c \right )}}{5 x^{5}} - \frac{3 b d e^{2} n \log{\left (x \right )}}{4 x^{4}} - \frac{3 b d e^{2} n}{16 x^{4}} - \frac{3 b d e^{2} \log{\left (c \right )}}{4 x^{4}} - \frac{b e^{3} n \log{\left (x \right )}}{3 x^{3}} - \frac{b e^{3} n}{9 x^{3}} - \frac{b e^{3} \log{\left (c \right )}}{3 x^{3}} \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**7,x)

[Out]

-a*d**3/(6*x**6) - 3*a*d**2*e/(5*x**5) - 3*a*d*e**2/(4*x**4) - a*e**3/(3*x**3) - b*d**3*n*log(x)/(6*x**6) - b*
d**3*n/(36*x**6) - b*d**3*log(c)/(6*x**6) - 3*b*d**2*e*n*log(x)/(5*x**5) - 3*b*d**2*e*n/(25*x**5) - 3*b*d**2*e
*log(c)/(5*x**5) - 3*b*d*e**2*n*log(x)/(4*x**4) - 3*b*d*e**2*n/(16*x**4) - 3*b*d*e**2*log(c)/(4*x**4) - b*e**3
*n*log(x)/(3*x**3) - b*e**3*n/(9*x**3) - b*e**3*log(c)/(3*x**3)

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Giac [A]  time = 1.28759, size = 213, normalized size = 1.6 \begin{align*} -\frac{1200 \, b n x^{3} e^{3} \log \left (x\right ) + 2700 \, b d n x^{2} e^{2} \log \left (x\right ) + 2160 \, b d^{2} n x e \log \left (x\right ) + 400 \, b n x^{3} e^{3} + 675 \, b d n x^{2} e^{2} + 432 \, b d^{2} n x e + 1200 \, b x^{3} e^{3} \log \left (c\right ) + 2700 \, b d x^{2} e^{2} \log \left (c\right ) + 2160 \, b d^{2} x e \log \left (c\right ) + 600 \, b d^{3} n \log \left (x\right ) + 100 \, b d^{3} n + 1200 \, a x^{3} e^{3} + 2700 \, a d x^{2} e^{2} + 2160 \, a d^{2} x e + 600 \, b d^{3} \log \left (c\right ) + 600 \, a d^{3}}{3600 \, x^{6}} \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^7,x, algorithm="giac")

[Out]

-1/3600*(1200*b*n*x^3*e^3*log(x) + 2700*b*d*n*x^2*e^2*log(x) + 2160*b*d^2*n*x*e*log(x) + 400*b*n*x^3*e^3 + 675
*b*d*n*x^2*e^2 + 432*b*d^2*n*x*e + 1200*b*x^3*e^3*log(c) + 2700*b*d*x^2*e^2*log(c) + 2160*b*d^2*x*e*log(c) + 6
00*b*d^3*n*log(x) + 100*b*d^3*n + 1200*a*x^3*e^3 + 2700*a*d*x^2*e^2 + 2160*a*d^2*x*e + 600*b*d^3*log(c) + 600*
a*d^3)/x^6