3.28 \(\int \frac{(d+e x)^3 (a+b \log (c x^n))}{x^6} \, dx\)

Optimal. Leaf size=142 \[ \frac{e (d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{20 d^2 x^4}-\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}-\frac{b e^5 n \log (x)}{20 d^2}+\frac{b d^2 e n}{80 x^4}-\frac{b n (d+e x)^5}{25 d^2 x^5}+\frac{b d e^2 n}{15 x^3}+\frac{b e^4 n}{5 d x}+\frac{3 b e^3 n}{20 x^2} \]

[Out]

(b*d^2*e*n)/(80*x^4) + (b*d*e^2*n)/(15*x^3) + (3*b*e^3*n)/(20*x^2) + (b*e^4*n)/(5*d*x) - (b*n*(d + e*x)^5)/(25
*d^2*x^5) - (b*e^5*n*Log[x])/(20*d^2) - ((d + e*x)^4*(a + b*Log[c*x^n]))/(5*d*x^5) + (e*(d + e*x)^4*(a + b*Log
[c*x^n]))/(20*d^2*x^4)

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Rubi [A]  time = 0.0983751, antiderivative size = 133, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {45, 37, 2334, 12, 78, 43} \[ -\frac{1}{20} \left (\frac{4 (d+e x)^4}{d x^5}-\frac{e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b e^5 n \log (x)}{20 d^2}+\frac{b d^2 e n}{80 x^4}-\frac{b n (d+e x)^5}{25 d^2 x^5}+\frac{b d e^2 n}{15 x^3}+\frac{b e^4 n}{5 d x}+\frac{3 b e^3 n}{20 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

(b*d^2*e*n)/(80*x^4) + (b*d*e^2*n)/(15*x^3) + (3*b*e^3*n)/(20*x^2) + (b*e^4*n)/(5*d*x) - (b*n*(d + e*x)^5)/(25
*d^2*x^5) - (b*e^5*n*Log[x])/(20*d^2) - (((4*(d + e*x)^4)/(d*x^5) - (e*(d + e*x)^4)/(d^2*x^4))*(a + b*Log[c*x^
n]))/20

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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])

Rubi steps

\begin{align*} \int \frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac{1}{20} \left (\frac{4 (d+e x)^4}{d x^5}-\frac{e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{(-4 d+e x) (d+e x)^4}{20 d^2 x^6} \, dx\\ &=-\frac{1}{20} \left (\frac{4 (d+e x)^4}{d x^5}-\frac{e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(b n) \int \frac{(-4 d+e x) (d+e x)^4}{x^6} \, dx}{20 d^2}\\ &=-\frac{b n (d+e x)^5}{25 d^2 x^5}-\frac{1}{20} \left (\frac{4 (d+e x)^4}{d x^5}-\frac{e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(b e n) \int \frac{(d+e x)^4}{x^5} \, dx}{20 d^2}\\ &=-\frac{b n (d+e x)^5}{25 d^2 x^5}-\frac{1}{20} \left (\frac{4 (d+e x)^4}{d x^5}-\frac{e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{(b e n) \int \left (\frac{d^4}{x^5}+\frac{4 d^3 e}{x^4}+\frac{6 d^2 e^2}{x^3}+\frac{4 d e^3}{x^2}+\frac{e^4}{x}\right ) \, dx}{20 d^2}\\ &=\frac{b d^2 e n}{80 x^4}+\frac{b d e^2 n}{15 x^3}+\frac{3 b e^3 n}{20 x^2}+\frac{b e^4 n}{5 d x}-\frac{b n (d+e x)^5}{25 d^2 x^5}-\frac{b e^5 n \log (x)}{20 d^2}-\frac{1}{20} \left (\frac{4 (d+e x)^4}{d x^5}-\frac{e (d+e x)^4}{d^2 x^4}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0524181, size = 113, normalized size = 0.8 \[ -\frac{60 a \left (15 d^2 e x+4 d^3+20 d e^2 x^2+10 e^3 x^3\right )+60 b \left (15 d^2 e x+4 d^3+20 d e^2 x^2+10 e^3 x^3\right ) \log \left (c x^n\right )+b n \left (225 d^2 e x+48 d^3+400 d e^2 x^2+300 e^3 x^3\right )}{1200 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(a + b*Log[c*x^n]))/x^6,x]

[Out]

-(60*a*(4*d^3 + 15*d^2*e*x + 20*d*e^2*x^2 + 10*e^3*x^3) + b*n*(48*d^3 + 225*d^2*e*x + 400*d*e^2*x^2 + 300*e^3*
x^3) + 60*b*(4*d^3 + 15*d^2*e*x + 20*d*e^2*x^2 + 10*e^3*x^3)*Log[c*x^n])/(1200*x^5)

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Maple [C]  time = 0.136, size = 571, normalized size = 4. \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^6,x)

[Out]

-1/20*b*(10*e^3*x^3+20*d*e^2*x^2+15*d^2*e*x+4*d^3)/x^5*ln(x^n)-1/1200*(240*a*d^3-300*I*Pi*b*e^3*x^3*csgn(I*c*x
^n)^3+600*ln(c)*b*e^3*x^3+1200*a*d*e^2*x^2+900*a*d^2*e*x+240*ln(c)*b*d^3+450*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I
*c*x^n)^2+450*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^2*csgn(I*c)+600*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+600*I*
Pi*b*d*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)-600*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-450*I*Pi*b*d
^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-300*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+120*I*Pi*b*d
^3*csgn(I*x^n)*csgn(I*c*x^n)^2+120*I*Pi*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)+1200*ln(c)*b*d*e^2*x^2+900*ln(c)*b*d^2
*e*x+600*a*e^3*x^3+48*b*d^3*n-450*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^3+300*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^
2-120*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-120*I*Pi*b*d^3*csgn(I*c*x^n)^3+300*I*Pi*b*e^3*x^3*csgn(I*
c*x^n)^2*csgn(I*c)-600*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^3+300*b*e^3*n*x^3+225*b*d^2*e*n*x+400*b*d*e^2*n*x^2)/x^5

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Maxima [A]  time = 1.14936, size = 193, normalized size = 1.36 \begin{align*} -\frac{b e^{3} n}{4 \, x^{2}} - \frac{b e^{3} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{b d e^{2} n}{3 \, x^{3}} - \frac{a e^{3}}{2 \, x^{2}} - \frac{b d e^{2} \log \left (c x^{n}\right )}{x^{3}} - \frac{3 \, b d^{2} e n}{16 \, x^{4}} - \frac{a d e^{2}}{x^{3}} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac{b d^{3} n}{25 \, x^{5}} - \frac{3 \, a d^{2} e}{4 \, x^{4}} - \frac{b d^{3} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{a d^{3}}{5 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.992106, size = 378, normalized size = 2.66 \begin{align*} -\frac{48 \, b d^{3} n + 240 \, a d^{3} + 300 \,{\left (b e^{3} n + 2 \, a e^{3}\right )} x^{3} + 400 \,{\left (b d e^{2} n + 3 \, a d e^{2}\right )} x^{2} + 225 \,{\left (b d^{2} e n + 4 \, a d^{2} e\right )} x + 60 \,{\left (10 \, b e^{3} x^{3} + 20 \, b d e^{2} x^{2} + 15 \, b d^{2} e x + 4 \, b d^{3}\right )} \log \left (c\right ) + 60 \,{\left (10 \, b e^{3} n x^{3} + 20 \, b d e^{2} n x^{2} + 15 \, b d^{2} e n x + 4 \, b d^{3} n\right )} \log \left (x\right )}{1200 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

-1/1200*(48*b*d^3*n + 240*a*d^3 + 300*(b*e^3*n + 2*a*e^3)*x^3 + 400*(b*d*e^2*n + 3*a*d*e^2)*x^2 + 225*(b*d^2*e
*n + 4*a*d^2*e)*x + 60*(10*b*e^3*x^3 + 20*b*d*e^2*x^2 + 15*b*d^2*e*x + 4*b*d^3)*log(c) + 60*(10*b*e^3*n*x^3 +
20*b*d*e^2*n*x^2 + 15*b*d^2*e*n*x + 4*b*d^3*n)*log(x))/x^5

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Sympy [A]  time = 9.25503, size = 219, normalized size = 1.54 \begin{align*} - \frac{a d^{3}}{5 x^{5}} - \frac{3 a d^{2} e}{4 x^{4}} - \frac{a d e^{2}}{x^{3}} - \frac{a e^{3}}{2 x^{2}} - \frac{b d^{3} n \log{\left (x \right )}}{5 x^{5}} - \frac{b d^{3} n}{25 x^{5}} - \frac{b d^{3} \log{\left (c \right )}}{5 x^{5}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{4 x^{4}} - \frac{3 b d^{2} e n}{16 x^{4}} - \frac{3 b d^{2} e \log{\left (c \right )}}{4 x^{4}} - \frac{b d e^{2} n \log{\left (x \right )}}{x^{3}} - \frac{b d e^{2} n}{3 x^{3}} - \frac{b d e^{2} \log{\left (c \right )}}{x^{3}} - \frac{b e^{3} n \log{\left (x \right )}}{2 x^{2}} - \frac{b e^{3} n}{4 x^{2}} - \frac{b e^{3} \log{\left (c \right )}}{2 x^{2}} \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**6,x)

[Out]

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

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Giac [A]  time = 1.21088, size = 213, normalized size = 1.5 \begin{align*} -\frac{600 \, b n x^{3} e^{3} \log \left (x\right ) + 1200 \, b d n x^{2} e^{2} \log \left (x\right ) + 900 \, b d^{2} n x e \log \left (x\right ) + 300 \, b n x^{3} e^{3} + 400 \, b d n x^{2} e^{2} + 225 \, b d^{2} n x e + 600 \, b x^{3} e^{3} \log \left (c\right ) + 1200 \, b d x^{2} e^{2} \log \left (c\right ) + 900 \, b d^{2} x e \log \left (c\right ) + 240 \, b d^{3} n \log \left (x\right ) + 48 \, b d^{3} n + 600 \, a x^{3} e^{3} + 1200 \, a d x^{2} e^{2} + 900 \, a d^{2} x e + 240 \, b d^{3} \log \left (c\right ) + 240 \, a d^{3}}{1200 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

-1/1200*(600*b*n*x^3*e^3*log(x) + 1200*b*d*n*x^2*e^2*log(x) + 900*b*d^2*n*x*e*log(x) + 300*b*n*x^3*e^3 + 400*b
*d*n*x^2*e^2 + 225*b*d^2*n*x*e + 600*b*x^3*e^3*log(c) + 1200*b*d*x^2*e^2*log(c) + 900*b*d^2*x*e*log(c) + 240*b
*d^3*n*log(x) + 48*b*d^3*n + 600*a*x^3*e^3 + 1200*a*d*x^2*e^2 + 900*a*d^2*x*e + 240*b*d^3*log(c) + 240*a*d^3)/
x^5