3.1.28 \(\int \frac {(d+e x)^3 (a+b \log (c x^n))}{x^6} \, dx\) [28]

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

[Out]

1/80*b*d^2*e*n/x^4+1/15*b*d*e^2*n/x^3+3/20*b*e^3*n/x^2+1/5*b*e^4*n/d/x-1/25*b*n*(e*x+d)^5/d^2/x^5-1/20*b*e^5*n
*ln(x)/d^2-1/5*(e*x+d)^4*(a+b*ln(c*x^n))/d/x^5+1/20*e*(e*x+d)^4*(a+b*ln(c*x^n))/d^2/x^4

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Rubi [A]
time = 0.07, antiderivative size = 142, normalized size of antiderivative = 1.00, 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 = {47, 37, 2372, 12, 79, 45} \begin {gather*} \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 n (d+e x)^5}{25 d^2 x^5}+\frac {b d^2 e n}{80 x^4}+\frac {b e^4 n}{5 d x}+\frac {b d e^2 n}{15 x^3}+\frac {3 b e^3 n}{20 x^2} \end {gather*}

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) - ((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)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 45

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 47

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 79

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] || L
tQ[p, n]))))

Rule 2372

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]}, Dist[a + b*Log[c*x^n], u, 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 {(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*}

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Mathematica [A]
time = 0.04, size = 113, normalized size = 0.80 \begin {gather*} -\frac {60 a \left (4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right )+b n \left (48 d^3+225 d^2 e x+400 d e^2 x^2+300 e^3 x^3\right )+60 b \left (4 d^3+15 d^2 e x+20 d e^2 x^2+10 e^3 x^3\right ) \log \left (c x^n\right )}{1200 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1200*(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 + 3
00*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])/x^5

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 571, normalized size = 4.02

method	result	size
risch	\(-\frac {b \left (10 e^{3} x^{3}+20 d \,e^{2} x^{2}+15 d^{2} e x +4 d^{3}\right ) \ln \left (x^{n}\right )}{20 x^{5}}-\frac {450 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+450 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+600 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+600 \ln \left (c \right ) b \,e^{3} x^{3}-300 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-600 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-450 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+240 a \,d^{3}+1200 a d \,e^{2} x^{2}+900 a \,d^{2} e x +600 a \,e^{3} x^{3}+48 b \,d^{3} n +240 d^{3} b \ln \left (c \right )+1200 \ln \left (c \right ) b d \,e^{2} x^{2}+900 \ln \left (c \right ) b \,d^{2} e x +600 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+300 b \,e^{3} n \,x^{3}-450 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+300 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+225 b \,d^{2} e n x +400 b d \,e^{2} n \,x^{2}-300 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+120 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+120 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-120 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-120 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+300 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-600 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{1200 x^{5}}\)	\(571\)

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,method=_RETURNVERBOSE)

[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*(-120*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*c
sgn(I*c*x^n)-450*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^3+600*ln(c)*b*e^3*x^3+240*a*d^3-120*I*Pi*b*d^3*csgn(I*c*x^n)^3+1
200*a*d*e^2*x^2+900*a*d^2*e*x+600*a*e^3*x^3-300*I*Pi*b*e^3*x^3*csgn(I*c*x^n)^3+120*I*Pi*b*d^3*csgn(I*c)*csgn(I
*c*x^n)^2+300*I*Pi*b*e^3*x^3*csgn(I*c)*csgn(I*c*x^n)^2+48*b*d^3*n+300*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)
^2-600*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^3+240*d^3*b*ln(c)-450*I*Pi*b*d^2*e*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)
-600*I*Pi*b*d*e^2*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1200*ln(c)*b*d*e^2*x^2+900*ln(c)*b*d^2*e*x+600*I*Pi*
b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+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)*csgn(I*c*x^n)^2-300*I*Pi*b*e^3*x^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+600*I*Pi*b*d*e^2*x^2*csgn(I*c)*cs
gn(I*c*x^n)^2+120*I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+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 = 0.29, size = 140, normalized size = 0.99 \begin {gather*} -\frac {b n e^{3}}{4 \, x^{2}} - \frac {b d n e^{2}}{3 \, x^{3}} - \frac {3 \, b d^{2} n e}{16 \, x^{4}} - \frac {b e^{3} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {b d e^{2} \log \left (c x^{n}\right )}{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 {a e^{3}}{2 \, x^{2}} - \frac {a d e^{2}}{x^{3}} - \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 {gather*}

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*n*e^3/x^2 - 1/3*b*d*n*e^2/x^3 - 3/16*b*d^2*n*e/x^4 - 1/2*b*e^3*log(c*x^n)/x^2 - b*d*e^2*log(c*x^n)/x^3
- 3/4*b*d^2*e*log(c*x^n)/x^4 - 1/25*b*d^3*n/x^5 - 1/2*a*e^3/x^2 - a*d*e^2/x^3 - 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 = 0.34, size = 145, normalized size = 1.02 \begin {gather*} -\frac {48 \, b d^{3} n + 300 \, {\left (b n + 2 \, a\right )} x^{3} e^{3} + 240 \, a d^{3} + 400 \, {\left (b d n + 3 \, a d\right )} x^{2} e^{2} + 225 \, {\left (b d^{2} n + 4 \, a d^{2}\right )} x e + 60 \, {\left (10 \, b x^{3} e^{3} + 20 \, b d x^{2} e^{2} + 15 \, b d^{2} x e + 4 \, b d^{3}\right )} \log \left (c\right ) + 60 \, {\left (10 \, b n x^{3} e^{3} + 20 \, b d n x^{2} e^{2} + 15 \, b d^{2} n x e + 4 \, b d^{3} n\right )} \log \left (x\right )}{1200 \, x^{5}} \end {gather*}

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

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Sympy [A]
time = 0.74, size = 168, normalized size = 1.18 \begin {gather*} - \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}{25 x^{5}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {3 b d^{2} e n}{16 x^{4}} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{4 x^{4}} - \frac {b d e^{2} n}{3 x^{3}} - \frac {b d e^{2} \log {\left (c x^{n} \right )}}{x^{3}} - \frac {b e^{3} n}{4 x^{2}} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{2 x^{2}} \end {gather*}

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

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Giac [A]
time = 1.82, size = 158, normalized size = 1.11 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 3.58, size = 120, normalized size = 0.85 \begin {gather*} -\frac {x^3\,\left (10\,a\,e^3+5\,b\,e^3\,n\right )+x\,\left (15\,a\,d^2\,e+\frac {15\,b\,d^2\,e\,n}{4}\right )+4\,a\,d^3+x^2\,\left (20\,a\,d\,e^2+\frac {20\,b\,d\,e^2\,n}{3}\right )+\frac {4\,b\,d^3\,n}{5}}{20\,x^5}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{5}+\frac {3\,b\,d^2\,e\,x}{4}+b\,d\,e^2\,x^2+\frac {b\,e^3\,x^3}{2}\right )}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*log(c*x^n))*(d + e*x)^3)/x^6,x)

[Out]

- (x^3*(10*a*e^3 + 5*b*e^3*n) + x*(15*a*d^2*e + (15*b*d^2*e*n)/4) + 4*a*d^3 + x^2*(20*a*d*e^2 + (20*b*d*e^2*n)
/3) + (4*b*d^3*n)/5)/(20*x^5) - (log(c*x^n)*((b*d^3)/5 + (b*e^3*x^3)/2 + (3*b*d^2*e*x)/4 + b*d*e^2*x^2))/x^5

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