3.1.29 \(\int \frac {(d+e x)^3 (a+b \log (c x^n))}{x^7} \, dx\) [29]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 2372, 12, 14} \begin {gather*} -\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\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 {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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36*(b*d^3*n)/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)

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]]

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 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^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*}

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Mathematica [A]
time = 0.04, size = 113, normalized size = 0.85 \begin {gather*} -\frac {60 a \left (10 d^3+36 d^2 e x+45 d e^2 x^2+20 e^3 x^3\right )+b n \left (100 d^3+432 d^2 e x+675 d e^2 x^2+400 e^3 x^3\right )+60 b \left (10 d^3+36 d^2 e x+45 d e^2 x^2+20 e^3 x^3\right ) \log \left (c x^n\right )}{3600 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3600*(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])/x^6

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 571, normalized size = 4.29

method	result	size
risch	\(-\frac {b \left (20 e^{3} x^{3}+45 d \,e^{2} x^{2}+36 d^{2} e x +10 d^{3}\right ) \ln \left (x^{n}\right )}{60 x^{6}}-\frac {1080 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+1080 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+1350 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+1200 \ln \left (c \right ) b \,e^{3} x^{3}-600 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-1350 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 )-1080 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 )+600 a \,d^{3}+2700 a d \,e^{2} x^{2}+2160 a \,d^{2} e x +1200 a \,e^{3} x^{3}+100 b \,d^{3} n +600 d^{3} b \ln \left (c \right )+2700 \ln \left (c \right ) b d \,e^{2} x^{2}+2160 \ln \left (c \right ) b \,d^{2} e x +1350 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+400 b \,e^{3} n \,x^{3}-1080 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+600 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+432 b \,d^{2} e n x +675 b d \,e^{2} n \,x^{2}-600 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 )+300 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+300 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-300 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 )-300 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+600 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-1350 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{3600 x^{6}}\)	\(571\)

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,method=_RETURNVERBOSE)

[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*(1350*I*Pi*b*d*e^2*x^2*csgn(I*c)*csgn(I
*c*x^n)^2+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*c)*csgn(I*c*x^n)^2+1200*l
n(c)*b*e^3*x^3+600*a*d^3+1350*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+2700*a*d*e^2*x^2+2160*a*d^2*e*x-300
*I*Pi*b*d^3*csgn(I*c*x^n)^3+1200*a*e^3*x^3-600*I*Pi*b*e^3*x^3*csgn(I*c*x^n)^3+600*I*Pi*b*e^3*x^3*csgn(I*c)*csg
n(I*c*x^n)^2+600*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-1350*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^3+100*b*d^3*n-
1080*I*Pi*b*d^2*e*x*csgn(I*c*x^n)^3-300*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+600*d^3*b*ln(c)-1350*I*
Pi*b*d*e^2*x^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+2700*ln(c)*b*d*e^2*x^2+2160*ln(c)*b*d^2*e*x-600*I*Pi*b*e^3*
x^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+300*I*Pi*b*d^3*csgn(I*c)*csgn(I*c*x^n)^2+300*I*Pi*b*d^3*csgn(I*x^n)*cs
gn(I*c*x^n)^2-1080*I*Pi*b*d^2*e*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+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 = 0.28, size = 140, normalized size = 1.05 \begin {gather*} -\frac {b n e^{3}}{9 \, x^{3}} - \frac {3 \, b d n e^{2}}{16 \, x^{4}} - \frac {3 \, b d^{2} n e}{25 \, x^{5}} - \frac {b e^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{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 {a e^{3}}{3 \, x^{3}} - \frac {3 \, a d e^{2}}{4 \, x^{4}} - \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 {gather*}

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*n*e^3/x^3 - 3/16*b*d*n*e^2/x^4 - 3/25*b*d^2*n*e/x^5 - 1/3*b*e^3*log(c*x^n)/x^3 - 3/4*b*d*e^2*log(c*x^n)
/x^4 - 3/5*b*d^2*e*log(c*x^n)/x^5 - 1/36*b*d^3*n/x^6 - 1/3*a*e^3/x^3 - 3/4*a*d*e^2/x^4 - 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 = 0.37, size = 145, normalized size = 1.09 \begin {gather*} -\frac {100 \, b d^{3} n + 400 \, {\left (b n + 3 \, a\right )} x^{3} e^{3} + 600 \, a d^{3} + 675 \, {\left (b d n + 4 \, a d\right )} x^{2} e^{2} + 432 \, {\left (b d^{2} n + 5 \, a d^{2}\right )} x e + 60 \, {\left (20 \, b x^{3} e^{3} + 45 \, b d x^{2} e^{2} + 36 \, b d^{2} x e + 10 \, b d^{3}\right )} \log \left (c\right ) + 60 \, {\left (20 \, b n x^{3} e^{3} + 45 \, b d n x^{2} e^{2} + 36 \, b d^{2} n x e + 10 \, b d^{3} n\right )} \log \left (x\right )}{3600 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 1.03, size = 177, normalized size = 1.33 \begin {gather*} - \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}{36 x^{6}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{6 x^{6}} - \frac {3 b d^{2} e n}{25 x^{5}} - \frac {3 b d^{2} e \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {3 b d e^{2} n}{16 x^{4}} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{4 x^{4}} - \frac {b e^{3} n}{9 x^{3}} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{3 x^{3}} \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**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/(36*x**6) - b*d**3*l
og(c*x**n)/(6*x**6) - 3*b*d**2*e*n/(25*x**5) - 3*b*d**2*e*log(c*x**n)/(5*x**5) - 3*b*d*e**2*n/(16*x**4) - 3*b*
d*e**2*log(c*x**n)/(4*x**4) - b*e**3*n/(9*x**3) - b*e**3*log(c*x**n)/(3*x**3)

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Giac [A]
time = 2.47, size = 158, normalized size = 1.19 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 3.74, size = 121, normalized size = 0.91 \begin {gather*} -\frac {x^3\,\left (20\,a\,e^3+\frac {20\,b\,e^3\,n}{3}\right )+x\,\left (36\,a\,d^2\,e+\frac {36\,b\,d^2\,e\,n}{5}\right )+10\,a\,d^3+x^2\,\left (45\,a\,d\,e^2+\frac {45\,b\,d\,e^2\,n}{4}\right )+\frac {5\,b\,d^3\,n}{3}}{60\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{6}+\frac {3\,b\,d^2\,e\,x}{5}+\frac {3\,b\,d\,e^2\,x^2}{4}+\frac {b\,e^3\,x^3}{3}\right )}{x^6} \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^7,x)

[Out]

- (x^3*(20*a*e^3 + (20*b*e^3*n)/3) + x*(36*a*d^2*e + (36*b*d^2*e*n)/5) + 10*a*d^3 + x^2*(45*a*d*e^2 + (45*b*d*
e^2*n)/4) + (5*b*d^3*n)/3)/(60*x^6) - (log(c*x^n)*((b*d^3)/6 + (b*e^3*x^3)/3 + (3*b*d^2*e*x)/5 + (3*b*d*e^2*x^
2)/4))/x^6

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