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

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

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 100, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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 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 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])

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*}

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Mathematica [A]  time = 0.06, size = 113, normalized size = 0.85 \[ -\frac {420 a \left (20 d^3+70 d^2 e x+84 d e^2 x^2+35 e^3 x^3\right )+420 b \left (20 d^3+70 d^2 e x+84 d e^2 x^2+35 e^3 x^3\right ) \log \left (c x^n\right )+b n \left (1200 d^3+4900 d^2 e x+7056 d e^2 x^2+3675 e^3 x^3\right )}{58800 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/58800*(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 + 3675*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])/x^7

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fricas [A]  time = 0.47, size = 155, normalized size = 1.17 \[ -\frac {1200 \, b d^{3} n + 8400 \, a d^{3} + 3675 \, {\left (b e^{3} n + 4 \, a e^{3}\right )} x^{3} + 7056 \, {\left (b d e^{2} n + 5 \, a d e^{2}\right )} x^{2} + 4900 \, {\left (b d^{2} e n + 6 \, a d^{2} e\right )} x + 420 \, {\left (35 \, b e^{3} x^{3} + 84 \, b d e^{2} x^{2} + 70 \, b d^{2} e x + 20 \, b d^{3}\right )} \log \relax (c) + 420 \, {\left (35 \, b e^{3} n x^{3} + 84 \, b d e^{2} n x^{2} + 70 \, b d^{2} e n x + 20 \, b d^{3} n\right )} \log \relax (x)}{58800 \, x^{7}} \]

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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giac [A]  time = 0.46, size = 158, normalized size = 1.19 \[ -\frac {14700 \, b n x^{3} e^{3} \log \relax (x) + 35280 \, b d n x^{2} e^{2} \log \relax (x) + 29400 \, b d^{2} n x e \log \relax (x) + 3675 \, b n x^{3} e^{3} + 7056 \, b d n x^{2} e^{2} + 4900 \, b d^{2} n x e + 14700 \, b x^{3} e^{3} \log \relax (c) + 35280 \, b d x^{2} e^{2} \log \relax (c) + 29400 \, b d^{2} x e \log \relax (c) + 8400 \, b d^{3} n \log \relax (x) + 1200 \, b d^{3} n + 14700 \, a x^{3} e^{3} + 35280 \, a d x^{2} e^{2} + 29400 \, a d^{2} x e + 8400 \, b d^{3} \log \relax (c) + 8400 \, a d^{3}}{58800 \, x^{7}} \]

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

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maple [C]  time = 0.18, size = 571, normalized size = 4.29 \[ -\frac {\left (35 e^{3} x^{3}+84 d \,e^{2} x^{2}+70 d^{2} e x +20 d^{3}\right ) b \ln \left (x^{n}\right )}{140 x^{7}}-\frac {35280 b d \,e^{2} x^{2} \ln \relax (c )+29400 b \,d^{2} e x \ln \relax (c )+35280 a d \,e^{2} x^{2}+29400 a \,d^{2} e x +8400 a \,d^{3}+14700 a \,e^{3} x^{3}+1200 b \,d^{3} n +8400 b \,d^{3} \ln \relax (c )+14700 b \,e^{3} x^{3} \ln \relax (c )-14700 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 )-17640 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 )-4200 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 )-7350 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 )+17640 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+17640 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3675 b \,e^{3} n \,x^{3}-4200 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+14700 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+14700 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-7350 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+7350 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+7350 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-17640 i \pi b d \,e^{2} x^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-14700 i \pi b \,d^{2} e x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4200 i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4200 i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+4900 b \,d^{2} e n x +7056 b d \,e^{2} n \,x^{2}}{58800 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(b*ln(c*x^n)+a)/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*(-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)+35280*b*d*e^2*x^2*ln(c)+29400
*b*d^2*e*x*ln(c)+14700*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2*e*x+35280*a*d*e^2*x^2+29400*a*d^2*e*x+8400*a*d^3
+17640*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)+14700*a*e^3*x^3+1200*b*d^3*n+8400*b*d^3*ln(c)+14700*b*e^3*x^
3*ln(c)-4200*I*Pi*b*d^3*csgn(I*c*x^n)^3+3675*b*e^3*n*x^3-7350*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)+17640*I*Pi*b*d*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2+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)^2-7350*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+4200*I*Pi*b*d^3*csgn(I*c*x^n)^2*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*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+4900*b*d^2*e*n*x+7056*b*d*e^2*n*x^2)/x^7

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maxima [A]  time = 0.57, size = 143, normalized size = 1.08 \[ -\frac {b e^{3} n}{16 \, x^{4}} - \frac {b e^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {3 \, b d e^{2} n}{25 \, x^{5}} - \frac {a e^{3}}{4 \, x^{4}} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {b d^{2} e n}{12 \, x^{6}} - \frac {3 \, a d e^{2}}{5 \, x^{5}} - \frac {b d^{2} e \log \left (c x^{n}\right )}{2 \, x^{6}} - \frac {b d^{3} n}{49 \, x^{7}} - \frac {a d^{2} e}{2 \, x^{6}} - \frac {b d^{3} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac {a d^{3}}{7 \, x^{7}} \]

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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mupad [B]  time = 3.59, size = 121, normalized size = 0.91 \[ -\frac {x^3\,\left (35\,a\,e^3+\frac {35\,b\,e^3\,n}{4}\right )+x\,\left (70\,a\,d^2\,e+\frac {35\,b\,d^2\,e\,n}{3}\right )+20\,a\,d^3+x^2\,\left (84\,a\,d\,e^2+\frac {84\,b\,d\,e^2\,n}{5}\right )+\frac {20\,b\,d^3\,n}{7}}{140\,x^7}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{7}+\frac {b\,d^2\,e\,x}{2}+\frac {3\,b\,d\,e^2\,x^2}{5}+\frac {b\,e^3\,x^3}{4}\right )}{x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (x^3*(35*a*e^3 + (35*b*e^3*n)/4) + x*(70*a*d^2*e + (35*b*d^2*e*n)/3) + 20*a*d^3 + x^2*(84*a*d*e^2 + (84*b*d*
e^2*n)/5) + (20*b*d^3*n)/7)/(140*x^7) - (log(c*x^n)*((b*d^3)/7 + (b*e^3*x^3)/4 + (b*d^2*e*x)/2 + (3*b*d*e^2*x^
2)/5))/x^7

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sympy [A]  time = 10.56, size = 224, normalized size = 1.68 \[ - \frac {a d^{3}}{7 x^{7}} - \frac {a d^{2} e}{2 x^{6}} - \frac {3 a d e^{2}}{5 x^{5}} - \frac {a e^{3}}{4 x^{4}} - \frac {b d^{3} n \log {\relax (x )}}{7 x^{7}} - \frac {b d^{3} n}{49 x^{7}} - \frac {b d^{3} \log {\relax (c )}}{7 x^{7}} - \frac {b d^{2} e n \log {\relax (x )}}{2 x^{6}} - \frac {b d^{2} e n}{12 x^{6}} - \frac {b d^{2} e \log {\relax (c )}}{2 x^{6}} - \frac {3 b d e^{2} n \log {\relax (x )}}{5 x^{5}} - \frac {3 b d e^{2} n}{25 x^{5}} - \frac {3 b d e^{2} \log {\relax (c )}}{5 x^{5}} - \frac {b e^{3} n \log {\relax (x )}}{4 x^{4}} - \frac {b e^{3} n}{16 x^{4}} - \frac {b e^{3} \log {\relax (c )}}{4 x^{4}} \]

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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