\(\int \frac {(d+e x)^3 (a+b \log (c x^n))}{x^8} \, dx\) [30]

Optimal result

Integrand size = 21, antiderivative size = 133 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b d^3 n}{49 x^7}-\frac {b d^2 e n}{12 x^6}-\frac {3 b d e^2 n}{25 x^5}-\frac {b e^3 n}{16 x^4}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{2 x^6}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4} \]

[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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 2372, 12, 14} \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{7 x^7}-\frac {d^2 e \left (a+b \log \left (c x^n\right )\right )}{2 x^6}-\frac {3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {b d^3 n}{49 x^7}-\frac {b d^2 e n}{12 x^6}-\frac {3 b d e^2 n}{25 x^5}-\frac {b e^3 n}{16 x^4} \]

[In]

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

[Out]

-1/49*(b*d^3*n)/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) - (d^3*(a + b*Log[c*x
^n]))/(7*x^7) - (d^2*e*(a + b*Log[c*x^n]))/(2*x^6) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log[c*
x^n]))/(4*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 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*} \text {integral}& = -\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}-(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 {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 {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 {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 {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 {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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {420 a \left (20 d^3+70 d^2 e x+84 d e^2 x^2+35 e^3 x^3\right )+b n \left (1200 d^3+4900 d^2 e x+7056 d e^2 x^2+3675 e^3 x^3\right )+420 b \left (20 d^3+70 d^2 e x+84 d e^2 x^2+35 e^3 x^3\right ) \log \left (c x^n\right )}{58800 x^7} \]

[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

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.01

method	result	size
parallelrisch	\(-\frac {14700 b \ln \left (c \,x^{n}\right ) e^{3} x^{3}+3675 b \,e^{3} n \,x^{3}+14700 a \,e^{3} x^{3}+35280 b \ln \left (c \,x^{n}\right ) d \,e^{2} x^{2}+7056 b d \,e^{2} n \,x^{2}+35280 a d \,e^{2} x^{2}+29400 b \ln \left (c \,x^{n}\right ) d^{2} e x +4900 b \,d^{2} e n x +29400 a \,d^{2} e x +8400 b \ln \left (c \,x^{n}\right ) d^{3}+1200 b \,d^{3} n +8400 a \,d^{3}}{58800 x^{7}}\)	\(134\)
risch	\(-\frac {b \left (35 e^{3} x^{3}+84 d \,e^{2} x^{2}+70 d^{2} e x +20 d^{3}\right ) \ln \left (x^{n}\right )}{140 x^{7}}-\frac {14700 \ln \left (c \right ) b \,e^{3} x^{3}-17640 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-14700 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) e x -7350 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+17640 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+17640 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+14700 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} e x +14700 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} e x -4200 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+8400 a \,d^{3}+8400 d^{3} b \ln \left (c \right )+14700 a \,e^{3} x^{3}-14700 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} e x +7350 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+7350 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-17640 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+1200 b \,d^{3} n -4200 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+35280 a d \,e^{2} x^{2}+29400 a \,d^{2} e x +35280 \ln \left (c \right ) b d \,e^{2} x^{2}+29400 \ln \left (c \right ) b \,d^{2} e x +4900 b \,d^{2} e n x +7056 b d \,e^{2} n \,x^{2}+3675 b \,e^{3} n \,x^{3}+4200 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-7350 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+4200 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{58800 x^{7}}\)	\(571\)

[In]

int((e*x+d)^3*(a+b*ln(c*x^n))/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/58800/x^7*(14700*b*ln(c*x^n)*e^3*x^3+3675*b*e^3*n*x^3+14700*a*e^3*x^3+35280*b*ln(c*x^n)*d*e^2*x^2+7056*b*d*
e^2*n*x^2+35280*a*d*e^2*x^2+29400*b*ln(c*x^n)*d^2*e*x+4900*b*d^2*e*n*x+29400*a*d^2*e*x+8400*b*ln(c*x^n)*d^3+12
00*b*d^3*n+8400*a*d^3)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {1200 \, b d^{3} n + 8400 \, a d^{3} + 3675 \, {\left (b e^{3} n + 4 \, a e^{3}\right )} x^{3} + 7056 \, {\left (b d e^{2} n + 5 \, a d e^{2}\right )} x^{2} + 4900 \, {\left (b d^{2} e n + 6 \, a d^{2} e\right )} x + 420 \, {\left (35 \, b e^{3} x^{3} + 84 \, b d e^{2} x^{2} + 70 \, b d^{2} e x + 20 \, b d^{3}\right )} \log \left (c\right ) + 420 \, {\left (35 \, b e^{3} n x^{3} + 84 \, b d e^{2} n x^{2} + 70 \, b d^{2} e n x + 20 \, b d^{3} n\right )} \log \left (x\right )}{58800 \, x^{7}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=- \frac {a d^{3}}{7 x^{7}} - \frac {a d^{2} e}{2 x^{6}} - \frac {3 a d e^{2}}{5 x^{5}} - \frac {a e^{3}}{4 x^{4}} - \frac {b d^{3} n}{49 x^{7}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{7 x^{7}} - \frac {b d^{2} e n}{12 x^{6}} - \frac {b d^{2} e \log {\left (c x^{n} \right )}}{2 x^{6}} - \frac {3 b d e^{2} n}{25 x^{5}} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{5 x^{5}} - \frac {b e^{3} n}{16 x^{4}} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{4 x^{4}} \]

[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/(49*x**7) - b*d**3*log
(c*x**n)/(7*x**7) - b*d**2*e*n/(12*x**6) - b*d**2*e*log(c*x**n)/(2*x**6) - 3*b*d*e**2*n/(25*x**5) - 3*b*d*e**2
*log(c*x**n)/(5*x**5) - b*e**3*n/(16*x**4) - b*e**3*log(c*x**n)/(4*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {b e^{3} n}{16 \, x^{4}} - \frac {b e^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {3 \, b d e^{2} n}{25 \, x^{5}} - \frac {a e^{3}}{4 \, x^{4}} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac {b d^{2} e n}{12 \, x^{6}} - \frac {3 \, a d e^{2}}{5 \, x^{5}} - \frac {b d^{2} e \log \left (c x^{n}\right )}{2 \, x^{6}} - \frac {b d^{3} n}{49 \, x^{7}} - \frac {a d^{2} e}{2 \, x^{6}} - \frac {b d^{3} \log \left (c x^{n}\right )}{7 \, x^{7}} - \frac {a d^{3}}{7 \, x^{7}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {{\left (35 \, b e^{3} n x^{3} + 84 \, b d e^{2} n x^{2} + 70 \, b d^{2} e n x + 20 \, b d^{3} n\right )} \log \left (x\right )}{140 \, x^{7}} - \frac {3675 \, b e^{3} n x^{3} + 14700 \, b e^{3} x^{3} \log \left (c\right ) + 7056 \, b d e^{2} n x^{2} + 14700 \, a e^{3} x^{3} + 35280 \, b d e^{2} x^{2} \log \left (c\right ) + 4900 \, b d^{2} e n x + 35280 \, a d e^{2} x^{2} + 29400 \, b d^{2} e x \log \left (c\right ) + 1200 \, b d^{3} n + 29400 \, a d^{2} e x + 8400 \, b d^{3} \log \left (c\right ) + 8400 \, a d^{3}}{58800 \, x^{7}} \]

[In]

integrate((e*x+d)^3*(a+b*log(c*x^n))/x^8,x, algorithm="giac")

[Out]

-1/140*(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 - 1/58800*(3675*b*e^3*n*x^
3 + 14700*b*e^3*x^3*log(c) + 7056*b*d*e^2*n*x^2 + 14700*a*e^3*x^3 + 35280*b*d*e^2*x^2*log(c) + 4900*b*d^2*e*n*
x + 35280*a*d*e^2*x^2 + 29400*b*d^2*e*x*log(c) + 1200*b*d^3*n + 29400*a*d^2*e*x + 8400*b*d^3*log(c) + 8400*a*d
^3)/x^7

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^8} \, dx=-\frac {x^3\,\left (35\,a\,e^3+\frac {35\,b\,e^3\,n}{4}\right )+x\,\left (70\,a\,d^2\,e+\frac {35\,b\,d^2\,e\,n}{3}\right )+20\,a\,d^3+x^2\,\left (84\,a\,d\,e^2+\frac {84\,b\,d\,e^2\,n}{5}\right )+\frac {20\,b\,d^3\,n}{7}}{140\,x^7}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{7}+\frac {b\,d^2\,e\,x}{2}+\frac {3\,b\,d\,e^2\,x^2}{5}+\frac {b\,e^3\,x^3}{4}\right )}{x^7} \]

[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