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

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

[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

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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 )}{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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 = 0.04, size = 113, normalized size = 0.85 \begin {gather*} -\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} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 571, normalized size = 4.29

method	result	size
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 i \pi b \,d^{2} e x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+14700 i \pi b \,d^{2} e x \,\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 c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+14700 \ln \left (c \right ) b \,e^{3} x^{3}-7350 i \pi b \,e^{3} x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-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 )-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 )+8400 a \,d^{3}+35280 a d \,e^{2} x^{2}+29400 a \,d^{2} e x +14700 a \,e^{3} x^{3}+1200 b \,d^{3} n +8400 d^{3} b \ln \left (c \right )+35280 \ln \left (c \right ) b d \,e^{2} x^{2}+29400 \ln \left (c \right ) b \,d^{2} e x +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}-14700 i \pi b \,d^{2} e x \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}+4900 b \,d^{2} e n x +7056 b d \,e^{2} n \,x^{2}-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 )+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}-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 )-4200 i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+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}}{58800 x^{7}}\)	\(571\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*(14700*I*Pi*b*d^2*e*x*csgn(I*c)*csgn(
I*c*x^n)^2+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)^3+14700*ln(c)
*b*e^3*x^3+7350*I*Pi*b*e^3*x^3*csgn(I*c)*csgn(I*c*x^n)^2-4200*I*Pi*b*d^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+8
400*a*d^3+35280*a*d*e^2*x^2+29400*a*d^2*e*x+14700*a*e^3*x^3-7350*I*Pi*b*e^3*x^3*csgn(I*c*x^n)^3+1200*b*d^3*n+7
350*I*Pi*b*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2-17640*I*Pi*b*d*e^2*x^2*csgn(I*c*x^n)^3+8400*d^3*b*ln(c)+35280*l
n(c)*b*d*e^2*x^2+29400*ln(c)*b*d^2*e*x+4200*I*Pi*b*d^3*csgn(I*c)*csgn(I*c*x^n)^2+4200*I*Pi*b*d^3*csgn(I*x^n)*c
sgn(I*c*x^n)^2-14700*I*Pi*b*d^2*e*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-17640*I*Pi*b*d*e^2*x^2*csgn(I*c)*csgn(
I*x^n)*csgn(I*c*x^n)-7350*I*Pi*b*e^3*x^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+17640*I*Pi*b*d*e^2*x^2*csgn(I*c)*
csgn(I*c*x^n)^2+14700*I*Pi*b*d^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-4200*I*Pi*b*d^3*csgn(I*c*x^n)^3+3675*b*e^3*n*
x^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 = 0.28, size = 140, normalized size = 1.05 \begin {gather*} -\frac {b n e^{3}}{16 \, x^{4}} - \frac {3 \, b d n e^{2}}{25 \, x^{5}} - \frac {b d^{2} n e}{12 \, x^{6}} - \frac {b e^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{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 e^{3}}{4 \, x^{4}} - \frac {3 \, a d e^{2}}{5 \, x^{5}} - \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}} \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^8,x, algorithm="maxima")

[Out]

-1/16*b*n*e^3/x^4 - 3/25*b*d*n*e^2/x^5 - 1/12*b*d^2*n*e/x^6 - 1/4*b*e^3*log(c*x^n)/x^4 - 3/5*b*d*e^2*log(c*x^n
)/x^5 - 1/2*b*d^2*e*log(c*x^n)/x^6 - 1/49*b*d^3*n/x^7 - 1/4*a*e^3/x^4 - 3/5*a*d*e^2/x^5 - 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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Fricas [A]
time = 0.35, size = 145, normalized size = 1.09 \begin {gather*} -\frac {1200 \, b d^{3} n + 3675 \, {\left (b n + 4 \, a\right )} x^{3} e^{3} + 8400 \, a d^{3} + 7056 \, {\left (b d n + 5 \, a d\right )} x^{2} e^{2} + 4900 \, {\left (b d^{2} n + 6 \, a d^{2}\right )} x e + 420 \, {\left (35 \, b x^{3} e^{3} + 84 \, b d x^{2} e^{2} + 70 \, b d^{2} x e + 20 \, b d^{3}\right )} \log \left (c\right ) + 420 \, {\left (35 \, b n x^{3} e^{3} + 84 \, b d n x^{2} e^{2} + 70 \, b d^{2} n x e + 20 \, b d^{3} n\right )} \log \left (x\right )}{58800 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

-1/58800*(1200*b*d^3*n + 3675*(b*n + 4*a)*x^3*e^3 + 8400*a*d^3 + 7056*(b*d*n + 5*a*d)*x^2*e^2 + 4900*(b*d^2*n
+ 6*a*d^2)*x*e + 420*(35*b*x^3*e^3 + 84*b*d*x^2*e^2 + 70*b*d^2*x*e + 20*b*d^3)*log(c) + 420*(35*b*n*x^3*e^3 +
84*b*d*n*x^2*e^2 + 70*b*d^2*n*x*e + 20*b*d^3*n)*log(x))/x^7

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Sympy [A]
time = 1.40, size = 172, normalized size = 1.29 \begin {gather*} - \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}} \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**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)

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Giac [A]
time = 2.17, size = 158, normalized size = 1.19 \begin {gather*} -\frac {14700 \, b n x^{3} e^{3} \log \left (x\right ) + 35280 \, b d n x^{2} e^{2} \log \left (x\right ) + 29400 \, b d^{2} n x e \log \left (x\right ) + 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 \left (c\right ) + 35280 \, b d x^{2} e^{2} \log \left (c\right ) + 29400 \, b d^{2} x e \log \left (c\right ) + 8400 \, b d^{3} n \log \left (x\right ) + 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 \left (c\right ) + 8400 \, a d^{3}}{58800 \, x^{7}} \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^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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Mupad [B]
time = 3.59, size = 121, normalized size = 0.91 \begin {gather*} -\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} \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^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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