3.66 \(\int \frac{x^4 (a+b \log (c x^n))}{(d+e x)^7} \, dx\)
Optimal. Leaf size=163 \[ \frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac{b d^2 n}{120 e^5 (d+e x)^4}-\frac{b n \log (d+e x)}{30 d^2 e^5}-\frac{b n x^5}{30 d^2 (d+e x)^5}-\frac{2 b n}{15 d e^5 (d+e x)}+\frac{b n}{10 e^5 (d+e x)^2}-\frac{2 b d n}{45 e^5 (d+e x)^3} \]
[Out]
-(b*n*x^5)/(30*d^2*(d + e*x)^5) + (b*d^2*n)/(120*e^5*(d + e*x)^4) - (2*b*d*n)/(45*e^5*(d + e*x)^3) + (b*n)/(10
*e^5*(d + e*x)^2) - (2*b*n)/(15*d*e^5*(d + e*x)) + (x^5*(a + b*Log[c*x^n]))/(6*d*(d + e*x)^6) + (x^5*(a + b*Lo
g[c*x^n]))/(30*d^2*(d + e*x)^5) - (b*n*Log[d + e*x])/(30*d^2*e^5)
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Rubi [A] time = 0.129124, antiderivative size = 163, normalized size of antiderivative = 1.,
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 =
{45, 37, 2350, 12, 78, 43} \[ \frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac{b d^2 n}{120 e^5 (d+e x)^4}-\frac{b n \log (d+e x)}{30 d^2 e^5}-\frac{b n x^5}{30 d^2 (d+e x)^5}-\frac{2 b n}{15 d e^5 (d+e x)}+\frac{b n}{10 e^5 (d+e x)^2}-\frac{2 b d n}{45 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
Int[(x^4*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
[Out]
-(b*n*x^5)/(30*d^2*(d + e*x)^5) + (b*d^2*n)/(120*e^5*(d + e*x)^4) - (2*b*d*n)/(45*e^5*(d + e*x)^3) + (b*n)/(10
*e^5*(d + e*x)^2) - (2*b*n)/(15*d*e^5*(d + e*x)) + (x^5*(a + b*Log[c*x^n]))/(6*d*(d + e*x)^6) + (x^5*(a + b*Lo
g[c*x^n]))/(30*d^2*(d + e*x)^5) - (b*n*Log[d + e*x])/(30*d^2*e^5)
Rule 45
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 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 2350
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*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] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 78
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] || LtQ
[p, n]))))
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])
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-(b n) \int \frac{x^4 (6 d+e x)}{30 d^2 (d+e x)^6} \, dx\\ &=\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac{(b n) \int \frac{x^4 (6 d+e x)}{(d+e x)^6} \, dx}{30 d^2}\\ &=-\frac{b n x^5}{30 d^2 (d+e x)^5}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac{(b n) \int \frac{x^4}{(d+e x)^5} \, dx}{30 d^2}\\ &=-\frac{b n x^5}{30 d^2 (d+e x)^5}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac{(b n) \int \left (\frac{d^4}{e^4 (d+e x)^5}-\frac{4 d^3}{e^4 (d+e x)^4}+\frac{6 d^2}{e^4 (d+e x)^3}-\frac{4 d}{e^4 (d+e x)^2}+\frac{1}{e^4 (d+e x)}\right ) \, dx}{30 d^2}\\ &=-\frac{b n x^5}{30 d^2 (d+e x)^5}+\frac{b d^2 n}{120 e^5 (d+e x)^4}-\frac{2 b d n}{45 e^5 (d+e x)^3}+\frac{b n}{10 e^5 (d+e x)^2}-\frac{2 b n}{15 d e^5 (d+e x)}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{6 d (d+e x)^6}+\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{30 d^2 (d+e x)^5}-\frac{b n \log (d+e x)}{30 d^2 e^5}\\ \end{align*}
Mathematica [A] time = 0.259977, size = 316, normalized size = 1.94 \[ -\frac{180 a d^4 e^2 x^2+240 a d^3 e^3 x^3+180 a d^2 e^4 x^4+72 a d^5 e x+12 a d^6+12 b d^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right ) \log \left (c x^n\right )+129 b d^4 e^2 n x^2+112 b d^3 e^3 n x^3+24 b d^2 e^4 n x^4+180 b d^4 e^2 n x^2 \log (d+e x)+240 b d^3 e^3 n x^3 \log (d+e x)+180 b d^2 e^4 n x^4 \log (d+e x)+66 b d^5 e n x+12 b d^6 n \log (d+e x)+72 b d^5 e n x \log (d+e x)+13 b d^6 n-12 b d e^5 n x^5+72 b d e^5 n x^5 \log (d+e x)+12 b e^6 n x^6 \log (d+e x)-12 b n \log (x) (d+e x)^6}{360 d^2 e^5 (d+e x)^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(x^4*(a + b*Log[c*x^n]))/(d + e*x)^7,x]
[Out]
-(12*a*d^6 + 13*b*d^6*n + 72*a*d^5*e*x + 66*b*d^5*e*n*x + 180*a*d^4*e^2*x^2 + 129*b*d^4*e^2*n*x^2 + 240*a*d^3*
e^3*x^3 + 112*b*d^3*e^3*n*x^3 + 180*a*d^2*e^4*x^4 + 24*b*d^2*e^4*n*x^4 - 12*b*d*e^5*n*x^5 - 12*b*n*(d + e*x)^6
*Log[x] + 12*b*d^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)*Log[c*x^n] + 12*b*d^6*n*Log[
d + e*x] + 72*b*d^5*e*n*x*Log[d + e*x] + 180*b*d^4*e^2*n*x^2*Log[d + e*x] + 240*b*d^3*e^3*n*x^3*Log[d + e*x] +
180*b*d^2*e^4*n*x^4*Log[d + e*x] + 72*b*d*e^5*n*x^5*Log[d + e*x] + 12*b*e^6*n*x^6*Log[d + e*x])/(360*d^2*e^5*
(d + e*x)^6)
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Maple [C] time = 0.158, size = 1022, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(a+b*ln(c*x^n))/(e*x+d)^7,x)
[Out]
-1/30*b*(15*e^4*x^4+20*d*e^3*x^3+15*d^2*e^2*x^2+6*d^3*e*x+d^4)/(e*x+d)^6/e^5*ln(x^n)+1/360*(-12*a*d^6-120*I*Pi
*b*d^3*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+12*b*d*e^5*n*x^5-24*b*d^2*e^4*n*x^4-112*b*d^3*e^3*n*x^3-129*b*d^4*e
^2*n*x^2-66*b*d^5*e*n*x-90*I*Pi*b*d^2*e^4*x^4*csgn(I*x^n)*csgn(I*c*x^n)^2-90*I*Pi*b*d^2*e^4*x^4*csgn(I*c*x^n)^
2*csgn(I*c)-240*a*d^3*e^3*x^3-180*a*d^4*e^2*x^2-72*a*d^5*e*x-120*I*Pi*b*d^3*e^3*x^3*csgn(I*c*x^n)^2*csgn(I*c)-
90*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-13*b*d^6*n-6*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2-6*I*Pi*b
*d^6*csgn(I*c*x^n)^2*csgn(I*c)-72*ln(e*x+d)*b*d*e^5*n*x^5-90*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)-36*I
*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-36*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^2*csgn(I*c)-180*ln(e*x+d)*b*d^2*e^4*
n*x^4-240*ln(e*x+d)*b*d^3*e^3*n*x^3-180*ln(e*x+d)*b*d^4*e^2*n*x^2-72*ln(e*x+d)*b*d^5*e*n*x+72*ln(-x)*b*d*e^5*n
*x^5+180*ln(-x)*b*d^2*e^4*n*x^4+240*ln(-x)*b*d^3*e^3*n*x^3+180*ln(-x)*b*d^4*e^2*n*x^2+72*ln(-x)*b*d^5*e*n*x+90
*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+36*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+
90*I*Pi*b*d^2*e^4*x^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+120*I*Pi*b*d^3*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)*csg
n(I*c)-12*ln(c)*b*d^6-180*a*d^2*e^4*x^4+90*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^3+36*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^
3+6*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-12*ln(e*x+d)*b*e^6*n*x^6+12*ln(-x)*b*e^6*n*x^6-180*ln(c)*b*
d^2*e^4*x^4-240*ln(c)*b*d^3*e^3*x^3-180*ln(c)*b*d^4*e^2*x^2+90*I*Pi*b*d^2*e^4*x^4*csgn(I*c*x^n)^3+120*I*Pi*b*d
^3*e^3*x^3*csgn(I*c*x^n)^3-72*ln(c)*b*d^5*e*x-12*ln(e*x+d)*b*d^6*n+12*ln(-x)*b*d^6*n+6*I*Pi*b*d^6*csgn(I*c*x^n
)^3)/d^2/e^5/(e*x+d)^6
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Maxima [B] time = 1.1953, size = 483, normalized size = 2.96 \begin{align*} \frac{1}{360} \, b n{\left (\frac{12 \, e^{4} x^{4} - 36 \, d e^{3} x^{3} - 76 \, d^{2} e^{2} x^{2} - 53 \, d^{3} e x - 13 \, d^{4}}{d e^{10} x^{5} + 5 \, d^{2} e^{9} x^{4} + 10 \, d^{3} e^{8} x^{3} + 10 \, d^{4} e^{7} x^{2} + 5 \, d^{5} e^{6} x + d^{6} e^{5}} - \frac{12 \, \log \left (e x + d\right )}{d^{2} e^{5}} + \frac{12 \, \log \left (x\right )}{d^{2} e^{5}}\right )} - \frac{{\left (15 \, e^{4} x^{4} + 20 \, d e^{3} x^{3} + 15 \, d^{2} e^{2} x^{2} + 6 \, d^{3} e x + d^{4}\right )} b \log \left (c x^{n}\right )}{30 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} - \frac{{\left (15 \, e^{4} x^{4} + 20 \, d e^{3} x^{3} + 15 \, d^{2} e^{2} x^{2} + 6 \, d^{3} e x + d^{4}\right )} a}{30 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")
[Out]
1/360*b*n*((12*e^4*x^4 - 36*d*e^3*x^3 - 76*d^2*e^2*x^2 - 53*d^3*e*x - 13*d^4)/(d*e^10*x^5 + 5*d^2*e^9*x^4 + 10
*d^3*e^8*x^3 + 10*d^4*e^7*x^2 + 5*d^5*e^6*x + d^6*e^5) - 12*log(e*x + d)/(d^2*e^5) + 12*log(x)/(d^2*e^5)) - 1/
30*(15*e^4*x^4 + 20*d*e^3*x^3 + 15*d^2*e^2*x^2 + 6*d^3*e*x + d^4)*b*log(c*x^n)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d
^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5) - 1/30*(15*e^4*x^4 + 20*d*e^3*x^3 + 15*d
^2*e^2*x^2 + 6*d^3*e*x + d^4)*a/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 +
6*d^5*e^6*x + d^6*e^5)
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Fricas [B] time = 1.1468, size = 790, normalized size = 4.85 \begin{align*} \frac{12 \, b d e^{5} n x^{5} - 13 \, b d^{6} n - 12 \, a d^{6} - 12 \,{\left (2 \, b d^{2} e^{4} n + 15 \, a d^{2} e^{4}\right )} x^{4} - 16 \,{\left (7 \, b d^{3} e^{3} n + 15 \, a d^{3} e^{3}\right )} x^{3} - 3 \,{\left (43 \, b d^{4} e^{2} n + 60 \, a d^{4} e^{2}\right )} x^{2} - 6 \,{\left (11 \, b d^{5} e n + 12 \, a d^{5} e\right )} x - 12 \,{\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 12 \,{\left (15 \, b d^{2} e^{4} x^{4} + 20 \, b d^{3} e^{3} x^{3} + 15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \left (c\right ) + 12 \,{\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5}\right )} \log \left (x\right )}{360 \,{\left (d^{2} e^{11} x^{6} + 6 \, d^{3} e^{10} x^{5} + 15 \, d^{4} e^{9} x^{4} + 20 \, d^{5} e^{8} x^{3} + 15 \, d^{6} e^{7} x^{2} + 6 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")
[Out]
1/360*(12*b*d*e^5*n*x^5 - 13*b*d^6*n - 12*a*d^6 - 12*(2*b*d^2*e^4*n + 15*a*d^2*e^4)*x^4 - 16*(7*b*d^3*e^3*n +
15*a*d^3*e^3)*x^3 - 3*(43*b*d^4*e^2*n + 60*a*d^4*e^2)*x^2 - 6*(11*b*d^5*e*n + 12*a*d^5*e)*x - 12*(b*e^6*n*x^6
+ 6*b*d*e^5*n*x^5 + 15*b*d^2*e^4*n*x^4 + 20*b*d^3*e^3*n*x^3 + 15*b*d^4*e^2*n*x^2 + 6*b*d^5*e*n*x + b*d^6*n)*lo
g(e*x + d) - 12*(15*b*d^2*e^4*x^4 + 20*b*d^3*e^3*x^3 + 15*b*d^4*e^2*x^2 + 6*b*d^5*e*x + b*d^6)*log(c) + 12*(b*
e^6*n*x^6 + 6*b*d*e^5*n*x^5)*log(x))/(d^2*e^11*x^6 + 6*d^3*e^10*x^5 + 15*d^4*e^9*x^4 + 20*d^5*e^8*x^3 + 15*d^6
*e^7*x^2 + 6*d^7*e^6*x + d^8*e^5)
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Sympy [A] time = 118.247, size = 2179, normalized size = 13.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4*(a+b*ln(c*x**n))/(e*x+d)**7,x)
[Out]
Piecewise((zoo*(-a/(2*x**2) - b*n*log(x)/(2*x**2) - b*n/(4*x**2) - b*log(c)/(2*x**2)), Eq(d, 0) & Eq(e, 0)), (
(-a/(2*x**2) - b*n*log(x)/(2*x**2) - b*n/(4*x**2) - b*log(c)/(2*x**2))/e**7, Eq(d, 0)), ((a*x**5/5 + b*n*x**5*
log(x)/5 - b*n*x**5/25 + b*x**5*log(c)/5)/d**7, Eq(e, 0)), (-12*a*d**6/(360*d**8*e**5 + 2160*d**7*e**6*x + 540
0*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 7
2*a*d**5*e*x/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x*
*4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 180*a*d**4*e**2*x**2/(360*d**8*e**5 + 2160*d**7*e**6*x + 54
00*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) -
240*a*d**3*e**3*x**3/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4
*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 180*a*d**2*e**4*x**4/(360*d**8*e**5 + 2160*d**7*e**
6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*
x**6) - 12*b*d**6*n*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3
+ 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 13*b*d**6*n/(360*d**8*e**5 + 2160*d**7*
e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**
11*x**6) - 72*b*d**5*e*n*x*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e*
*8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 66*b*d**5*e*n*x/(360*d**8*e**5 +
2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 3
60*d**2*e**11*x**6) - 180*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x
**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 129*b*d**4*e**
2*n*x**2/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 +
2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 240*b*d**3*e**3*n*x**3*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*
e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**
11*x**6) - 112*b*d**3*e**3*n*x**3/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**
3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 180*b*d**2*e**4*n*x**4*log(d/e + x)/(3
60*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*
e**10*x**5 + 360*d**2*e**11*x**6) - 24*b*d**2*e**4*n*x**4/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x
**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) + 72*b*d*e**5*n*
x**5*log(x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**
4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 72*b*d*e**5*n*x**5*log(d/e + x)/(360*d**8*e**5 + 2160*d**7*e
**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**1
1*x**6) + 12*b*d*e**5*n*x**5/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5
400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) + 72*b*d*e**5*x**5*log(c)/(360*d**8*e**5 + 21
60*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*
d**2*e**11*x**6) + 12*b*e**6*n*x**6*log(x)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5
*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6) - 12*b*e**6*n*x**6*log(d/e + x)
/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d*
*3*e**10*x**5 + 360*d**2*e**11*x**6) + 12*b*e**6*x**6*log(c)/(360*d**8*e**5 + 2160*d**7*e**6*x + 5400*d**6*e**
7*x**2 + 7200*d**5*e**8*x**3 + 5400*d**4*e**9*x**4 + 2160*d**3*e**10*x**5 + 360*d**2*e**11*x**6), True))
________________________________________________________________________________________
Giac [B] time = 1.24328, size = 516, normalized size = 3.17 \begin{align*} -\frac{12 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 72 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 180 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 240 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 180 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 72 \, b d^{5} n x e \log \left (x e + d\right ) - 12 \, b n x^{6} e^{6} \log \left (x\right ) - 72 \, b d n x^{5} e^{5} \log \left (x\right ) - 12 \, b d n x^{5} e^{5} + 24 \, b d^{2} n x^{4} e^{4} + 112 \, b d^{3} n x^{3} e^{3} + 129 \, b d^{4} n x^{2} e^{2} + 66 \, b d^{5} n x e + 12 \, b d^{6} n \log \left (x e + d\right ) + 180 \, b d^{2} x^{4} e^{4} \log \left (c\right ) + 240 \, b d^{3} x^{3} e^{3} \log \left (c\right ) + 180 \, b d^{4} x^{2} e^{2} \log \left (c\right ) + 72 \, b d^{5} x e \log \left (c\right ) + 13 \, b d^{6} n + 180 \, a d^{2} x^{4} e^{4} + 240 \, a d^{3} x^{3} e^{3} + 180 \, a d^{4} x^{2} e^{2} + 72 \, a d^{5} x e + 12 \, b d^{6} \log \left (c\right ) + 12 \, a d^{6}}{360 \,{\left (d^{2} x^{6} e^{11} + 6 \, d^{3} x^{5} e^{10} + 15 \, d^{4} x^{4} e^{9} + 20 \, d^{5} x^{3} e^{8} + 15 \, d^{6} x^{2} e^{7} + 6 \, d^{7} x e^{6} + d^{8} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")
[Out]
-1/360*(12*b*n*x^6*e^6*log(x*e + d) + 72*b*d*n*x^5*e^5*log(x*e + d) + 180*b*d^2*n*x^4*e^4*log(x*e + d) + 240*b
*d^3*n*x^3*e^3*log(x*e + d) + 180*b*d^4*n*x^2*e^2*log(x*e + d) + 72*b*d^5*n*x*e*log(x*e + d) - 12*b*n*x^6*e^6*
log(x) - 72*b*d*n*x^5*e^5*log(x) - 12*b*d*n*x^5*e^5 + 24*b*d^2*n*x^4*e^4 + 112*b*d^3*n*x^3*e^3 + 129*b*d^4*n*x
^2*e^2 + 66*b*d^5*n*x*e + 12*b*d^6*n*log(x*e + d) + 180*b*d^2*x^4*e^4*log(c) + 240*b*d^3*x^3*e^3*log(c) + 180*
b*d^4*x^2*e^2*log(c) + 72*b*d^5*x*e*log(c) + 13*b*d^6*n + 180*a*d^2*x^4*e^4 + 240*a*d^3*x^3*e^3 + 180*a*d^4*x^
2*e^2 + 72*a*d^5*x*e + 12*b*d^6*log(c) + 12*a*d^6)/(d^2*x^6*e^11 + 6*d^3*x^5*e^10 + 15*d^4*x^4*e^9 + 20*d^5*x^
3*e^8 + 15*d^6*x^2*e^7 + 6*d^7*x*e^6 + d^8*e^5)