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]

-1/30*b*n*x^5/d^2/(e*x+d)^5+1/120*b*d^2*n/e^5/(e*x+d)^4-2/45*b*d*n/e^5/(e*x+d)^3+1/10*b*n/e^5/(e*x+d)^2-2/15*b
*n/d/e^5/(e*x+d)+1/6*x^5*(a+b*ln(c*x^n))/d/(e*x+d)^6+1/30*x^5*(a+b*ln(c*x^n))/d^2/(e*x+d)^5-1/30*b*n*ln(e*x+d)
/d^2/e^5

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Rubi [A]  time = 0.13, antiderivative size = 163, normalized size of antiderivative = 1.00, 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 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 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 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])

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*}

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Mathematica [A]  time = 0.32, size = 316, normalized size = 1.94 \[ -\frac {12 a d^6+72 a d^5 e x+180 a d^4 e^2 x^2+240 a d^3 e^3 x^3+180 a d^2 e^4 x^4+12 b d^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right ) \log \left (c x^n\right )+12 b d^6 n \log (d+e x)+13 b d^6 n+66 b d^5 e n x+72 b d^5 e n x \log (d+e x)+129 b d^4 e^2 n x^2+180 b d^4 e^2 n x^2 \log (d+e x)+112 b d^3 e^3 n x^3+240 b d^3 e^3 n x^3 \log (d+e x)+24 b d^2 e^4 n x^4+180 b d^2 e^4 n x^4 \log (d+e x)+12 b e^6 n x^6 \log (d+e x)-12 b d e^5 n x^5+72 b d e^5 n x^5 \log (d+e x)-12 b n \log (x) (d+e x)^6}{360 d^2 e^5 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

-1/360*(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])/(d^2*e^
5*(d + e*x)^6)

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fricas [B]  time = 0.77, size = 356, normalized size = 2.18 \[ \frac {12 \, b d e^{5} n x^{5} - 13 \, b d^{6} n - 12 \, a d^{6} - 12 \, {\left (2 \, b d^{2} e^{4} n + 15 \, a d^{2} e^{4}\right )} x^{4} - 16 \, {\left (7 \, b d^{3} e^{3} n + 15 \, a d^{3} e^{3}\right )} x^{3} - 3 \, {\left (43 \, b d^{4} e^{2} n + 60 \, a d^{4} e^{2}\right )} x^{2} - 6 \, {\left (11 \, b d^{5} e n + 12 \, a d^{5} e\right )} x - 12 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 12 \, {\left (15 \, b d^{2} e^{4} x^{4} + 20 \, b d^{3} e^{3} x^{3} + 15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \relax (c) + 12 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5}\right )} \log \relax (x)}{360 \, {\left (d^{2} e^{11} x^{6} + 6 \, d^{3} e^{10} x^{5} + 15 \, d^{4} e^{9} x^{4} + 20 \, d^{5} e^{8} x^{3} + 15 \, d^{6} e^{7} x^{2} + 6 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \]

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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giac [B]  time = 0.34, size = 382, normalized size = 2.34 \[ -\frac {12 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 72 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 180 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 240 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 180 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 72 \, b d^{5} n x e \log \left (x e + d\right ) - 12 \, b n x^{6} e^{6} \log \relax (x) - 72 \, b d n x^{5} e^{5} \log \relax (x) - 12 \, b d n x^{5} e^{5} + 24 \, b d^{2} n x^{4} e^{4} + 112 \, b d^{3} n x^{3} e^{3} + 129 \, b d^{4} n x^{2} e^{2} + 66 \, b d^{5} n x e + 12 \, b d^{6} n \log \left (x e + d\right ) + 180 \, b d^{2} x^{4} e^{4} \log \relax (c) + 240 \, b d^{3} x^{3} e^{3} \log \relax (c) + 180 \, b d^{4} x^{2} e^{2} \log \relax (c) + 72 \, b d^{5} x e \log \relax (c) + 13 \, b d^{6} n + 180 \, a d^{2} x^{4} e^{4} + 240 \, a d^{3} x^{3} e^{3} + 180 \, a d^{4} x^{2} e^{2} + 72 \, a d^{5} x e + 12 \, b d^{6} \log \relax (c) + 12 \, a d^{6}}{360 \, {\left (d^{2} x^{6} e^{11} + 6 \, d^{3} x^{5} e^{10} + 15 \, d^{4} x^{4} e^{9} + 20 \, d^{5} x^{3} e^{8} + 15 \, d^{6} x^{2} e^{7} + 6 \, d^{7} x e^{6} + d^{8} e^{5}\right )}} \]

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)

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maple [C]  time = 0.27, size = 1022, normalized size = 6.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*ln(c*x^n)+a)/(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*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+12*b*d^6*n*ln(e*x+d)-12*b*d^6*n*ln(-
x)+180*a*d^2*e^4*x^4+240*a*d^3*e^3*x^3+180*a*d^4*e^2*x^2+72*a*d^5*e*x+12*a*d^6+12*b*d^6*ln(c)+13*b*d^6*n+36*I*
Pi*b*d^5*e*x*csgn(I*c*x^n)^2*csgn(I*c)+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^4*e^2*x^2
*csgn(I*x^n)*csgn(I*c*x^n)^2+12*b*e^6*n*x^6*ln(e*x+d)-12*b*e^6*n*x^6*ln(-x)+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)^2-90*I*Pi*b*d^2*e^4*x^4*csgn(I*c*x^n)^3+90*I*Pi*b*d^
2*e^4*x^4*csgn(I*c*x^n)^2*csgn(I*c)+120*I*Pi*b*d^3*e^3*x^3*csgn(I*x^n)*csgn(I*c*x^n)^2+120*I*Pi*b*d^3*e^3*x^3*
csgn(I*c*x^n)^2*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)+180*b*d^4*e^2*x^2*ln(c)+72
*b*d^5*e*x*ln(c)+180*b*d^2*e^4*x^4*ln(c)+240*b*d^3*e^3*x^3*ln(c)-6*I*Pi*b*d^6*csgn(I*c*x^n)^3-36*I*Pi*b*d^5*e*
x*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)*csgn(I*c)-90*I*Pi*b*d^4
*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+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*b*d*e^5*n*x^5*ln(e*x+d)+180*b*d^2*e^4*n*x^4*ln(e*x+d)+240*b*d^3*e^3*n*x^3*ln(e*x+d)+180*b*d^
4*e^2*n*x^2*ln(e*x+d)+72*b*d^5*e*n*x*ln(e*x+d)-72*b*d*e^5*n*x^5*ln(-x)-180*b*d^2*e^4*n*x^4*ln(-x)-240*b*d^3*e^
3*n*x^3*ln(-x)-180*b*d^4*e^2*n*x^2*ln(-x)-72*b*d^5*e*n*x*ln(-x)-120*I*Pi*b*d^3*e^3*x^3*csgn(I*c*x^n)^3-90*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))/d^2/e^5/(e*x+d)^6

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maxima [B]  time = 0.76, size = 358, normalized size = 2.20 \[ \frac {1}{360} \, b n {\left (\frac {12 \, e^{4} x^{4} - 36 \, d e^{3} x^{3} - 76 \, d^{2} e^{2} x^{2} - 53 \, d^{3} e x - 13 \, d^{4}}{d e^{10} x^{5} + 5 \, d^{2} e^{9} x^{4} + 10 \, d^{3} e^{8} x^{3} + 10 \, d^{4} e^{7} x^{2} + 5 \, d^{5} e^{6} x + d^{6} e^{5}} - \frac {12 \, \log \left (e x + d\right )}{d^{2} e^{5}} + \frac {12 \, \log \relax (x)}{d^{2} e^{5}}\right )} - \frac {{\left (15 \, e^{4} x^{4} + 20 \, d e^{3} x^{3} + 15 \, d^{2} e^{2} x^{2} + 6 \, d^{3} e x + d^{4}\right )} b \log \left (c x^{n}\right )}{30 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} - \frac {{\left (15 \, e^{4} x^{4} + 20 \, d e^{3} x^{3} + 15 \, d^{2} e^{2} x^{2} + 6 \, d^{3} e x + d^{4}\right )} a}{30 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

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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mupad [B]  time = 4.26, size = 320, normalized size = 1.96 \[ -\frac {x^4\,\left (15\,a\,e^4+2\,b\,e^4\,n\right )+x\,\left (6\,a\,d^3\,e+\frac {11\,b\,d^3\,e\,n}{2}\right )+a\,d^4+x^2\,\left (15\,a\,d^2\,e^2+\frac {43\,b\,d^2\,e^2\,n}{4}\right )+x^3\,\left (20\,a\,d\,e^3+\frac {28\,b\,d\,e^3\,n}{3}\right )+\frac {13\,b\,d^4\,n}{12}-\frac {b\,e^5\,n\,x^5}{d}}{30\,d^6\,e^5+180\,d^5\,e^6\,x+450\,d^4\,e^7\,x^2+600\,d^3\,e^8\,x^3+450\,d^2\,e^9\,x^4+180\,d\,e^{10}\,x^5+30\,e^{11}\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^4}{30\,e^5}+\frac {b\,x^4}{2\,e}+\frac {b\,d^2\,x^2}{2\,e^3}+\frac {2\,b\,d\,x^3}{3\,e^2}+\frac {b\,d^3\,x}{5\,e^4}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{15\,d^2\,e^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*log(c*x^n)))/(d + e*x)^7,x)

[Out]

- (x^4*(15*a*e^4 + 2*b*e^4*n) + x*(6*a*d^3*e + (11*b*d^3*e*n)/2) + a*d^4 + x^2*(15*a*d^2*e^2 + (43*b*d^2*e^2*n
)/4) + x^3*(20*a*d*e^3 + (28*b*d*e^3*n)/3) + (13*b*d^4*n)/12 - (b*e^5*n*x^5)/d)/(30*d^6*e^5 + 30*e^11*x^6 + 18
0*d^5*e^6*x + 180*d*e^10*x^5 + 450*d^4*e^7*x^2 + 600*d^3*e^8*x^3 + 450*d^2*e^9*x^4) - (log(c*x^n)*((b*d^4)/(30
*e^5) + (b*x^4)/(2*e) + (b*d^2*x^2)/(2*e^3) + (2*b*d*x^3)/(3*e^2) + (b*d^3*x)/(5*e^4)))/(d^6 + e^6*x^6 + 6*d*e
^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x) - (b*n*atanh((2*e*x)/d + 1))/(15*d^2*e^
5)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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