3.64 \(\int \frac{x^6 (a+b \log (c x^n))}{(d+e x)^7} \, dx\)

Optimal. Leaf size=243 \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^7}-\frac{x^5 \left (6 a+6 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac{x^4 \left (30 a+30 b \log \left (c x^n\right )+11 b n\right )}{120 e^3 (d+e x)^4}-\frac{x^3 \left (60 a+60 b \log \left (c x^n\right )+37 b n\right )}{180 e^4 (d+e x)^3}-\frac{x^2 \left (20 a+20 b \log \left (c x^n\right )+19 b n\right )}{40 e^5 (d+e x)^2}-\frac{x \left (20 a+20 b \log \left (c x^n\right )+29 b n\right )}{20 e^6 (d+e x)}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (20 a+20 b \log \left (c x^n\right )+49 b n\right )}{20 e^7}-\frac{x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6} \]

[Out]

-(x^6*(a + b*Log[c*x^n]))/(6*e*(d + e*x)^6) - (x^5*(6*a + b*n + 6*b*Log[c*x^n]))/(30*e^2*(d + e*x)^5) - (x^2*(
20*a + 19*b*n + 20*b*Log[c*x^n]))/(40*e^5*(d + e*x)^2) - (x*(20*a + 29*b*n + 20*b*Log[c*x^n]))/(20*e^6*(d + e*
x)) - (x^4*(30*a + 11*b*n + 30*b*Log[c*x^n]))/(120*e^3*(d + e*x)^4) - (x^3*(60*a + 37*b*n + 60*b*Log[c*x^n]))/
(180*e^4*(d + e*x)^3) + ((20*a + 49*b*n + 20*b*Log[c*x^n])*Log[1 + (e*x)/d])/(20*e^7) + (b*n*PolyLog[2, -((e*x
)/d)])/e^7

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Rubi [A]  time = 0.538136, antiderivative size = 316, normalized size of antiderivative = 1.3, number of steps used = 21, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {43, 2351, 2319, 44, 2314, 31, 2317, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^7}-\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{6 e^7 (d+e x)^6}+\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{5 e^7 (d+e x)^5}-\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{4 e^7 (d+e x)^4}+\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^7 (d+e x)^3}-\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7 (d+e x)^2}-\frac{6 x \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^7}+\frac{b d^5 n}{30 e^7 (d+e x)^5}-\frac{31 b d^4 n}{120 e^7 (d+e x)^4}+\frac{163 b d^3 n}{180 e^7 (d+e x)^3}-\frac{79 b d^2 n}{40 e^7 (d+e x)^2}+\frac{71 b d n}{20 e^7 (d+e x)}+\frac{49 b n \log (d+e x)}{20 e^7}+\frac{71 b n \log (x)}{20 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(b*d^5*n)/(30*e^7*(d + e*x)^5) - (31*b*d^4*n)/(120*e^7*(d + e*x)^4) + (163*b*d^3*n)/(180*e^7*(d + e*x)^3) - (7
9*b*d^2*n)/(40*e^7*(d + e*x)^2) + (71*b*d*n)/(20*e^7*(d + e*x)) + (71*b*n*Log[x])/(20*e^7) - (d^6*(a + b*Log[c
*x^n]))/(6*e^7*(d + e*x)^6) + (6*d^5*(a + b*Log[c*x^n]))/(5*e^7*(d + e*x)^5) - (15*d^4*(a + b*Log[c*x^n]))/(4*
e^7*(d + e*x)^4) + (20*d^3*(a + b*Log[c*x^n]))/(3*e^7*(d + e*x)^3) - (15*d^2*(a + b*Log[c*x^n]))/(2*e^7*(d + e
*x)^2) - (6*x*(a + b*Log[c*x^n]))/(e^6*(d + e*x)) + (49*b*n*Log[d + e*x])/(20*e^7) + ((a + b*Log[c*x^n])*Log[1
+ (e*x)/d])/e^7 + (b*n*PolyLog[2, -((e*x)/d)])/e^7

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\int \left (\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^7}-\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^6}+\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^5}-\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^4}+\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^3}-\frac{6 d \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^2}+\frac{a+b \log \left (c x^n\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^6}-\frac{(6 d) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^6}+\frac{\left (15 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^6}-\frac{\left (20 d^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^6}+\frac{\left (15 d^4\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{e^6}-\frac{\left (6 d^5\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{e^6}+\frac{d^6 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{e^6}\\ &=-\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{6 e^7 (d+e x)^6}+\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{5 e^7 (d+e x)^5}-\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{4 e^7 (d+e x)^4}+\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^7 (d+e x)^3}-\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7 (d+e x)^2}-\frac{6 x \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^7}-\frac{(b n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^7}+\frac{\left (15 b d^2 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{2 e^7}-\frac{\left (20 b d^3 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 e^7}+\frac{\left (15 b d^4 n\right ) \int \frac{1}{x (d+e x)^4} \, dx}{4 e^7}-\frac{\left (6 b d^5 n\right ) \int \frac{1}{x (d+e x)^5} \, dx}{5 e^7}+\frac{\left (b d^6 n\right ) \int \frac{1}{x (d+e x)^6} \, dx}{6 e^7}+\frac{(6 b n) \int \frac{1}{d+e x} \, dx}{e^6}\\ &=-\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{6 e^7 (d+e x)^6}+\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{5 e^7 (d+e x)^5}-\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{4 e^7 (d+e x)^4}+\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^7 (d+e x)^3}-\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7 (d+e x)^2}-\frac{6 x \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)}+\frac{6 b n \log (d+e x)}{e^7}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^7}+\frac{b n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^7}+\frac{\left (15 b d^2 n\right ) \int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx}{2 e^7}-\frac{\left (20 b d^3 n\right ) \int \left (\frac{1}{d^3 x}-\frac{e}{d (d+e x)^3}-\frac{e}{d^2 (d+e x)^2}-\frac{e}{d^3 (d+e x)}\right ) \, dx}{3 e^7}+\frac{\left (15 b d^4 n\right ) \int \left (\frac{1}{d^4 x}-\frac{e}{d (d+e x)^4}-\frac{e}{d^2 (d+e x)^3}-\frac{e}{d^3 (d+e x)^2}-\frac{e}{d^4 (d+e x)}\right ) \, dx}{4 e^7}-\frac{\left (6 b d^5 n\right ) \int \left (\frac{1}{d^5 x}-\frac{e}{d (d+e x)^5}-\frac{e}{d^2 (d+e x)^4}-\frac{e}{d^3 (d+e x)^3}-\frac{e}{d^4 (d+e x)^2}-\frac{e}{d^5 (d+e x)}\right ) \, dx}{5 e^7}+\frac{\left (b d^6 n\right ) \int \left (\frac{1}{d^6 x}-\frac{e}{d (d+e x)^6}-\frac{e}{d^2 (d+e x)^5}-\frac{e}{d^3 (d+e x)^4}-\frac{e}{d^4 (d+e x)^3}-\frac{e}{d^5 (d+e x)^2}-\frac{e}{d^6 (d+e x)}\right ) \, dx}{6 e^7}\\ &=\frac{b d^5 n}{30 e^7 (d+e x)^5}-\frac{31 b d^4 n}{120 e^7 (d+e x)^4}+\frac{163 b d^3 n}{180 e^7 (d+e x)^3}-\frac{79 b d^2 n}{40 e^7 (d+e x)^2}+\frac{71 b d n}{20 e^7 (d+e x)}+\frac{71 b n \log (x)}{20 e^7}-\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{6 e^7 (d+e x)^6}+\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{5 e^7 (d+e x)^5}-\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{4 e^7 (d+e x)^4}+\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^7 (d+e x)^3}-\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7 (d+e x)^2}-\frac{6 x \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)}+\frac{49 b n \log (d+e x)}{20 e^7}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^7}+\frac{b n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.447716, size = 333, normalized size = 1.37 \[ \frac{360 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{432 a d^5 (d+e x)-1350 a d^4 (d+e x)^2+2400 a d^3 (d+e x)^3-2700 a d^2 (d+e x)^4-60 a d^6+2160 a d (d+e x)^5+360 a (d+e x)^6 \log \left (\frac{e x}{d}+1\right )+432 b d^5 (d+e x) \log \left (c x^n\right )-1350 b d^4 (d+e x)^2 \log \left (c x^n\right )+2400 b d^3 (d+e x)^3 \log \left (c x^n\right )-2700 b d^2 (d+e x)^4 \log \left (c x^n\right )-60 b d^6 \log \left (c x^n\right )+2160 b d (d+e x)^5 \log \left (c x^n\right )+360 b (d+e x)^6 \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )+12 b d^5 n (d+e x)-93 b d^4 n (d+e x)^2+326 b d^3 n (d+e x)^3-711 b d^2 n (d+e x)^4+1278 b d n (d+e x)^5+882 b n (d+e x)^6 \log (d+e x)}{(d+e x)^6}-882 b n \log (x)}{360 e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(-882*b*n*Log[x] + (-60*a*d^6 + 432*a*d^5*(d + e*x) + 12*b*d^5*n*(d + e*x) - 1350*a*d^4*(d + e*x)^2 - 93*b*d^4
*n*(d + e*x)^2 + 2400*a*d^3*(d + e*x)^3 + 326*b*d^3*n*(d + e*x)^3 - 2700*a*d^2*(d + e*x)^4 - 711*b*d^2*n*(d +
e*x)^4 + 2160*a*d*(d + e*x)^5 + 1278*b*d*n*(d + e*x)^5 - 60*b*d^6*Log[c*x^n] + 432*b*d^5*(d + e*x)*Log[c*x^n]
- 1350*b*d^4*(d + e*x)^2*Log[c*x^n] + 2400*b*d^3*(d + e*x)^3*Log[c*x^n] - 2700*b*d^2*(d + e*x)^4*Log[c*x^n] +
2160*b*d*(d + e*x)^5*Log[c*x^n] + 882*b*n*(d + e*x)^6*Log[d + e*x] + 360*a*(d + e*x)^6*Log[1 + (e*x)/d] + 360*
b*(d + e*x)^6*Log[c*x^n]*Log[1 + (e*x)/d])/(d + e*x)^6 + 360*b*n*PolyLog[2, -((e*x)/d)])/(360*e^7)

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Maple [C]  time = 0.174, size = 1416, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a+b*ln(c*x^n))/(e*x+d)^7,x)

[Out]

-1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^6/e^7/(e*x+d)^6-1/12*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*d^6/e^7/(e*x+
d)^6+15/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*d^4/e^7/(e*x+d)^4-b*n/e^7*dilog(-e*x/d)+a/e^7*ln(e*x+d)-1
/2*I*b*Pi*csgn(I*c*x^n)^3/e^7*ln(e*x+d)-15/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^2/e^7/(e*x+d)^2-15/4*I*b*Pi*
csgn(I*c*x^n)^2*csgn(I*c)*d^2/e^7/(e*x+d)^2-15/2*b*ln(c)*d^2/e^7/(e*x+d)^2+20/3*b*ln(c)*d^3/e^7/(e*x+d)^3+10/3
*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^3/e^7/(e*x+d)^3+3/5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^5/e^7/(e*x+d)^5
+10/3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*d^3/e^7/(e*x+d)^3+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*d/e^7/(e*x+d)-15/8
*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*d^4/e^7/(e*x+d)^4-15/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^4/e^7/(e*x+d)^4+
3/5*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*d^5/e^7/(e*x+d)^5+3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d/e^7/(e*x+d)-1/6*
b*ln(c)*d^6/e^7/(e*x+d)^6+6/5*b*ln(c)*d^5/e^7/(e*x+d)^5+6*b*ln(c)*d/e^7/(e*x+d)-15/4*b*ln(c)*d^4/e^7/(e*x+d)^4
+15/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*d^2/e^7/(e*x+d)^2+6/5*b*ln(x^n)*d^5/e^7/(e*x+d)^5+6*b*ln(x^n)
*d/e^7/(e*x+d)-15/4*b*ln(x^n)*d^4/e^7/(e*x+d)^4-3/5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*d^5/e^7/(e*x+d)
^5-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*d/e^7/(e*x+d)-10/3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*
d^3/e^7/(e*x+d)^3-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^7*ln(e*x+d)+71/20*b*n*d/e^7/(e*x+d)-79/40*b
*n*d^2/e^7/(e*x+d)^2+163/180*b*n*d^3/e^7/(e*x+d)^3-31/120*b*n*d^4/e^7/(e*x+d)^4+1/30*b*n*d^5/e^7/(e*x+d)^5-b*n
/e^7*ln(e*x+d)*ln(-e*x/d)+15/4*I*b*Pi*csgn(I*c*x^n)^3*d^2/e^7/(e*x+d)^2+1/12*I*b*Pi*csgn(I*c*x^n)^3*d^6/e^7/(e
*x+d)^6-3*I*b*Pi*csgn(I*c*x^n)^3*d/e^7/(e*x+d)+15/8*I*b*Pi*csgn(I*c*x^n)^3*d^4/e^7/(e*x+d)^4-3/5*I*b*Pi*csgn(I
*c*x^n)^3*d^5/e^7/(e*x+d)^5-15/2*b*ln(x^n)*d^2/e^7/(e*x+d)^2+20/3*b*ln(x^n)*d^3/e^7/(e*x+d)^3-10/3*I*b*Pi*csgn
(I*c*x^n)^3*d^3/e^7/(e*x+d)^3+1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^7*ln(e*x+d)+1/2*I*b*Pi*csgn(I*x^n)*csgn(I
*c*x^n)^2/e^7*ln(e*x+d)+20/3*a*d^3/e^7/(e*x+d)^3+6/5*a*d^5/e^7/(e*x+d)^5+6*a*d/e^7/(e*x+d)-1/6*b*ln(x^n)*d^6/e
^7/(e*x+d)^6+1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*d^6/e^7/(e*x+d)^6+b*ln(x^n)/e^7*ln(e*x+d)-49/20*b
*n/e^7*ln(e*x)+49/20*b*n/e^7*ln(e*x+d)-15/4*a*d^4/e^7/(e*x+d)^4-1/6*a*d^6/e^7/(e*x+d)^6-15/2*a*d^2/e^7/(e*x+d)
^2+b*ln(c)/e^7*ln(e*x+d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{60} \, a{\left (\frac{360 \, d e^{5} x^{5} + 1350 \, d^{2} e^{4} x^{4} + 2200 \, d^{3} e^{3} x^{3} + 1875 \, d^{4} e^{2} x^{2} + 822 \, d^{5} e x + 147 \, d^{6}}{e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}} + \frac{60 \, \log \left (e x + d\right )}{e^{7}}\right )} + b \int \frac{x^{6} \log \left (c\right ) + x^{6} \log \left (x^{n}\right )}{e^{7} x^{7} + 7 \, d e^{6} x^{6} + 21 \, d^{2} e^{5} x^{5} + 35 \, d^{3} e^{4} x^{4} + 35 \, d^{4} e^{3} x^{3} + 21 \, d^{5} e^{2} x^{2} + 7 \, d^{6} e x + d^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/60*a*((360*d*e^5*x^5 + 1350*d^2*e^4*x^4 + 2200*d^3*e^3*x^3 + 1875*d^4*e^2*x^2 + 822*d^5*e*x + 147*d^6)/(e^13
*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + 60*log(e*x
+ d)/e^7) + b*integrate((x^6*log(c) + x^6*log(x^n))/(e^7*x^7 + 7*d*e^6*x^6 + 21*d^2*e^5*x^5 + 35*d^3*e^4*x^4
+ 35*d^4*e^3*x^3 + 21*d^5*e^2*x^2 + 7*d^6*e*x + d^7), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{6} \log \left (c x^{n}\right ) + a x^{6}}{e^{7} x^{7} + 7 \, d e^{6} x^{6} + 21 \, d^{2} e^{5} x^{5} + 35 \, d^{3} e^{4} x^{4} + 35 \, d^{4} e^{3} x^{3} + 21 \, d^{5} e^{2} x^{2} + 7 \, d^{6} e x + d^{7}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

integral((b*x^6*log(c*x^n) + a*x^6)/(e^7*x^7 + 7*d*e^6*x^6 + 21*d^2*e^5*x^5 + 35*d^3*e^4*x^4 + 35*d^4*e^3*x^3
+ 21*d^5*e^2*x^2 + 7*d^6*e*x + d^7), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(a+b*ln(c*x**n))/(e*x+d)**7,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{6}}{{\left (e x + d\right )}^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^6/(e*x + d)^7, x)