3.1.63 \(\int \frac {x^7 (a+b \log (c x^n))}{(d+e x)^7} \, dx\) [63]

Optimal. Leaf size=285 \[ -\frac {7 b n x}{e^7}+\frac {(140 a+223 b n) x}{20 e^7}+\frac {7 b x \log \left (c x^n\right )}{e^7}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}-\frac {x^6 \left (7 a+b n+7 b \log \left (c x^n\right )\right )}{30 e^2 (d+e x)^5}-\frac {x^5 \left (42 a+13 b n+42 b \log \left (c x^n\right )\right )}{120 e^3 (d+e x)^4}-\frac {x^2 \left (140 a+153 b n+140 b \log \left (c x^n\right )\right )}{40 e^6 (d+e x)}-\frac {x^4 \left (210 a+107 b n+210 b \log \left (c x^n\right )\right )}{360 e^4 (d+e x)^3}-\frac {x^3 \left (420 a+319 b n+420 b \log \left (c x^n\right )\right )}{360 e^5 (d+e x)^2}-\frac {d \left (140 a+223 b n+140 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{20 e^8}-\frac {7 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^8} \]

[Out]

-7*b*n*x/e^7+1/20*(223*b*n+140*a)*x/e^7+7*b*x*ln(c*x^n)/e^7-1/6*x^7*(a+b*ln(c*x^n))/e/(e*x+d)^6-1/30*x^6*(7*a+
b*n+7*b*ln(c*x^n))/e^2/(e*x+d)^5-1/120*x^5*(42*a+13*b*n+42*b*ln(c*x^n))/e^3/(e*x+d)^4-1/40*x^2*(140*a+153*b*n+
140*b*ln(c*x^n))/e^6/(e*x+d)-1/360*x^4*(210*a+107*b*n+210*b*ln(c*x^n))/e^4/(e*x+d)^3-1/360*x^3*(420*a+319*b*n+
420*b*ln(c*x^n))/e^5/(e*x+d)^2-1/20*d*(140*a+223*b*n+140*b*ln(c*x^n))*ln(1+e*x/d)/e^8-7*b*d*n*polylog(2,-e*x/d
)/e^8

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Rubi [A]
time = 0.57, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2384, 45, 2393, 2332, 2354, 2438} \begin {gather*} -\frac {7 b d n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^8}-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (140 a+140 b \log \left (c x^n\right )+223 b n\right )}{20 e^8}-\frac {x^2 \left (140 a+140 b \log \left (c x^n\right )+153 b n\right )}{40 e^6 (d+e x)}-\frac {x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{360 e^5 (d+e x)^2}-\frac {x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{360 e^4 (d+e x)^3}-\frac {x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{120 e^3 (d+e x)^4}-\frac {x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac {x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac {x (140 a+223 b n)}{20 e^7}+\frac {7 b x \log \left (c x^n\right )}{e^7}-\frac {7 b n x}{e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b*n*x)/e^7 + ((140*a + 223*b*n)*x)/(20*e^7) + (7*b*x*Log[c*x^n])/e^7 - (x^7*(a + b*Log[c*x^n]))/(6*e*(d +
e*x)^6) - (x^6*(7*a + b*n + 7*b*Log[c*x^n]))/(30*e^2*(d + e*x)^5) - (x^5*(42*a + 13*b*n + 42*b*Log[c*x^n]))/(1
20*e^3*(d + e*x)^4) - (x^2*(140*a + 153*b*n + 140*b*Log[c*x^n]))/(40*e^6*(d + e*x)) - (x^4*(210*a + 107*b*n +
210*b*Log[c*x^n]))/(360*e^4*(d + e*x)^3) - (x^3*(420*a + 319*b*n + 420*b*Log[c*x^n]))/(360*e^5*(d + e*x)^2) -
(d*(140*a + 223*b*n + 140*b*Log[c*x^n])*Log[1 + (e*x)/d])/(20*e^8) - (7*b*d*n*PolyLog[2, -((e*x)/d)])/e^8

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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2354

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 2384

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

Rule 2393

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 2438

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^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{e^7}-\frac {d^7 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^7}+\frac {7 d^6 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^6}-\frac {21 d^5 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^5}+\frac {35 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^4}-\frac {35 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^3}+\frac {21 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^2}-\frac {7 d \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^7}-\frac {(7 d) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^7}+\frac {\left (21 d^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^7}-\frac {\left (35 d^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^7}+\frac {\left (35 d^4\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^7}-\frac {\left (21 d^5\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{e^7}+\frac {\left (7 d^6\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{e^7}-\frac {d^7 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{e^7}\\ &=\frac {a x}{e^7}+\frac {d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac {7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac {21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac {35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac {35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac {21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac {7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^8}+\frac {b \int \log \left (c x^n\right ) \, dx}{e^7}+\frac {(7 b d n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^8}-\frac {\left (35 b d^3 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 e^8}+\frac {\left (35 b d^4 n\right ) \int \frac {1}{x (d+e x)^3} \, dx}{3 e^8}-\frac {\left (21 b d^5 n\right ) \int \frac {1}{x (d+e x)^4} \, dx}{4 e^8}+\frac {\left (7 b d^6 n\right ) \int \frac {1}{x (d+e x)^5} \, dx}{5 e^8}-\frac {\left (b d^7 n\right ) \int \frac {1}{x (d+e x)^6} \, dx}{6 e^8}-\frac {(21 b d n) \int \frac {1}{d+e x} \, dx}{e^7}\\ &=\frac {a x}{e^7}-\frac {b n x}{e^7}+\frac {b x \log \left (c x^n\right )}{e^7}+\frac {d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac {7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac {21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac {35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac {35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac {21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac {21 b d n \log (d+e x)}{e^8}-\frac {7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^8}-\frac {7 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^8}-\frac {\left (35 b d^3 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^8}+\frac {\left (35 b d^4 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^8}-\frac {\left (21 b d^5 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^8}+\frac {\left (7 b d^6 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^8}-\frac {\left (b d^7 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^8}\\ &=\frac {a x}{e^7}-\frac {b n x}{e^7}-\frac {b d^6 n}{30 e^8 (d+e x)^5}+\frac {37 b d^5 n}{120 e^8 (d+e x)^4}-\frac {241 b d^4 n}{180 e^8 (d+e x)^3}+\frac {153 b d^3 n}{40 e^8 (d+e x)^2}-\frac {197 b d^2 n}{20 e^8 (d+e x)}-\frac {197 b d n \log (x)}{20 e^8}+\frac {b x \log \left (c x^n\right )}{e^7}+\frac {d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac {7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac {21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac {35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac {35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac {21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac {223 b d n \log (d+e x)}{20 e^8}-\frac {7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^8}-\frac {7 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^8}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 356, normalized size = 1.25 \begin {gather*} -\frac {-360 a e x+360 b e n x-\frac {60 a d^7}{(d+e x)^6}+\frac {504 a d^6}{(d+e x)^5}+\frac {12 b d^6 n}{(d+e x)^5}-\frac {1890 a d^5}{(d+e x)^4}-\frac {111 b d^5 n}{(d+e x)^4}+\frac {4200 a d^4}{(d+e x)^3}+\frac {482 b d^4 n}{(d+e x)^3}-\frac {6300 a d^3}{(d+e x)^2}-\frac {1377 b d^3 n}{(d+e x)^2}+\frac {7560 a d^2}{d+e x}+\frac {3546 b d^2 n}{d+e x}-4014 b d n \log (x)-360 b e x \log \left (c x^n\right )-\frac {60 b d^7 \log \left (c x^n\right )}{(d+e x)^6}+\frac {504 b d^6 \log \left (c x^n\right )}{(d+e x)^5}-\frac {1890 b d^5 \log \left (c x^n\right )}{(d+e x)^4}+\frac {4200 b d^4 \log \left (c x^n\right )}{(d+e x)^3}-\frac {6300 b d^3 \log \left (c x^n\right )}{(d+e x)^2}+\frac {7560 b d^2 \log \left (c x^n\right )}{d+e x}+4014 b d n \log (d+e x)+2520 a d \log \left (1+\frac {e x}{d}\right )+2520 b d \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )+2520 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{360 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/360*(-360*a*e*x + 360*b*e*n*x - (60*a*d^7)/(d + e*x)^6 + (504*a*d^6)/(d + e*x)^5 + (12*b*d^6*n)/(d + e*x)^5
- (1890*a*d^5)/(d + e*x)^4 - (111*b*d^5*n)/(d + e*x)^4 + (4200*a*d^4)/(d + e*x)^3 + (482*b*d^4*n)/(d + e*x)^3
- (6300*a*d^3)/(d + e*x)^2 - (1377*b*d^3*n)/(d + e*x)^2 + (7560*a*d^2)/(d + e*x) + (3546*b*d^2*n)/(d + e*x) -
4014*b*d*n*Log[x] - 360*b*e*x*Log[c*x^n] - (60*b*d^7*Log[c*x^n])/(d + e*x)^6 + (504*b*d^6*Log[c*x^n])/(d + e*
x)^5 - (1890*b*d^5*Log[c*x^n])/(d + e*x)^4 + (4200*b*d^4*Log[c*x^n])/(d + e*x)^3 - (6300*b*d^3*Log[c*x^n])/(d
+ e*x)^2 + (7560*b*d^2*Log[c*x^n])/(d + e*x) + 4014*b*d*n*Log[d + e*x] + 2520*a*d*Log[1 + (e*x)/d] + 2520*b*d*
Log[c*x^n]*Log[1 + (e*x)/d] + 2520*b*d*n*PolyLog[2, -((e*x)/d)])/e^8

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 1584, normalized size = 5.56

method	result	size
risch	\(\text {Expression too large to display}\)	\(1584\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

223/20*b*n/e^8*d*ln(e*x)-223/20*b*n/e^8*d*ln(e*x+d)-197/20*b*n/e^8*d^2/(e*x+d)+153/40*b*n/e^8*d^3/(e*x+d)^2-24
1/180*b*n/e^8*d^4/(e*x+d)^3+37/120*b*n/e^8*d^5/(e*x+d)^4-1/30*b*n/e^8*d^6/(e*x+d)^5+7*b*n/e^8*d*dilog(-e*x/d)+
7/10*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/e^8*d^6/(e*x+d)^5-21/8*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x
^n)/e^8*d^5/(e*x+d)^4+7/10*I*b*Pi*csgn(I*c*x^n)^3/e^8*d^6/(e*x+d)^5-1/12*I*b*Pi*csgn(I*c*x^n)^3*d^7/e^8/(e*x+d
)^6-35/4*I*b*Pi*csgn(I*c*x^n)^3/e^8*d^3/(e*x+d)^2+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/e^7*x+1/2*I*b*Pi*csgn(I
*x^n)*csgn(I*c*x^n)^2/e^7*x-1/2*I*b*Pi*csgn(I*c*x^n)^3/e^7*x+7/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/e^
8*d*ln(e*x+d)+1/6*b*ln(c)*d^7/e^8/(e*x+d)^6-7*b*ln(c)/e^8*d*ln(e*x+d)-35/3*b*ln(c)/e^8*d^4/(e*x+d)^3-7/5*b*ln(
c)/e^8*d^6/(e*x+d)^5+21/4*b*ln(c)/e^8*d^5/(e*x+d)^4-21*b*ln(c)/e^8*d^2/(e*x+d)+35/2*b*ln(c)/e^8*d^3/(e*x+d)^2+
1/6*a*d^7/e^8/(e*x+d)^6+35/2*a/e^8*d^3/(e*x+d)^2-35/3*a/e^8*d^4/(e*x+d)^3-7*a/e^8*d*ln(e*x+d)-7/5*a/e^8*d^6/(e
*x+d)^5+21/4*a/e^8*d^5/(e*x+d)^4-21*a/e^8*d^2/(e*x+d)+a/e^7*x+7*b*n/e^8*d*ln(e*x+d)*ln(-e*x/d)-35/3*b*ln(x^n)/
e^8*d^4/(e*x+d)^3-b*n*x/e^7-7*b*ln(x^n)/e^8*d*ln(e*x+d)-7/5*b*ln(x^n)/e^8*d^6/(e*x+d)^5+21/4*b*ln(x^n)/e^8*d^5
/(e*x+d)^4-21*b*ln(x^n)/e^8*d^2/(e*x+d)+35/2*b*ln(x^n)/e^8*d^3/(e*x+d)^2+1/6*b*ln(x^n)*d^7/e^8/(e*x+d)^6+b*ln(
x^n)/e^7*x-b*n/e^8*d-35/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d^4/(e*x+d)^3+1/12*I*b*Pi*csgn(I*c)*csgn(I*c*
x^n)^2*d^7/e^8/(e*x+d)^6+35/4*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/e^8*d^3/(e*x+d)^2+21/8*I*b*Pi*csgn(I*c)*csgn(I*
c*x^n)^2/e^8*d^5/(e*x+d)^4-21/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/e^8*d^2/(e*x+d)-21/8*I*b*Pi*csgn(I*c*x^n)^3/e
^8*d^5/(e*x+d)^4+21/2*I*b*Pi*csgn(I*c*x^n)^3/e^8*d^2/(e*x+d)+35/6*I*b*Pi*csgn(I*c*x^n)^3/e^8*d^4/(e*x+d)^3+7/2
*I*b*Pi*csgn(I*c*x^n)^3/e^8*d*ln(e*x+d)-21/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d^2/(e*x+d)+35/4*I*b*Pi*cs
gn(I*x^n)*csgn(I*c*x^n)^2/e^8*d^3/(e*x+d)^2+1/12*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^7/e^8/(e*x+d)^6-7/2*I*b*
Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d*ln(e*x+d)-7/10*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d^6/(e*x+d)^5+b*ln(
c)/e^7*x+21/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^8*d^5/(e*x+d)^4-35/6*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/e^8*d
^4/(e*x+d)^3-7/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/e^8*d*ln(e*x+d)-7/10*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/e^8*d^
6/(e*x+d)^5-1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/e^7*x+21/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^
n)/e^8*d^2/(e*x+d)-35/4*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/e^8*d^3/(e*x+d)^2-1/12*I*b*Pi*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)*d^7/e^8/(e*x+d)^6+35/6*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/e^8*d^4/(e*x+d)^3

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(420*d*e^(-8)*log(x*e + d) - 60*x*e^(-7) + (1260*d^2*x^5*e^5 + 5250*d^3*x^4*e^4 + 9100*d^4*x^3*e^3 + 808
5*d^5*x^2*e^2 + 3654*d^6*x*e + 669*d^7)/(x^6*e^14 + 6*d*x^5*e^13 + 15*d^2*x^4*e^12 + 20*d^3*x^3*e^11 + 15*d^4*
x^2*e^10 + 6*d^5*x*e^9 + d^6*e^8))*a + b*integrate((x^7*log(c) + x^7*log(x^n))/(x^7*e^7 + 7*d*x^6*e^6 + 21*d^2
*x^5*e^5 + 35*d^3*x^4*e^4 + 35*d^4*x^3*e^3 + 21*d^5*x^2*e^2 + 7*d^6*x*e + d^7), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 109.88, size = 1632, normalized size = 5.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*d**7*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/e**7 + 7*a*d**6*Piecewise((x/d**6, Eq(e,
0)), (-1/(5*e*(d + e*x)**5), True))/e**7 - 21*a*d**5*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), Tru
e))/e**7 + 35*a*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/e**7 - 35*a*d**3*Piecewise((
x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/e**7 + 21*a*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e*
*2*x), True))/e**7 - 7*a*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**7 + a*x/e**7 + b*d**7*n*Piece
wise((x/d**7, Eq(e, 0)), (-137*d**4/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**
3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 385*d**3*e*x/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*
x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 470*d**2*e**2*x**2/(360*d**10*e + 180
0*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - 270*d*
e**3*x**3/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3 + 1800*d**6*e**5*x**4 +
360*d**5*e**6*x**5) - 60*e**4*x**4/(360*d**10*e + 1800*d**9*e**2*x + 3600*d**8*e**3*x**2 + 3600*d**7*e**4*x**3
+ 1800*d**6*e**5*x**4 + 360*d**5*e**6*x**5) - log(x)/(6*d**6*e) + log(d/e + x)/(6*d**6*e), True))/e**7 - b*d*
*7*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))*log(c*x**n)/e**7 - 7*b*d**6*n*Piecewise((x/d**
6, Eq(e, 0)), (-25*d**3/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*x**3 + 60*d**4*e**5*
x**4) - 52*d**2*e*x/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*x**3 + 60*d**4*e**5*x**4
) - 42*d*e**2*x**2/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*x**3 + 60*d**4*e**5*x**4)
- 12*e**3*x**3/(60*d**8*e + 240*d**7*e**2*x + 360*d**6*e**3*x**2 + 240*d**5*e**4*x**3 + 60*d**4*e**5*x**4) -
log(x)/(5*d**5*e) + log(d/e + x)/(5*d**5*e), True))/e**7 + 7*b*d**6*Piecewise((x/d**6, Eq(e, 0)), (-1/(5*e*(d
+ e*x)**5), True))*log(c*x**n)/e**7 + 21*b*d**5*n*Piecewise((x/d**5, Eq(e, 0)), (-11*d**2/(24*d**6*e + 72*d**5
*e**2*x + 72*d**4*e**3*x**2 + 24*d**3*e**4*x**3) - 15*d*e*x/(24*d**6*e + 72*d**5*e**2*x + 72*d**4*e**3*x**2 +
24*d**3*e**4*x**3) - 6*e**2*x**2/(24*d**6*e + 72*d**5*e**2*x + 72*d**4*e**3*x**2 + 24*d**3*e**4*x**3) - log(x)
/(4*d**4*e) + log(d/e + x)/(4*d**4*e), True))/e**7 - 21*b*d**5*Piecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x
)**4), True))*log(c*x**n)/e**7 - 35*b*d**4*n*Piecewise((x/d**4, Eq(e, 0)), (-3*d/(6*d**4*e + 12*d**3*e**2*x +
6*d**2*e**3*x**2) - 2*e*x/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3*x**2) - log(x)/(3*d**3*e) + log(d/e + x)/(3
*d**3*e), True))/e**7 + 35*b*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))*log(c*x**n)/e**
7 + 35*b*d**3*n*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2*x) - log(x)/(2*d**2*e) + log(d/e + x)/(
2*d**2*e), True))/e**7 - 35*b*d**3*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))*log(c*x**n)/e*
*7 - 21*b*d**2*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e + x)/(d*e), True))/e**7 + 21*b*d**2*Pi
ecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/e**7 + 7*b*d*n*Piecewise((x/d, Eq(e, 0)), (
Piecewise((-polylog(2, e*x*exp_polar(I*pi)/d), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) - polylog(2, e*x
*exp_polar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/d), 1/Abs(x) < 1), (-meij
erg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp
_polar(I*pi)/d), True))/e, True))/e**7 - 7*b*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/
e**7 - b*n*x/e**7 + b*x*log(c*x**n)/e**7

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^7\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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