3.68 \(\int \frac{x^2 (a+b \log (c x^n))}{(d+e x)^7} \, dx\)

Optimal. Leaf size=199 \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac{a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}+\frac{b n}{120 d^2 e^3 (d+e x)^2}+\frac{b n}{60 d^3 e^3 (d+e x)}+\frac{b n \log (x)}{60 d^4 e^3}-\frac{b n \log (d+e x)}{60 d^4 e^3}+\frac{b d n}{30 e^3 (d+e x)^5}-\frac{7 b n}{120 e^3 (d+e x)^4}+\frac{b n}{180 d e^3 (d+e x)^3} \]

[Out]

(b*d*n)/(30*e^3*(d + e*x)^5) - (7*b*n)/(120*e^3*(d + e*x)^4) + (b*n)/(180*d*e^3*(d + e*x)^3) + (b*n)/(120*d^2*
e^3*(d + e*x)^2) + (b*n)/(60*d^3*e^3*(d + e*x)) + (b*n*Log[x])/(60*d^4*e^3) - (d^2*(a + b*Log[c*x^n]))/(6*e^3*
(d + e*x)^6) + (2*d*(a + b*Log[c*x^n]))/(5*e^3*(d + e*x)^5) - (a + b*Log[c*x^n])/(4*e^3*(d + e*x)^4) - (b*n*Lo
g[d + e*x])/(60*d^4*e^3)

________________________________________________________________________________________

Rubi [A]  time = 0.164803, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {43, 2350, 12, 893} \[ -\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac{a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}+\frac{b n}{120 d^2 e^3 (d+e x)^2}+\frac{b n}{60 d^3 e^3 (d+e x)}+\frac{b n \log (x)}{60 d^4 e^3}-\frac{b n \log (d+e x)}{60 d^4 e^3}+\frac{b d n}{30 e^3 (d+e x)^5}-\frac{7 b n}{120 e^3 (d+e x)^4}+\frac{b n}{180 d e^3 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(b*d*n)/(30*e^3*(d + e*x)^5) - (7*b*n)/(120*e^3*(d + e*x)^4) + (b*n)/(180*d*e^3*(d + e*x)^3) + (b*n)/(120*d^2*
e^3*(d + e*x)^2) + (b*n)/(60*d^3*e^3*(d + e*x)) + (b*n*Log[x])/(60*d^4*e^3) - (d^2*(a + b*Log[c*x^n]))/(6*e^3*
(d + e*x)^6) + (2*d*(a + b*Log[c*x^n]))/(5*e^3*(d + e*x)^5) - (a + b*Log[c*x^n])/(4*e^3*(d + e*x)^4) - (b*n*Lo
g[d + e*x])/(60*d^4*e^3)

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 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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac{a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-(b n) \int \frac{-d^2-6 d e x-15 e^2 x^2}{60 e^3 x (d+e x)^6} \, dx\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac{a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-\frac{(b n) \int \frac{-d^2-6 d e x-15 e^2 x^2}{x (d+e x)^6} \, dx}{60 e^3}\\ &=-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac{a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-\frac{(b n) \int \left (-\frac{1}{d^4 x}+\frac{10 d e}{(d+e x)^6}-\frac{14 e}{(d+e x)^5}+\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}{60 e^3}\\ &=\frac{b d n}{30 e^3 (d+e x)^5}-\frac{7 b n}{120 e^3 (d+e x)^4}+\frac{b n}{180 d e^3 (d+e x)^3}+\frac{b n}{120 d^2 e^3 (d+e x)^2}+\frac{b n}{60 d^3 e^3 (d+e x)}+\frac{b n \log (x)}{60 d^4 e^3}-\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)^6}+\frac{2 d \left (a+b \log \left (c x^n\right )\right )}{5 e^3 (d+e x)^5}-\frac{a+b \log \left (c x^n\right )}{4 e^3 (d+e x)^4}-\frac{b n \log (d+e x)}{60 d^4 e^3}\\ \end{align*}

Mathematica [A]  time = 0.192912, size = 192, normalized size = 0.96 \[ \frac{144 a d^5 (d+e x)-90 a d^4 (d+e x)^2-60 a d^6+144 b d^5 (d+e x) \log \left (c x^n\right )-90 b d^4 (d+e x)^2 \log \left (c x^n\right )-60 b d^6 \log \left (c x^n\right )+12 b d^5 n (d+e x)-21 b d^4 n (d+e x)^2+2 b d^3 n (d+e x)^3+3 b d^2 n (d+e x)^4+6 b d n (d+e x)^5+6 b n \log (x) (d+e x)^6-6 b n (d+e x)^6 \log (d+e x)}{360 d^4 e^3 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x)^7,x]

[Out]

(-60*a*d^6 + 144*a*d^5*(d + e*x) + 12*b*d^5*n*(d + e*x) - 90*a*d^4*(d + e*x)^2 - 21*b*d^4*n*(d + e*x)^2 + 2*b*
d^3*n*(d + e*x)^3 + 3*b*d^2*n*(d + e*x)^4 + 6*b*d*n*(d + e*x)^5 + 6*b*n*(d + e*x)^6*Log[x] - 60*b*d^6*Log[c*x^
n] + 144*b*d^5*(d + e*x)*Log[c*x^n] - 90*b*d^4*(d + e*x)^2*Log[c*x^n] - 6*b*n*(d + e*x)^6*Log[d + e*x])/(360*d
^4*e^3*(d + e*x)^6)

________________________________________________________________________________________

Maple [C]  time = 0.133, size = 712, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))/(e*x+d)^7,x)

[Out]

-1/60*b*(15*e^2*x^2+6*d*e*x+d^2)/(e*x+d)^6/e^3*ln(x^n)+1/360*(18*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)+45*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-6*a*d^6+6*b*d*e^5*n*x^5+33*b*d^2*e^4*n*x^4+74*
b*d^3*e^3*n*x^3+63*b*d^4*e^2*n*x^2+18*b*d^5*e*n*x-18*I*Pi*b*d^5*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-18*I*Pi*b*d^5*
e*x*csgn(I*c*x^n)^2*csgn(I*c)-45*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^2*csgn(I*c)-90*a*d^4*e^2*x^2-36*a*d^5*e*x-45
*I*Pi*b*d^4*e^2*x^2*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*Pi*b*d^6*csgn(I*c
*x^n)^2*csgn(I*c)+2*b*d^6*n+3*I*Pi*b*d^6*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-36*ln(e*x+d)*b*d*e^5*n*x^5-90*ln(
e*x+d)*b*d^2*e^4*n*x^4-120*ln(e*x+d)*b*d^3*e^3*n*x^3-90*ln(e*x+d)*b*d^4*e^2*n*x^2-36*ln(e*x+d)*b*d^5*e*n*x+36*
ln(-x)*b*d*e^5*n*x^5+90*ln(-x)*b*d^2*e^4*n*x^4+120*ln(-x)*b*d^3*e^3*n*x^3+90*ln(-x)*b*d^4*e^2*n*x^2+36*ln(-x)*
b*d^5*e*n*x+45*I*Pi*b*d^4*e^2*x^2*csgn(I*c*x^n)^3+18*I*Pi*b*d^5*e*x*csgn(I*c*x^n)^3-6*ln(c)*b*d^6+3*I*Pi*b*d^6
*csgn(I*c*x^n)^3-6*ln(e*x+d)*b*e^6*n*x^6+6*ln(-x)*b*e^6*n*x^6-90*ln(c)*b*d^4*e^2*x^2-36*ln(c)*b*d^5*e*x-6*ln(e
*x+d)*b*d^6*n+6*ln(-x)*b*d^6*n)/d^4/e^3/(e*x+d)^6

________________________________________________________________________________________

Maxima [A]  time = 1.2428, size = 427, normalized size = 2.15 \begin{align*} \frac{1}{360} \, b n{\left (\frac{6 \, e^{4} x^{4} + 27 \, d e^{3} x^{3} + 47 \, d^{2} e^{2} x^{2} + 16 \, d^{3} e x + 2 \, d^{4}}{d^{3} e^{8} x^{5} + 5 \, d^{4} e^{7} x^{4} + 10 \, d^{5} e^{6} x^{3} + 10 \, d^{6} e^{5} x^{2} + 5 \, d^{7} e^{4} x + d^{8} e^{3}} - \frac{6 \, \log \left (e x + d\right )}{d^{4} e^{3}} + \frac{6 \, \log \left (x\right )}{d^{4} e^{3}}\right )} - \frac{{\left (15 \, e^{2} x^{2} + 6 \, d e x + d^{2}\right )} b \log \left (c x^{n}\right )}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} - \frac{{\left (15 \, e^{2} x^{2} + 6 \, d e x + d^{2}\right )} a}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/360*b*n*((6*e^4*x^4 + 27*d*e^3*x^3 + 47*d^2*e^2*x^2 + 16*d^3*e*x + 2*d^4)/(d^3*e^8*x^5 + 5*d^4*e^7*x^4 + 10*
d^5*e^6*x^3 + 10*d^6*e^5*x^2 + 5*d^7*e^4*x + d^8*e^3) - 6*log(e*x + d)/(d^4*e^3) + 6*log(x)/(d^4*e^3)) - 1/60*
(15*e^2*x^2 + 6*d*e*x + d^2)*b*log(c*x^n)/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^7*x^4 + 20*d^3*e^6*x^3 + 15*d^4*e^
5*x^2 + 6*d^5*e^4*x + d^6*e^3) - 1/60*(15*e^2*x^2 + 6*d*e*x + d^2)*a/(e^9*x^6 + 6*d*e^8*x^5 + 15*d^2*e^7*x^4 +
20*d^3*e^6*x^3 + 15*d^4*e^5*x^2 + 6*d^5*e^4*x + d^6*e^3)

________________________________________________________________________________________

Fricas [A]  time = 1.45163, size = 729, normalized size = 3.66 \begin{align*} \frac{6 \, b d e^{5} n x^{5} + 33 \, b d^{2} e^{4} n x^{4} + 74 \, b d^{3} e^{3} n x^{3} + 2 \, b d^{6} n - 6 \, a d^{6} + 9 \,{\left (7 \, b d^{4} e^{2} n - 10 \, a d^{4} e^{2}\right )} x^{2} + 18 \,{\left (b d^{5} e n - 2 \, a d^{5} e\right )} x - 6 \,{\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 ) - 6 \,{\left (15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \left (c\right ) + 6 \,{\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}\right )} \log \left (x\right )}{360 \,{\left (d^{4} e^{9} x^{6} + 6 \, d^{5} e^{8} x^{5} + 15 \, d^{6} e^{7} x^{4} + 20 \, d^{7} e^{6} x^{3} + 15 \, d^{8} e^{5} x^{2} + 6 \, d^{9} e^{4} x + d^{10} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/360*(6*b*d*e^5*n*x^5 + 33*b*d^2*e^4*n*x^4 + 74*b*d^3*e^3*n*x^3 + 2*b*d^6*n - 6*a*d^6 + 9*(7*b*d^4*e^2*n - 10
*a*d^4*e^2)*x^2 + 18*(b*d^5*e*n - 2*a*d^5*e)*x - 6*(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)*log(e*x + d) - 6*(15*b*d^4*e^2*x^2 + 6*b*d^5*e*x
+ b*d^6)*log(c) + 6*(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)*log(x))/(d^4*e^
9*x^6 + 6*d^5*e^8*x^5 + 15*d^6*e^7*x^4 + 20*d^7*e^6*x^3 + 15*d^8*e^5*x^2 + 6*d^9*e^4*x + d^10*e^3)

________________________________________________________________________________________

Sympy [A]  time = 141.94, size = 2380, normalized size = 11.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))/(e*x+d)**7,x)

[Out]

Piecewise((zoo*(-a/(4*x**4) - b*n*log(x)/(4*x**4) - b*n/(16*x**4) - b*log(c)/(4*x**4)), Eq(d, 0) & Eq(e, 0)),
((-a/(4*x**4) - b*n*log(x)/(4*x**4) - b*n/(16*x**4) - b*log(c)/(4*x**4))/e**7, Eq(d, 0)), ((a*x**3/3 + b*n*x**
3*log(x)/3 - b*n*x**3/9 + b*x**3*log(c)/3)/d**7, Eq(e, 0)), (-6*a*d**6/(360*d**10*e**3 + 2160*d**9*e**4*x + 54
00*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 36
*a*d**5*e*x/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x*
*4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 90*a*d**4*e**2*x**2/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400
*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 6*b*
d**6*n*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6
*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 2*b*d**6*n/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*
d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 36*b*
d**5*e*n*x*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*
d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 18*b*d**5*e*n*x/(360*d**10*e**3 + 2160*d**9*e**4*
x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6
) - 90*b*d**4*e**2*n*x**2*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e*
*6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 63*b*d**4*e**2*n*x**2/(360*d**10*e
**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5
+ 360*d**4*e**9*x**6) + 120*b*d**3*e**3*n*x**3*log(x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**
2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 120*b*d**3*e**3*n*
x**3*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e
**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 74*b*d**3*e**3*n*x**3/(360*d**10*e**3 + 2160*d**9*e**4*
x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6
) + 120*b*d**3*e**3*x**3*log(c)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3
+ 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 90*b*d**2*e**4*n*x**4*log(x)/(360*d**10*e
**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5
+ 360*d**4*e**9*x**6) - 90*b*d**2*e**4*n*x**4*log(d/e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**
5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 33*b*d**2*e**
4*n*x**4/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4
+ 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 90*b*d**2*e**4*x**4*log(c)/(360*d**10*e**3 + 2160*d**9*e**4*x +
5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) +
36*b*d*e**5*n*x**5*log(x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 540
0*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 36*b*d*e**5*n*x**5*log(d/e + x)/(360*d**10*e**3
+ 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 +
360*d**4*e**9*x**6) + 6*b*d*e**5*n*x**5/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e
**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 36*b*d*e**5*x**5*log(c)/(360*d**1
0*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x
**5 + 360*d**4*e**9*x**6) + 6*b*e**6*n*x**6*log(x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 +
7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) - 6*b*e**6*n*x**6*log(d/
e + x)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 +
2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6) + 6*b*e**6*x**6*log(c)/(360*d**10*e**3 + 2160*d**9*e**4*x + 5400*d**
8*e**5*x**2 + 7200*d**7*e**6*x**3 + 5400*d**6*e**7*x**4 + 2160*d**5*e**8*x**5 + 360*d**4*e**9*x**6), True))

________________________________________________________________________________________

Giac [B]  time = 1.24182, size = 489, normalized size = 2.46 \begin{align*} -\frac{6 \, b n x^{6} e^{6} \log \left (x e + d\right ) + 36 \, b d n x^{5} e^{5} \log \left (x e + d\right ) + 90 \, b d^{2} n x^{4} e^{4} \log \left (x e + d\right ) + 120 \, b d^{3} n x^{3} e^{3} \log \left (x e + d\right ) + 90 \, b d^{4} n x^{2} e^{2} \log \left (x e + d\right ) + 36 \, b d^{5} n x e \log \left (x e + d\right ) - 6 \, b n x^{6} e^{6} \log \left (x\right ) - 36 \, b d n x^{5} e^{5} \log \left (x\right ) - 90 \, b d^{2} n x^{4} e^{4} \log \left (x\right ) - 120 \, b d^{3} n x^{3} e^{3} \log \left (x\right ) - 6 \, b d n x^{5} e^{5} - 33 \, b d^{2} n x^{4} e^{4} - 74 \, b d^{3} n x^{3} e^{3} - 63 \, b d^{4} n x^{2} e^{2} - 18 \, b d^{5} n x e + 6 \, b d^{6} n \log \left (x e + d\right ) + 90 \, b d^{4} x^{2} e^{2} \log \left (c\right ) + 36 \, b d^{5} x e \log \left (c\right ) - 2 \, b d^{6} n + 90 \, a d^{4} x^{2} e^{2} + 36 \, a d^{5} x e + 6 \, b d^{6} \log \left (c\right ) + 6 \, a d^{6}}{360 \,{\left (d^{4} x^{6} e^{9} + 6 \, d^{5} x^{5} e^{8} + 15 \, d^{6} x^{4} e^{7} + 20 \, d^{7} x^{3} e^{6} + 15 \, d^{8} x^{2} e^{5} + 6 \, d^{9} x e^{4} + d^{10} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/360*(6*b*n*x^6*e^6*log(x*e + d) + 36*b*d*n*x^5*e^5*log(x*e + d) + 90*b*d^2*n*x^4*e^4*log(x*e + d) + 120*b*d
^3*n*x^3*e^3*log(x*e + d) + 90*b*d^4*n*x^2*e^2*log(x*e + d) + 36*b*d^5*n*x*e*log(x*e + d) - 6*b*n*x^6*e^6*log(
x) - 36*b*d*n*x^5*e^5*log(x) - 90*b*d^2*n*x^4*e^4*log(x) - 120*b*d^3*n*x^3*e^3*log(x) - 6*b*d*n*x^5*e^5 - 33*b
*d^2*n*x^4*e^4 - 74*b*d^3*n*x^3*e^3 - 63*b*d^4*n*x^2*e^2 - 18*b*d^5*n*x*e + 6*b*d^6*n*log(x*e + d) + 90*b*d^4*
x^2*e^2*log(c) + 36*b*d^5*x*e*log(c) - 2*b*d^6*n + 90*a*d^4*x^2*e^2 + 36*a*d^5*x*e + 6*b*d^6*log(c) + 6*a*d^6)
/(d^4*x^6*e^9 + 6*d^5*x^5*e^8 + 15*d^6*x^4*e^7 + 20*d^7*x^3*e^6 + 15*d^8*x^2*e^5 + 6*d^9*x*e^4 + d^10*e^3)