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3.91 \(\int \frac {\log (d (a+b x+c x^2)^n)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=519 \[ \frac {n \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{8 e \left (a e^2-b d e+c d^2\right )^4}-\frac {n \log (d+e x) \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right )}{4 e \left (a e^2-b d e+c d^2\right )^4}+\frac {n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{4 e (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac {n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{8 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac {n \sqrt {b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 \left (a e^2-b d e+c d^2\right )^4}+\frac {n (2 c d-b e)}{12 e (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4} \]

[Out]

1/12*(-b*e+2*c*d)*n/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^3+1/8*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))*n/e/(a*e^2-b*d*e+c
*d^2)^2/(e*x+d)^2+1/4*(-b*e+2*c*d)*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*n/e/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)-1/4*(2*
c^4*d^4+b^4*e^4-4*b^2*c*e^3*(a*e+b*d)-4*c^3*d^2*e*(3*a*e+b*d)+2*c^2*e^2*(a^2*e^2+6*a*b*d*e+3*b^2*d^2))*n*ln(e*
x+d)/e/(a*e^2-b*d*e+c*d^2)^4+1/8*(2*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(a*e+b*d)-4*c^3*d^2*e*(3*a*e+b*d)+2*c^2*e^2*(a
^2*e^2+6*a*b*d*e+3*b^2*d^2))*n*ln(c*x^2+b*x+a)/e/(a*e^2-b*d*e+c*d^2)^4-1/4*ln(d*(c*x^2+b*x+a)^n)/e/(e*x+d)^4+1
/4*(-b*e+2*c*d)*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))*n*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))*(-4*a*c+b^2)^(1/2)
/(a*e^2-b*d*e+c*d^2)^4

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Rubi [A]  time = 1.01, antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2525, 800, 634, 618, 206, 628} \[ \frac {n \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{8 e \left (a e^2-b d e+c d^2\right )^4}-\frac {n \log (d+e x) \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right )}{4 e \left (a e^2-b d e+c d^2\right )^4}+\frac {n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{4 e (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac {n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{8 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac {n \sqrt {b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 \left (a e^2-b d e+c d^2\right )^4}+\frac {n (2 c d-b e)}{12 e (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^5,x]

[Out]

((2*c*d - b*e)*n)/(12*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n)/(
8*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*n)/(4*e*(c
*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)
)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*(c*d^2 - b*d*e + a*e^2)^4) - ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^
3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[d + e*x])/(4*e*
(c*d^2 - b*d*e + a*e^2)^4) + ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c
^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[a + b*x + c*x^2])/(8*e*(c*d^2 - b*d*e + a*e^2)^4) - Log[d*(a +
 b*x + c*x^2)^n]/(4*e*(d + e*x)^4)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
 b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
 1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^5} \, dx &=-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}+\frac {n \int \frac {b+2 c x}{(d+e x)^4 \left (a+b x+c x^2\right )} \, dx}{4 e}\\ &=-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}+\frac {n \int \left (\frac {e (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac {e \left (-2 c^2 d^2-b^2 e^2+2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac {e (2 c d-b e) \left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e \left (-2 c^4 d^4-b^4 e^4+4 b^2 c e^3 (b d+a e)+4 c^3 d^2 e (b d+3 a e)-2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right )}{\left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {-4 b^4 c d e^3+b^5 e^4+b^3 c e^2 \left (6 c d^2-5 a e^2\right )-4 b^2 c^2 d e \left (c d^2-4 a e^2\right )+8 a c^3 d e \left (c d^2-a e^2\right )+b c^2 \left (c^2 d^4-18 a c d^2 e^2+5 a^2 e^4\right )+c \left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{\left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}\right ) \, dx}{4 e}\\ &=\frac {(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}+\frac {n \int \frac {-4 b^4 c d e^3+b^5 e^4+b^3 c e^2 \left (6 c d^2-5 a e^2\right )-4 b^2 c^2 d e \left (c d^2-4 a e^2\right )+8 a c^3 d e \left (c d^2-a e^2\right )+b c^2 \left (c^2 d^4-18 a c d^2 e^2+5 a^2 e^4\right )+c \left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{4 e \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac {(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{8 \left (c d^2-b d e+a e^2\right )^4}+\frac {\left (\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{8 e \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac {(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}+\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 e \left (c d^2-b d e+a e^2\right )^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}+\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{4 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac {(2 c d-b e) n}{12 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{8 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n}{4 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\sqrt {b^2-4 a c} (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 \left (c d^2-b d e+a e^2\right )^4}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log (d+e x)}{4 e \left (c d^2-b d e+a e^2\right )^4}+\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 e \left (c d^2-b d e+a e^2\right )^4}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e (d+e x)^4}\\ \end {align*}

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Mathematica [A]  time = 2.03, size = 469, normalized size = 0.90 \[ \frac {\frac {n (d+e x) \left (-6 (d+e x)^3 \log (d+e x) \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right )+3 (d+e x)^3 \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log (a+x (b+c x))+3 (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2+6 (d+e x)^2 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )+6 e \sqrt {b^2-4 a c} (d+e x)^3 (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+2 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3\right )}{\left (e (a e-b d)+c d^2\right )^4}-6 \log \left (d (a+x (b+c x))^n\right )}{24 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^5,x]

[Out]

((n*(d + e*x)*(2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3 + 3*(c*d^2 + e*(-(b*d) + a*e))^2*(2*c^2*d^2 + b^2*
e^2 - 2*c*e*(b*d + a*e))*(d + e*x) + 6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(c^2*d^2 + b^2*e^2 - c*e*(b*d
+ 3*a*e))*(d + e*x)^2 + 6*Sqrt[b^2 - 4*a*c]*e*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*(d + e*x
)^3*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] - 6*(2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b
*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*(d + e*x)^3*Log[d + e*x] + 3*(2*c^4*d^4 + b^4*e^4 -
 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*(d + e*x)^
3*Log[a + x*(b + c*x)]))/(c*d^2 + e*(-(b*d) + a*e))^4 - 6*Log[d*(a + x*(b + c*x))^n])/(24*e*(d + e*x)^4)

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fricas [B]  time = 92.03, size = 5824, normalized size = 11.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^5,x, algorithm="fricas")

[Out]

[1/24*(6*(2*c^4*d^5*e^3 - 5*b*c^3*d^4*e^4 + 2*(3*b^2*c^2 - 2*a*c^3)*d^3*e^5 - 2*(2*b^3*c - 3*a*b*c^2)*d^2*e^6
+ (b^4 - 6*a^2*c^2)*d*e^7 - (a*b^3 - 3*a^2*b*c)*e^8)*n*x^3 + 3*(14*c^4*d^6*e^2 - 36*b*c^3*d^5*e^3 + (43*b^2*c^
2 - 22*a*c^3)*d^4*e^4 - 4*(7*b^3*c - 8*a*b*c^2)*d^3*e^5 + (7*b^4 + 4*a*b^2*c - 38*a^2*c^2)*d^2*e^6 - 4*(2*a*b^
3 - 5*a^2*b*c)*d*e^7 + (a^2*b^2 - 2*a^3*c)*e^8)*n*x^2 + 2*(26*c^4*d^7*e - 70*b*c^3*d^6*e^2 - a^3*b*e^8 + 12*(7
*b^2*c^2 - 2*a*c^3)*d^5*e^3 - (53*b^3*c - 27*a*b*c^2)*d^4*e^4 + (13*b^4 + 24*a*b^2*c - 54*a^2*c^2)*d^3*e^5 - 6
*(3*a*b^3 - 4*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2 - 2*a^3*c)*d*e^7)*n*x + 3*((4*c^3*d^3*e^5 - 6*b*c^2*d^2*e^6 + 4*
(b^2*c - a*c^2)*d*e^7 - (b^3 - 2*a*b*c)*e^8)*n*x^4 + 4*(4*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 + 4*(b^2*c - a*c^2)*d^
2*e^6 - (b^3 - 2*a*b*c)*d*e^7)*n*x^3 + 6*(4*c^3*d^5*e^3 - 6*b*c^2*d^4*e^4 + 4*(b^2*c - a*c^2)*d^3*e^5 - (b^3 -
 2*a*b*c)*d^2*e^6)*n*x^2 + 4*(4*c^3*d^6*e^2 - 6*b*c^2*d^5*e^3 + 4*(b^2*c - a*c^2)*d^4*e^4 - (b^3 - 2*a*b*c)*d^
3*e^5)*n*x + (4*c^3*d^7*e - 6*b*c^2*d^6*e^2 + 4*(b^2*c - a*c^2)*d^5*e^3 - (b^3 - 2*a*b*c)*d^4*e^4)*n)*sqrt(b^2
 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + (22*c^4
*d^8 - 62*b*c^3*d^7*e - 2*a^3*b*d*e^7 + 3*(25*b^2*c^2 - 2*a*c^3)*d^6*e^2 - 2*(23*b^3*c + 3*a*b*c^2)*d^5*e^3 +
(11*b^4 + 36*a*b^2*c - 30*a^2*c^2)*d^4*e^4 - 6*(3*a*b^3 - a^2*b*c)*d^3*e^5 + (9*a^2*b^2 - 2*a^3*c)*d^2*e^6)*n
+ 3*((2*c^4*d^4*e^4 - 4*b*c^3*d^3*e^5 + 6*(b^2*c^2 - 2*a*c^3)*d^2*e^6 - 4*(b^3*c - 3*a*b*c^2)*d*e^7 + (b^4 - 4
*a*b^2*c + 2*a^2*c^2)*e^8)*n*x^4 + 4*(2*c^4*d^5*e^3 - 4*b*c^3*d^4*e^4 + 6*(b^2*c^2 - 2*a*c^3)*d^3*e^5 - 4*(b^3
*c - 3*a*b*c^2)*d^2*e^6 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d*e^7)*n*x^3 + 6*(2*c^4*d^6*e^2 - 4*b*c^3*d^5*e^3 + 6*
(b^2*c^2 - 2*a*c^3)*d^4*e^4 - 4*(b^3*c - 3*a*b*c^2)*d^3*e^5 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^2*e^6)*n*x^2 + 4
*(2*c^4*d^7*e - 4*b*c^3*d^6*e^2 + 6*(b^2*c^2 - 2*a*c^3)*d^5*e^3 - 4*(b^3*c - 3*a*b*c^2)*d^4*e^4 + (b^4 - 4*a*b
^2*c + 2*a^2*c^2)*d^3*e^5)*n*x + (4*b*c^3*d^7*e + 8*a^3*b*d*e^7 - 2*a^4*e^8 - 2*(3*b^2*c^2 + 10*a*c^3)*d^6*e^2
 + 4*(b^3*c + 9*a*b*c^2)*d^5*e^3 - (b^4 + 28*a*b^2*c + 10*a^2*c^2)*d^4*e^4 + 8*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 4
*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*n)*log(c*x^2 + b*x + a) - 6*((2*c^4*d^4*e^4 - 4*b*c^3*d^3*e^5 + 6*(b^2*c^2 - 2
*a*c^3)*d^2*e^6 - 4*(b^3*c - 3*a*b*c^2)*d*e^7 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^8)*n*x^4 + 4*(2*c^4*d^5*e^3 -
4*b*c^3*d^4*e^4 + 6*(b^2*c^2 - 2*a*c^3)*d^3*e^5 - 4*(b^3*c - 3*a*b*c^2)*d^2*e^6 + (b^4 - 4*a*b^2*c + 2*a^2*c^2
)*d*e^7)*n*x^3 + 6*(2*c^4*d^6*e^2 - 4*b*c^3*d^5*e^3 + 6*(b^2*c^2 - 2*a*c^3)*d^4*e^4 - 4*(b^3*c - 3*a*b*c^2)*d^
3*e^5 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^2*e^6)*n*x^2 + 4*(2*c^4*d^7*e - 4*b*c^3*d^6*e^2 + 6*(b^2*c^2 - 2*a*c^3
)*d^5*e^3 - 4*(b^3*c - 3*a*b*c^2)*d^4*e^4 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3*e^5)*n*x + (2*c^4*d^8 - 4*b*c^3*
d^7*e + 6*(b^2*c^2 - 2*a*c^3)*d^6*e^2 - 4*(b^3*c - 3*a*b*c^2)*d^5*e^3 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^4*e^4)
*n)*log(e*x + d) - 6*(c^4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 4*
(b^3*c + 3*a*b*c^2)*d^5*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^
2*b^2 + 2*a^3*c)*d^2*e^6)*log(d))/(c^4*d^12*e - 4*b*c^3*d^11*e^2 - 4*a^3*b*d^5*e^8 + a^4*d^4*e^9 + 2*(3*b^2*c^
2 + 2*a*c^3)*d^10*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^8*e^5 - 4*(a*b^3 + 3*
a^2*b*c)*d^7*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^7 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 - 4*a^3*b*d*e^12 + a^4*e^1
3 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^8 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^9 -
 4*(a*b^3 + 3*a^2*b*c)*d^3*e^10 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^11)*x^4 + 4*(c^4*d^9*e^4 - 4*b*c^3*d^8*e^5 - 4
*a^3*b*d^2*e^11 + a^4*d*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^7 + (b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d^5*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^9 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^10)*x^3 + 6*(c^4*d^1
0*e^3 - 4*b*c^3*d^9*e^4 - 4*a^3*b*d^3*e^10 + a^4*d^2*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^5 - 4*(b^3*c + 3*a*b
*c^2)*d^7*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^8 + 2*(3*a^2*b^2 + 2*a^3*
c)*d^4*e^9)*x^2 + 4*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 - 4*a^3*b*d^4*e^9 + a^4*d^3*e^10 + 2*(3*b^2*c^2 + 2*a*c^3
)*d^9*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^6
*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^8)*x), 1/24*(6*(2*c^4*d^5*e^3 - 5*b*c^3*d^4*e^4 + 2*(3*b^2*c^2 - 2*a*c^3)
*d^3*e^5 - 2*(2*b^3*c - 3*a*b*c^2)*d^2*e^6 + (b^4 - 6*a^2*c^2)*d*e^7 - (a*b^3 - 3*a^2*b*c)*e^8)*n*x^3 + 3*(14*
c^4*d^6*e^2 - 36*b*c^3*d^5*e^3 + (43*b^2*c^2 - 22*a*c^3)*d^4*e^4 - 4*(7*b^3*c - 8*a*b*c^2)*d^3*e^5 + (7*b^4 +
4*a*b^2*c - 38*a^2*c^2)*d^2*e^6 - 4*(2*a*b^3 - 5*a^2*b*c)*d*e^7 + (a^2*b^2 - 2*a^3*c)*e^8)*n*x^2 + 2*(26*c^4*d
^7*e - 70*b*c^3*d^6*e^2 - a^3*b*e^8 + 12*(7*b^2*c^2 - 2*a*c^3)*d^5*e^3 - (53*b^3*c - 27*a*b*c^2)*d^4*e^4 + (13
*b^4 + 24*a*b^2*c - 54*a^2*c^2)*d^3*e^5 - 6*(3*a*b^3 - 4*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2 - 2*a^3*c)*d*e^7)*n*x
 + 6*((4*c^3*d^3*e^5 - 6*b*c^2*d^2*e^6 + 4*(b^2*c - a*c^2)*d*e^7 - (b^3 - 2*a*b*c)*e^8)*n*x^4 + 4*(4*c^3*d^4*e
^4 - 6*b*c^2*d^3*e^5 + 4*(b^2*c - a*c^2)*d^2*e^6 - (b^3 - 2*a*b*c)*d*e^7)*n*x^3 + 6*(4*c^3*d^5*e^3 - 6*b*c^2*d
^4*e^4 + 4*(b^2*c - a*c^2)*d^3*e^5 - (b^3 - 2*a*b*c)*d^2*e^6)*n*x^2 + 4*(4*c^3*d^6*e^2 - 6*b*c^2*d^5*e^3 + 4*(
b^2*c - a*c^2)*d^4*e^4 - (b^3 - 2*a*b*c)*d^3*e^5)*n*x + (4*c^3*d^7*e - 6*b*c^2*d^6*e^2 + 4*(b^2*c - a*c^2)*d^5
*e^3 - (b^3 - 2*a*b*c)*d^4*e^4)*n)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) +
(22*c^4*d^8 - 62*b*c^3*d^7*e - 2*a^3*b*d*e^7 + 3*(25*b^2*c^2 - 2*a*c^3)*d^6*e^2 - 2*(23*b^3*c + 3*a*b*c^2)*d^5
*e^3 + (11*b^4 + 36*a*b^2*c - 30*a^2*c^2)*d^4*e^4 - 6*(3*a*b^3 - a^2*b*c)*d^3*e^5 + (9*a^2*b^2 - 2*a^3*c)*d^2*
e^6)*n + 3*((2*c^4*d^4*e^4 - 4*b*c^3*d^3*e^5 + 6*(b^2*c^2 - 2*a*c^3)*d^2*e^6 - 4*(b^3*c - 3*a*b*c^2)*d*e^7 + (
b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^8)*n*x^4 + 4*(2*c^4*d^5*e^3 - 4*b*c^3*d^4*e^4 + 6*(b^2*c^2 - 2*a*c^3)*d^3*e^5 -
 4*(b^3*c - 3*a*b*c^2)*d^2*e^6 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d*e^7)*n*x^3 + 6*(2*c^4*d^6*e^2 - 4*b*c^3*d^5*e
^3 + 6*(b^2*c^2 - 2*a*c^3)*d^4*e^4 - 4*(b^3*c - 3*a*b*c^2)*d^3*e^5 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^2*e^6)*n*
x^2 + 4*(2*c^4*d^7*e - 4*b*c^3*d^6*e^2 + 6*(b^2*c^2 - 2*a*c^3)*d^5*e^3 - 4*(b^3*c - 3*a*b*c^2)*d^4*e^4 + (b^4
- 4*a*b^2*c + 2*a^2*c^2)*d^3*e^5)*n*x + (4*b*c^3*d^7*e + 8*a^3*b*d*e^7 - 2*a^4*e^8 - 2*(3*b^2*c^2 + 10*a*c^3)*
d^6*e^2 + 4*(b^3*c + 9*a*b*c^2)*d^5*e^3 - (b^4 + 28*a*b^2*c + 10*a^2*c^2)*d^4*e^4 + 8*(a*b^3 + 3*a^2*b*c)*d^3*
e^5 - 4*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*n)*log(c*x^2 + b*x + a) - 6*((2*c^4*d^4*e^4 - 4*b*c^3*d^3*e^5 + 6*(b^2*
c^2 - 2*a*c^3)*d^2*e^6 - 4*(b^3*c - 3*a*b*c^2)*d*e^7 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^8)*n*x^4 + 4*(2*c^4*d^5
*e^3 - 4*b*c^3*d^4*e^4 + 6*(b^2*c^2 - 2*a*c^3)*d^3*e^5 - 4*(b^3*c - 3*a*b*c^2)*d^2*e^6 + (b^4 - 4*a*b^2*c + 2*
a^2*c^2)*d*e^7)*n*x^3 + 6*(2*c^4*d^6*e^2 - 4*b*c^3*d^5*e^3 + 6*(b^2*c^2 - 2*a*c^3)*d^4*e^4 - 4*(b^3*c - 3*a*b*
c^2)*d^3*e^5 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^2*e^6)*n*x^2 + 4*(2*c^4*d^7*e - 4*b*c^3*d^6*e^2 + 6*(b^2*c^2 -
2*a*c^3)*d^5*e^3 - 4*(b^3*c - 3*a*b*c^2)*d^4*e^4 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^3*e^5)*n*x + (2*c^4*d^8 - 4
*b*c^3*d^7*e + 6*(b^2*c^2 - 2*a*c^3)*d^6*e^2 - 4*(b^3*c - 3*a*b*c^2)*d^5*e^3 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*d
^4*e^4)*n)*log(e*x + d) - 6*(c^4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e
^2 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 +
2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)*log(d))/(c^4*d^12*e - 4*b*c^3*d^11*e^2 - 4*a^3*b*d^5*e^8 + a^4*d^4*e^9 + 2*(3
*b^2*c^2 + 2*a*c^3)*d^10*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^8*e^5 - 4*(a*b
^3 + 3*a^2*b*c)*d^7*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^7 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 - 4*a^3*b*d*e^12 +
a^4*e^13 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^7 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^8 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^
4*e^9 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^10 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^11)*x^4 + 4*(c^4*d^9*e^4 - 4*b*c^3*d^8*
e^5 - 4*a^3*b*d^2*e^11 + a^4*d*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^7 + (b^4 +
 12*a*b^2*c + 6*a^2*c^2)*d^5*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^9 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^10)*x^3 + 6*(
c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 - 4*a^3*b*d^3*e^10 + a^4*d^2*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^5 - 4*(b^3*c
+ 3*a*b*c^2)*d^7*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^8 + 2*(3*a^2*b^2 +
 2*a^3*c)*d^4*e^9)*x^2 + 4*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 - 4*a^3*b*d^4*e^9 + a^4*d^3*e^10 + 2*(3*b^2*c^2 +
2*a*c^3)*d^9*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^6 - 4*(a*b^3 + 3*a^2*b
*c)*d^6*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^8)*x)]

________________________________________________________________________________________

giac [B]  time = 1.17, size = 3759, normalized size = 7.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^5,x, algorithm="giac")

[Out]

1/8*(2*c^4*d^4*n - 4*b*c^3*d^3*n*e + 6*b^2*c^2*d^2*n*e^2 - 12*a*c^3*d^2*n*e^2 - 4*b^3*c*d*n*e^3 + 12*a*b*c^2*d
*n*e^3 + b^4*n*e^4 - 4*a*b^2*c*n*e^4 + 2*a^2*c^2*n*e^4)*log(c*x^2 + b*x + a)/(c^4*d^8*e - 4*b*c^3*d^7*e^2 + 6*
b^2*c^2*d^6*e^3 + 4*a*c^3*d^6*e^3 - 4*b^3*c*d^5*e^4 - 12*a*b*c^2*d^5*e^4 + b^4*d^4*e^5 + 12*a*b^2*c*d^4*e^5 +
6*a^2*c^2*d^4*e^5 - 4*a*b^3*d^3*e^6 - 12*a^2*b*c*d^3*e^6 + 6*a^2*b^2*d^2*e^7 + 4*a^3*c*d^2*e^7 - 4*a^3*b*d*e^8
 + a^4*e^9) - 1/4*(4*b^2*c^3*d^3*n - 16*a*c^4*d^3*n - 6*b^3*c^2*d^2*n*e + 24*a*b*c^3*d^2*n*e + 4*b^4*c*d*n*e^2
 - 20*a*b^2*c^2*d*n*e^2 + 16*a^2*c^3*d*n*e^2 - b^5*n*e^3 + 6*a*b^3*c*n*e^3 - 8*a^2*b*c^2*n*e^3)*arctan((2*c*x
+ b)/sqrt(-b^2 + 4*a*c))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 1
2*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^
5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sqrt(-b^2 + 4*a*c)) - 1/24*(12*c^4*d^4*n*x^
4*e^4*log(x*e + d) + 48*c^4*d^5*n*x^3*e^3*log(x*e + d) + 72*c^4*d^6*n*x^2*e^2*log(x*e + d) + 48*c^4*d^7*n*x*e*
log(x*e + d) - 12*c^4*d^5*n*x^3*e^3 - 42*c^4*d^6*n*x^2*e^2 - 52*c^4*d^7*n*x*e + 6*c^4*d^8*n*log(c*x^2 + b*x +
a) - 24*b*c^3*d^7*n*e*log(c*x^2 + b*x + a) + 12*c^4*d^8*n*log(x*e + d) - 24*b*c^3*d^3*n*x^4*e^5*log(x*e + d) -
 96*b*c^3*d^4*n*x^3*e^4*log(x*e + d) - 144*b*c^3*d^5*n*x^2*e^3*log(x*e + d) - 96*b*c^3*d^6*n*x*e^2*log(x*e + d
) - 24*b*c^3*d^7*n*e*log(x*e + d) - 22*c^4*d^8*n + 30*b*c^3*d^4*n*x^3*e^4 + 108*b*c^3*d^5*n*x^2*e^3 + 140*b*c^
3*d^6*n*x*e^2 + 62*b*c^3*d^7*n*e + 36*b^2*c^2*d^6*n*e^2*log(c*x^2 + b*x + a) + 24*a*c^3*d^6*n*e^2*log(c*x^2 +
b*x + a) + 36*b^2*c^2*d^2*n*x^4*e^6*log(x*e + d) - 72*a*c^3*d^2*n*x^4*e^6*log(x*e + d) + 144*b^2*c^2*d^3*n*x^3
*e^5*log(x*e + d) - 288*a*c^3*d^3*n*x^3*e^5*log(x*e + d) + 216*b^2*c^2*d^4*n*x^2*e^4*log(x*e + d) - 432*a*c^3*
d^4*n*x^2*e^4*log(x*e + d) + 144*b^2*c^2*d^5*n*x*e^3*log(x*e + d) - 288*a*c^3*d^5*n*x*e^3*log(x*e + d) + 36*b^
2*c^2*d^6*n*e^2*log(x*e + d) - 72*a*c^3*d^6*n*e^2*log(x*e + d) + 6*c^4*d^8*log(d) - 24*b*c^3*d^7*e*log(d) - 36
*b^2*c^2*d^3*n*x^3*e^5 + 24*a*c^3*d^3*n*x^3*e^5 - 129*b^2*c^2*d^4*n*x^2*e^4 + 66*a*c^3*d^4*n*x^2*e^4 - 168*b^2
*c^2*d^5*n*x*e^3 + 48*a*c^3*d^5*n*x*e^3 - 75*b^2*c^2*d^6*n*e^2 + 6*a*c^3*d^6*n*e^2 - 24*b^3*c*d^5*n*e^3*log(c*
x^2 + b*x + a) - 72*a*b*c^2*d^5*n*e^3*log(c*x^2 + b*x + a) - 24*b^3*c*d*n*x^4*e^7*log(x*e + d) + 72*a*b*c^2*d*
n*x^4*e^7*log(x*e + d) - 96*b^3*c*d^2*n*x^3*e^6*log(x*e + d) + 288*a*b*c^2*d^2*n*x^3*e^6*log(x*e + d) - 144*b^
3*c*d^3*n*x^2*e^5*log(x*e + d) + 432*a*b*c^2*d^3*n*x^2*e^5*log(x*e + d) - 96*b^3*c*d^4*n*x*e^4*log(x*e + d) +
288*a*b*c^2*d^4*n*x*e^4*log(x*e + d) - 24*b^3*c*d^5*n*e^3*log(x*e + d) + 72*a*b*c^2*d^5*n*e^3*log(x*e + d) + 3
6*b^2*c^2*d^6*e^2*log(d) + 24*a*c^3*d^6*e^2*log(d) + 24*b^3*c*d^2*n*x^3*e^6 - 36*a*b*c^2*d^2*n*x^3*e^6 + 84*b^
3*c*d^3*n*x^2*e^5 - 96*a*b*c^2*d^3*n*x^2*e^5 + 106*b^3*c*d^4*n*x*e^4 - 54*a*b*c^2*d^4*n*x*e^4 + 46*b^3*c*d^5*n
*e^3 + 6*a*b*c^2*d^5*n*e^3 + 6*b^4*d^4*n*e^4*log(c*x^2 + b*x + a) + 72*a*b^2*c*d^4*n*e^4*log(c*x^2 + b*x + a)
+ 36*a^2*c^2*d^4*n*e^4*log(c*x^2 + b*x + a) + 6*b^4*n*x^4*e^8*log(x*e + d) - 24*a*b^2*c*n*x^4*e^8*log(x*e + d)
 + 12*a^2*c^2*n*x^4*e^8*log(x*e + d) + 24*b^4*d*n*x^3*e^7*log(x*e + d) - 96*a*b^2*c*d*n*x^3*e^7*log(x*e + d) +
 48*a^2*c^2*d*n*x^3*e^7*log(x*e + d) + 36*b^4*d^2*n*x^2*e^6*log(x*e + d) - 144*a*b^2*c*d^2*n*x^2*e^6*log(x*e +
 d) + 72*a^2*c^2*d^2*n*x^2*e^6*log(x*e + d) + 24*b^4*d^3*n*x*e^5*log(x*e + d) - 96*a*b^2*c*d^3*n*x*e^5*log(x*e
 + d) + 48*a^2*c^2*d^3*n*x*e^5*log(x*e + d) + 6*b^4*d^4*n*e^4*log(x*e + d) - 24*a*b^2*c*d^4*n*e^4*log(x*e + d)
 + 12*a^2*c^2*d^4*n*e^4*log(x*e + d) - 24*b^3*c*d^5*e^3*log(d) - 72*a*b*c^2*d^5*e^3*log(d) - 6*b^4*d*n*x^3*e^7
 + 36*a^2*c^2*d*n*x^3*e^7 - 21*b^4*d^2*n*x^2*e^6 - 12*a*b^2*c*d^2*n*x^2*e^6 + 114*a^2*c^2*d^2*n*x^2*e^6 - 26*b
^4*d^3*n*x*e^5 - 48*a*b^2*c*d^3*n*x*e^5 + 108*a^2*c^2*d^3*n*x*e^5 - 11*b^4*d^4*n*e^4 - 36*a*b^2*c*d^4*n*e^4 +
30*a^2*c^2*d^4*n*e^4 - 24*a*b^3*d^3*n*e^5*log(c*x^2 + b*x + a) - 72*a^2*b*c*d^3*n*e^5*log(c*x^2 + b*x + a) + 6
*b^4*d^4*e^4*log(d) + 72*a*b^2*c*d^4*e^4*log(d) + 36*a^2*c^2*d^4*e^4*log(d) + 6*a*b^3*n*x^3*e^8 - 18*a^2*b*c*n
*x^3*e^8 + 24*a*b^3*d*n*x^2*e^7 - 60*a^2*b*c*d*n*x^2*e^7 + 36*a*b^3*d^2*n*x*e^6 - 48*a^2*b*c*d^2*n*x*e^6 + 18*
a*b^3*d^3*n*e^5 - 6*a^2*b*c*d^3*n*e^5 + 36*a^2*b^2*d^2*n*e^6*log(c*x^2 + b*x + a) + 24*a^3*c*d^2*n*e^6*log(c*x
^2 + b*x + a) - 24*a*b^3*d^3*e^5*log(d) - 72*a^2*b*c*d^3*e^5*log(d) - 3*a^2*b^2*n*x^2*e^8 + 6*a^3*c*n*x^2*e^8
- 12*a^2*b^2*d*n*x*e^7 + 8*a^3*c*d*n*x*e^7 - 9*a^2*b^2*d^2*n*e^6 + 2*a^3*c*d^2*n*e^6 - 24*a^3*b*d*n*e^7*log(c*
x^2 + b*x + a) + 36*a^2*b^2*d^2*e^6*log(d) + 24*a^3*c*d^2*e^6*log(d) + 2*a^3*b*n*x*e^8 + 2*a^3*b*d*n*e^7 + 6*a
^4*n*e^8*log(c*x^2 + b*x + a) - 24*a^3*b*d*e^7*log(d) + 6*a^4*e^8*log(d))/(c^4*d^8*x^4*e^5 + 4*c^4*d^9*x^3*e^4
 + 6*c^4*d^10*x^2*e^3 + 4*c^4*d^11*x*e^2 + c^4*d^12*e - 4*b*c^3*d^7*x^4*e^6 - 16*b*c^3*d^8*x^3*e^5 - 24*b*c^3*
d^9*x^2*e^4 - 16*b*c^3*d^10*x*e^3 - 4*b*c^3*d^11*e^2 + 6*b^2*c^2*d^6*x^4*e^7 + 4*a*c^3*d^6*x^4*e^7 + 24*b^2*c^
2*d^7*x^3*e^6 + 16*a*c^3*d^7*x^3*e^6 + 36*b^2*c^2*d^8*x^2*e^5 + 24*a*c^3*d^8*x^2*e^5 + 24*b^2*c^2*d^9*x*e^4 +
16*a*c^3*d^9*x*e^4 + 6*b^2*c^2*d^10*e^3 + 4*a*c^3*d^10*e^3 - 4*b^3*c*d^5*x^4*e^8 - 12*a*b*c^2*d^5*x^4*e^8 - 16
*b^3*c*d^6*x^3*e^7 - 48*a*b*c^2*d^6*x^3*e^7 - 24*b^3*c*d^7*x^2*e^6 - 72*a*b*c^2*d^7*x^2*e^6 - 16*b^3*c*d^8*x*e
^5 - 48*a*b*c^2*d^8*x*e^5 - 4*b^3*c*d^9*e^4 - 12*a*b*c^2*d^9*e^4 + b^4*d^4*x^4*e^9 + 12*a*b^2*c*d^4*x^4*e^9 +
6*a^2*c^2*d^4*x^4*e^9 + 4*b^4*d^5*x^3*e^8 + 48*a*b^2*c*d^5*x^3*e^8 + 24*a^2*c^2*d^5*x^3*e^8 + 6*b^4*d^6*x^2*e^
7 + 72*a*b^2*c*d^6*x^2*e^7 + 36*a^2*c^2*d^6*x^2*e^7 + 4*b^4*d^7*x*e^6 + 48*a*b^2*c*d^7*x*e^6 + 24*a^2*c^2*d^7*
x*e^6 + b^4*d^8*e^5 + 12*a*b^2*c*d^8*e^5 + 6*a^2*c^2*d^8*e^5 - 4*a*b^3*d^3*x^4*e^10 - 12*a^2*b*c*d^3*x^4*e^10
- 16*a*b^3*d^4*x^3*e^9 - 48*a^2*b*c*d^4*x^3*e^9 - 24*a*b^3*d^5*x^2*e^8 - 72*a^2*b*c*d^5*x^2*e^8 - 16*a*b^3*d^6
*x*e^7 - 48*a^2*b*c*d^6*x*e^7 - 4*a*b^3*d^7*e^6 - 12*a^2*b*c*d^7*e^6 + 6*a^2*b^2*d^2*x^4*e^11 + 4*a^3*c*d^2*x^
4*e^11 + 24*a^2*b^2*d^3*x^3*e^10 + 16*a^3*c*d^3*x^3*e^10 + 36*a^2*b^2*d^4*x^2*e^9 + 24*a^3*c*d^4*x^2*e^9 + 24*
a^2*b^2*d^5*x*e^8 + 16*a^3*c*d^5*x*e^8 + 6*a^2*b^2*d^6*e^7 + 4*a^3*c*d^6*e^7 - 4*a^3*b*d*x^4*e^12 - 16*a^3*b*d
^2*x^3*e^11 - 24*a^3*b*d^3*x^2*e^10 - 16*a^3*b*d^4*x*e^9 - 4*a^3*b*d^5*e^8 + a^4*x^4*e^13 + 4*a^4*d*x^3*e^12 +
 6*a^4*d^2*x^2*e^11 + 4*a^4*d^3*x*e^10 + a^4*d^4*e^9)

________________________________________________________________________________________

maple [B]  time = 1.30, size = 1137077, normalized size = 2190.90 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(c*x^2+b*x+a)^n)/(e*x+d)^5,x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 18.95, size = 4334, normalized size = 8.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(d*(a + b*x + c*x^2)^n)/(d + e*x)^5,x)

[Out]

(log(10*d*e^5*(b^2 - 4*a*c)^(7/2) + 3*e^6*x*(b^2 - 4*a*c)^(7/2) - 6*a*e^6*(4*a*c - b^2)^3 + 96*c^5*d^6*(4*a*c
- b^2) - 10*b*e^6*x*(4*a*c - b^2)^3 - 10*b^5*e^6*x*(4*a*c - b^2) + 29*b^2*e^6*x*(b^2 - 4*a*c)^(5/2) + 29*b^4*e
^6*x*(b^2 - 4*a*c)^(3/2) + 3*b^6*e^6*x*(b^2 - 4*a*c)^(1/2) + 192*c^6*d^6*x*(b^2 - 4*a*c)^(1/2) + 44*a*b^2*e^6*
(4*a*c - b^2)^2 - 16*b^3*d*e^5*(4*a*c - b^2)^2 + 58*c*d^2*e^4*(4*a*c - b^2)^3 + 176*c^2*d^3*e^3*(b^2 - 4*a*c)^
(5/2) + 44*b^3*e^6*x*(4*a*c - b^2)^2 + 14*a*b*e^6*(b^2 - 4*a*c)^(5/2) - 232*c^3*d^4*e^2*(4*a*c - b^2)^2 - 14*a
*b^4*e^6*(4*a*c - b^2) + 44*a*b^3*e^6*(b^2 - 4*a*c)^(3/2) + 6*a*b^5*e^6*(b^2 - 4*a*c)^(1/2) + 96*b*c^5*d^6*(b^
2 - 4*a*c)^(1/2) - 48*b*d*e^5*(4*a*c - b^2)^3 + 32*b^5*d*e^5*(4*a*c - b^2) + 74*b^2*d*e^5*(b^2 - 4*a*c)^(5/2)
- 66*b^4*d*e^5*(b^2 - 4*a*c)^(3/2) - 18*b^6*d*e^5*(b^2 - 4*a*c)^(1/2) + 160*c^4*d^5*e*(b^2 - 4*a*c)^(3/2) + 28
8*b*c^2*d^3*e^3*(4*a*c - b^2)^2 - 84*b^2*c*d^2*e^4*(4*a*c - b^2)^2 - 40*b^2*c^3*d^4*e^2*(4*a*c - b^2) + 160*b^
3*c^2*d^3*e^3*(4*a*c - b^2) - 64*b^2*c^2*d^3*e^3*(b^2 - 4*a*c)^(3/2) + 360*b^3*c^3*d^4*e^2*(b^2 - 4*a*c)^(1/2)
 - 240*b^4*c^2*d^3*e^3*(b^2 - 4*a*c)^(1/2) - 352*c^3*d^3*e^3*x*(4*a*c - b^2)^2 - 128*b*c^4*d^5*e*(4*a*c - b^2)
 - 206*b*c*d^2*e^4*(b^2 - 4*a*c)^(5/2) + 20*c*d*e^5*x*(4*a*c - b^2)^3 + 320*c^5*d^5*e*x*(4*a*c - b^2) - 110*b^
4*c*d^2*e^4*(4*a*c - b^2) - 168*b*c^3*d^4*e^2*(b^2 - 4*a*c)^(3/2) - 288*b^2*c^4*d^5*e*(b^2 - 4*a*c)^(1/2) + 14
8*b^3*c*d^2*e^4*(b^2 - 4*a*c)^(3/2) + 90*b^5*c*d^2*e^4*(b^2 - 4*a*c)^(1/2) + 116*c^2*d^2*e^4*x*(b^2 - 4*a*c)^(
5/2) + 464*c^4*d^4*e^2*x*(b^2 - 4*a*c)^(3/2) - 264*b^2*c*d*e^5*x*(4*a*c - b^2)^2 - 800*b*c^4*d^4*e^2*x*(4*a*c
- b^2) - 928*b*c^3*d^3*e^3*x*(b^2 - 4*a*c)^(3/2) - 116*b*c*d*e^5*x*(b^2 - 4*a*c)^(5/2) + 528*b*c^2*d^2*e^4*x*(
4*a*c - b^2)^2 + 800*b^2*c^3*d^3*e^3*x*(4*a*c - b^2) - 400*b^3*c^2*d^2*e^4*x*(4*a*c - b^2) + 696*b^2*c^2*d^2*e
^4*x*(b^2 - 4*a*c)^(3/2) + 720*b^2*c^4*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) - 480*b^3*c^3*d^3*e^3*x*(b^2 - 4*a*c)^(1/
2) + 180*b^4*c^2*d^2*e^4*x*(b^2 - 4*a*c)^(1/2) + 100*b^4*c*d*e^5*x*(4*a*c - b^2) - 576*b*c^5*d^5*e*x*(b^2 - 4*
a*c)^(1/2) - 232*b^3*c*d*e^5*x*(b^2 - 4*a*c)^(3/2) - 36*b^5*c*d*e^5*x*(b^2 - 4*a*c)^(1/2))*(e^4*((b^4*n)/8 + (
b^3*n*(b^2 - 4*a*c)^(1/2))/8 + (a^2*c^2*n)/4 - (a*b^2*c*n)/2 - (a*b*c*n*(b^2 - 4*a*c)^(1/2))/4) - e^3*((b^3*c*
d*n)/2 - (3*a*b*c^2*d*n)/2 - (a*c^2*d*n*(b^2 - 4*a*c)^(1/2))/2 + (b^2*c*d*n*(b^2 - 4*a*c)^(1/2))/2) + e^2*((3*
b^2*c^2*d^2*n)/4 - (3*a*c^3*d^2*n)/2 + (3*b*c^2*d^2*n*(b^2 - 4*a*c)^(1/2))/4) - e*((b*c^3*d^3*n)/2 + (c^3*d^3*
n*(b^2 - 4*a*c)^(1/2))/2) + (c^4*d^4*n)/4))/(a^4*e^9 + c^4*d^8*e + b^4*d^4*e^5 - 4*a*b^3*d^3*e^6 + 4*a*c^3*d^6
*e^3 + 4*a^3*c*d^2*e^7 - 4*b*c^3*d^7*e^2 - 4*b^3*c*d^5*e^4 + 6*a^2*b^2*d^2*e^7 + 6*a^2*c^2*d^4*e^5 + 6*b^2*c^2
*d^6*e^3 - 4*a^3*b*d*e^8 - 12*a*b*c^2*d^5*e^4 + 12*a*b^2*c*d^4*e^5 - 12*a^2*b*c*d^3*e^6) - (log(d + e*x)*(e^2*
(6*b^2*c^2*d^2*n - 12*a*c^3*d^2*n) - e^3*(4*b^3*c*d*n - 12*a*b*c^2*d*n) + e^4*(b^4*n + 2*a^2*c^2*n - 4*a*b^2*c
*n) + 2*c^4*d^4*n - 4*b*c^3*d^3*e*n))/(4*a^4*e^9 + 4*c^4*d^8*e + 4*b^4*d^4*e^5 - 16*a*b^3*d^3*e^6 + 16*a*c^3*d
^6*e^3 + 16*a^3*c*d^2*e^7 - 16*b*c^3*d^7*e^2 - 16*b^3*c*d^5*e^4 + 24*a^2*b^2*d^2*e^7 + 24*a^2*c^2*d^4*e^5 + 24
*b^2*c^2*d^6*e^3 - 16*a^3*b*d*e^8 - 48*a*b*c^2*d^5*e^4 + 48*a*b^2*c*d^4*e^5 - 48*a^2*b*c*d^3*e^6) - log(d*(a +
 b*x + c*x^2)^n)/(4*e*(d^4 + e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x)) - (log(10*d*e^5*(b^2 - 4*a*c)
^(7/2) + 3*e^6*x*(b^2 - 4*a*c)^(7/2) + 6*a*e^6*(4*a*c - b^2)^3 - 96*c^5*d^6*(4*a*c - b^2) + 10*b*e^6*x*(4*a*c
- b^2)^3 + 10*b^5*e^6*x*(4*a*c - b^2) + 29*b^2*e^6*x*(b^2 - 4*a*c)^(5/2) + 29*b^4*e^6*x*(b^2 - 4*a*c)^(3/2) +
3*b^6*e^6*x*(b^2 - 4*a*c)^(1/2) + 192*c^6*d^6*x*(b^2 - 4*a*c)^(1/2) - 44*a*b^2*e^6*(4*a*c - b^2)^2 + 16*b^3*d*
e^5*(4*a*c - b^2)^2 - 58*c*d^2*e^4*(4*a*c - b^2)^3 + 176*c^2*d^3*e^3*(b^2 - 4*a*c)^(5/2) - 44*b^3*e^6*x*(4*a*c
 - b^2)^2 + 14*a*b*e^6*(b^2 - 4*a*c)^(5/2) + 232*c^3*d^4*e^2*(4*a*c - b^2)^2 + 14*a*b^4*e^6*(4*a*c - b^2) + 44
*a*b^3*e^6*(b^2 - 4*a*c)^(3/2) + 6*a*b^5*e^6*(b^2 - 4*a*c)^(1/2) + 96*b*c^5*d^6*(b^2 - 4*a*c)^(1/2) + 48*b*d*e
^5*(4*a*c - b^2)^3 - 32*b^5*d*e^5*(4*a*c - b^2) + 74*b^2*d*e^5*(b^2 - 4*a*c)^(5/2) - 66*b^4*d*e^5*(b^2 - 4*a*c
)^(3/2) - 18*b^6*d*e^5*(b^2 - 4*a*c)^(1/2) + 160*c^4*d^5*e*(b^2 - 4*a*c)^(3/2) - 288*b*c^2*d^3*e^3*(4*a*c - b^
2)^2 + 84*b^2*c*d^2*e^4*(4*a*c - b^2)^2 + 40*b^2*c^3*d^4*e^2*(4*a*c - b^2) - 160*b^3*c^2*d^3*e^3*(4*a*c - b^2)
 - 64*b^2*c^2*d^3*e^3*(b^2 - 4*a*c)^(3/2) + 360*b^3*c^3*d^4*e^2*(b^2 - 4*a*c)^(1/2) - 240*b^4*c^2*d^3*e^3*(b^2
 - 4*a*c)^(1/2) + 352*c^3*d^3*e^3*x*(4*a*c - b^2)^2 + 128*b*c^4*d^5*e*(4*a*c - b^2) - 206*b*c*d^2*e^4*(b^2 - 4
*a*c)^(5/2) - 20*c*d*e^5*x*(4*a*c - b^2)^3 - 320*c^5*d^5*e*x*(4*a*c - b^2) + 110*b^4*c*d^2*e^4*(4*a*c - b^2) -
 168*b*c^3*d^4*e^2*(b^2 - 4*a*c)^(3/2) - 288*b^2*c^4*d^5*e*(b^2 - 4*a*c)^(1/2) + 148*b^3*c*d^2*e^4*(b^2 - 4*a*
c)^(3/2) + 90*b^5*c*d^2*e^4*(b^2 - 4*a*c)^(1/2) + 116*c^2*d^2*e^4*x*(b^2 - 4*a*c)^(5/2) + 464*c^4*d^4*e^2*x*(b
^2 - 4*a*c)^(3/2) + 264*b^2*c*d*e^5*x*(4*a*c - b^2)^2 + 800*b*c^4*d^4*e^2*x*(4*a*c - b^2) - 928*b*c^3*d^3*e^3*
x*(b^2 - 4*a*c)^(3/2) - 116*b*c*d*e^5*x*(b^2 - 4*a*c)^(5/2) - 528*b*c^2*d^2*e^4*x*(4*a*c - b^2)^2 - 800*b^2*c^
3*d^3*e^3*x*(4*a*c - b^2) + 400*b^3*c^2*d^2*e^4*x*(4*a*c - b^2) + 696*b^2*c^2*d^2*e^4*x*(b^2 - 4*a*c)^(3/2) +
720*b^2*c^4*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) - 480*b^3*c^3*d^3*e^3*x*(b^2 - 4*a*c)^(1/2) + 180*b^4*c^2*d^2*e^4*x*
(b^2 - 4*a*c)^(1/2) - 100*b^4*c*d*e^5*x*(4*a*c - b^2) - 576*b*c^5*d^5*e*x*(b^2 - 4*a*c)^(1/2) - 232*b^3*c*d*e^
5*x*(b^2 - 4*a*c)^(3/2) - 36*b^5*c*d*e^5*x*(b^2 - 4*a*c)^(1/2))*(e^3*((b^3*c*d*n)/2 - (3*a*b*c^2*d*n)/2 + (a*c
^2*d*n*(b^2 - 4*a*c)^(1/2))/2 - (b^2*c*d*n*(b^2 - 4*a*c)^(1/2))/2) - e^4*((b^4*n)/8 - (b^3*n*(b^2 - 4*a*c)^(1/
2))/8 + (a^2*c^2*n)/4 - (a*b^2*c*n)/2 + (a*b*c*n*(b^2 - 4*a*c)^(1/2))/4) + e^2*((3*a*c^3*d^2*n)/2 - (3*b^2*c^2
*d^2*n)/4 + (3*b*c^2*d^2*n*(b^2 - 4*a*c)^(1/2))/4) + e*((b*c^3*d^3*n)/2 - (c^3*d^3*n*(b^2 - 4*a*c)^(1/2))/2) -
 (c^4*d^4*n)/4))/(a^4*e^9 + c^4*d^8*e + b^4*d^4*e^5 - 4*a*b^3*d^3*e^6 + 4*a*c^3*d^6*e^3 + 4*a^3*c*d^2*e^7 - 4*
b*c^3*d^7*e^2 - 4*b^3*c*d^5*e^4 + 6*a^2*b^2*d^2*e^7 + 6*a^2*c^2*d^4*e^5 + 6*b^2*c^2*d^6*e^3 - 4*a^3*b*d*e^8 -
12*a*b*c^2*d^5*e^4 + 12*a*b^2*c*d^4*e^5 - 12*a^2*b*c*d^3*e^6) - ((11*b^3*d^2*e^3*n - 22*c^3*d^5*n + 2*a^2*b*e^
5*n - 7*a*b^2*d*e^4*n + 2*a^2*c*d*e^4*n + 40*b*c^2*d^4*e*n + 28*a*c^2*d^3*e^2*n - 35*b^2*c*d^3*e^2*n - 6*a*b*c
*d^2*e^3*n)/(6*(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*
c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3)) - (x*(a*b^2*e^5*n - 2*a^2*c*e^5*n - 5*b^3*d*e^4*
n + 10*c^3*d^4*e*n - 24*a*c^2*d^2*e^3*n - 16*b*c^2*d^3*e^2*n + 15*b^2*c*d^2*e^3*n + 12*a*b*c*d*e^4*n))/(2*(a^3
*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b
*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3)) + (n*x^2*(b^3*e^5 - 2*c^3*d^3*e^2 + 3*b*c^2*d^2*e^3 - 3*a*b*c*e^5 +
 6*a*c^2*d*e^4 - 3*b^2*c*d*e^4))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*
c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))/(4*d^3*e + 4*e^4*x^3 + 12*d^2*
e^2*x + 12*d*e^3*x^2)

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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(ln(d*(c*x**2+b*x+a)**n)/(e*x+d)**5,x)

[Out]

Timed out

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