3.82 \(\int (d+e x)^4 \log (d (a+b x+c x^2)^n) \, dx\)

Optimal. Leaf size=485 \[ -\frac{n (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{10 c^5 e}-\frac{n x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )}{5 c^4}+\frac{n \sqrt{b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{5 c^5}-\frac{e^2 n x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )}{15 c^2}-\frac{e n x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )}{10 c^3}+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{e^3 n x^4 (10 c d-b e)}{20 c}-\frac{2}{25} e^4 n x^5 \]

[Out]

-((10*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + 2*a*e) - b^2*c*e^3*(5*b*d + 4*a*e) + c^2*e^2*(10*b^2*d^2 + 15*a*
b*d*e + 2*a^2*e^2))*n*x)/(5*c^4) - (e*(20*c^3*d^3 - b^3*e^3 - 10*c^2*d*e*(b*d + a*e) + b*c*e^2*(5*b*d + 3*a*e)
)*n*x^2)/(10*c^3) - (e^2*(20*c^2*d^2 + b^2*e^2 - c*e*(5*b*d + 2*a*e))*n*x^3)/(15*c^2) - (e^3*(10*c*d - b*e)*n*
x^4)/(20*c) - (2*e^4*n*x^5)/25 + (Sqrt[b^2 - 4*a*c]*(5*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + a*e) - b^2*c*e^
3*(5*b*d + 3*a*e) + c^2*e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2))*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(5*
c^5) - ((2*c*d - b*e)*(c^4*d^4 + b^4*e^4 - 2*c^3*d^2*e*(b*d + 5*a*e) - b^2*c*e^3*(3*b*d + 5*a*e) + c^2*e^2*(4*
b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2))*n*Log[a + b*x + c*x^2])/(10*c^5*e) + ((d + e*x)^5*Log[d*(a + b*x + c*x^2)^n
])/(5*e)

________________________________________________________________________________________

Rubi [A]  time = 2.05768, antiderivative size = 485, normalized size of antiderivative = 1., 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} \[ -\frac{n (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{10 c^5 e}-\frac{n x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )}{5 c^4}+\frac{n \sqrt{b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{5 c^5}-\frac{e^2 n x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )}{15 c^2}-\frac{e n x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )}{10 c^3}+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{e^3 n x^4 (10 c d-b e)}{20 c}-\frac{2}{25} e^4 n x^5 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*Log[d*(a + b*x + c*x^2)^n],x]

[Out]

-((10*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + 2*a*e) - b^2*c*e^3*(5*b*d + 4*a*e) + c^2*e^2*(10*b^2*d^2 + 15*a*
b*d*e + 2*a^2*e^2))*n*x)/(5*c^4) - (e*(20*c^3*d^3 - b^3*e^3 - 10*c^2*d*e*(b*d + a*e) + b*c*e^2*(5*b*d + 3*a*e)
)*n*x^2)/(10*c^3) - (e^2*(20*c^2*d^2 + b^2*e^2 - c*e*(5*b*d + 2*a*e))*n*x^3)/(15*c^2) - (e^3*(10*c*d - b*e)*n*
x^4)/(20*c) - (2*e^4*n*x^5)/25 + (Sqrt[b^2 - 4*a*c]*(5*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + a*e) - b^2*c*e^
3*(5*b*d + 3*a*e) + c^2*e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2))*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(5*
c^5) - ((2*c*d - b*e)*(c^4*d^4 + b^4*e^4 - 2*c^3*d^2*e*(b*d + 5*a*e) - b^2*c*e^3*(3*b*d + 5*a*e) + c^2*e^2*(4*
b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2))*n*Log[a + b*x + c*x^2])/(10*c^5*e) + ((d + e*x)^5*Log[d*(a + b*x + c*x^2)^n
])/(5*e)

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]

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 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 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 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 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]

Rubi steps

\begin{align*} \int (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{n \int \frac{(b+2 c x) (d+e x)^5}{a+b x+c x^2} \, dx}{5 e}\\ &=\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{n \int \left (\frac{e \left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right )}{c^4}+\frac{e^2 \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) x}{c^3}+\frac{e^3 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) x^2}{c^2}+\frac{e^4 (10 c d-b e) x^3}{c}+2 e^5 x^4+\frac{5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )+(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx}{5 e}\\ &=-\frac{\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac{e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac{e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac{e^3 (10 c d-b e) n x^4}{20 c}-\frac{2}{25} e^4 n x^5+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{n \int \frac{5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )+(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{5 c^4 e}\\ &=-\frac{\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac{e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac{e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac{e^3 (10 c d-b e) n x^4}{20 c}-\frac{2}{25} e^4 n x^5+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{\left ((2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{10 c^5 e}-\frac{\left (\left (-b (2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right )+2 c \left (5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )\right )\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{10 c^5 e}\\ &=-\frac{\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac{e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac{e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac{e^3 (10 c d-b e) n x^4}{20 c}-\frac{2}{25} e^4 n x^5-\frac{(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}+\frac{\left (\left (-b (2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right )+2 c \left (5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )\right )\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{5 c^5 e}\\ &=-\frac{\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac{e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac{e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac{e^3 (10 c d-b e) n x^4}{20 c}-\frac{2}{25} e^4 n x^5+\frac{\sqrt{b^2-4 a c} \left (5 c^4 d^4-10 b c^3 d^3 e+10 b^2 c^2 d^2 e^2-10 a c^3 d^2 e^2-5 b^3 c d e^3+10 a b c^2 d e^3+b^4 e^4-3 a b^2 c e^4+a^2 c^2 e^4\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{5 c^5}-\frac{(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}\\ \end{align*}

Mathematica [A]  time = 1.90509, size = 468, normalized size = 0.96 \[ \frac{(d+e x)^5 \log \left (d (a+x (b+c x))^n\right )-\frac{n \left (60 c e x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )+30 (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log (a+x (b+c x))-60 e \sqrt{b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+20 c^3 e^3 x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )+30 c^2 e^2 x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )+15 c^4 e^4 x^4 (10 c d-b e)+24 c^5 e^5 x^5\right )}{60 c^5}}{5 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*Log[d*(a + b*x + c*x^2)^n],x]

[Out]

(-(n*(60*c*e*(10*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + 2*a*e) - b^2*c*e^3*(5*b*d + 4*a*e) + c^2*e^2*(10*b^2*
d^2 + 15*a*b*d*e + 2*a^2*e^2))*x + 30*c^2*e^2*(20*c^3*d^3 - b^3*e^3 - 10*c^2*d*e*(b*d + a*e) + b*c*e^2*(5*b*d
+ 3*a*e))*x^2 + 20*c^3*e^3*(20*c^2*d^2 + b^2*e^2 - c*e*(5*b*d + 2*a*e))*x^3 + 15*c^4*e^4*(10*c*d - b*e)*x^4 +
24*c^5*e^5*x^5 - 60*Sqrt[b^2 - 4*a*c]*e*(5*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + a*e) - b^2*c*e^3*(5*b*d + 3
*a*e) + c^2*e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] + 30*(2*c*d - b*e)
*(c^4*d^4 + b^4*e^4 - 2*c^3*d^2*e*(b*d + 5*a*e) - b^2*c*e^3*(3*b*d + 5*a*e) + c^2*e^2*(4*b^2*d^2 + 10*a*b*d*e
+ 5*a^2*e^2))*Log[a + x*(b + c*x)]))/(60*c^5) + (d + e*x)^5*Log[d*(a + x*(b + c*x))^n])/(5*e)

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Maple [C]  time = 0.23, size = 31895, normalized size = 65.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*ln(d*(c*x^2+b*x+a)^n),x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*log(d*(c*x^2+b*x+a)^n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.99206, size = 2700, normalized size = 5.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*log(d*(c*x^2+b*x+a)^n),x, algorithm="fricas")

[Out]

[-1/300*(24*c^5*e^4*n*x^5 + 15*(10*c^5*d*e^3 - b*c^4*e^4)*n*x^4 + 20*(20*c^5*d^2*e^2 - 5*b*c^4*d*e^3 + (b^2*c^
3 - 2*a*c^4)*e^4)*n*x^3 + 30*(20*c^5*d^3*e - 10*b*c^4*d^2*e^2 + 5*(b^2*c^3 - 2*a*c^4)*d*e^3 - (b^3*c^2 - 3*a*b
*c^3)*e^4)*n*x^2 - 30*(5*c^4*d^4 - 10*b*c^3*d^3*e + 10*(b^2*c^2 - a*c^3)*d^2*e^2 - 5*(b^3*c - 2*a*b*c^2)*d*e^3
+ (b^4 - 3*a*b^2*c + a^2*c^2)*e^4)*sqrt(b^2 - 4*a*c)*n*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)) + 60*(10*c^5*d^4 - 10*b*c^4*d^3*e + 10*(b^2*c^3 - 2*a*c^4)*d^2*e^2 - 5*(b
^3*c^2 - 3*a*b*c^3)*d*e^3 + (b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*e^4)*n*x - 30*(2*c^5*e^4*n*x^5 + 10*c^5*d*e^3*n*
x^4 + 20*c^5*d^2*e^2*n*x^3 + 20*c^5*d^3*e*n*x^2 + 10*c^5*d^4*n*x + (5*b*c^4*d^4 - 10*(b^2*c^3 - 2*a*c^4)*d^3*e
+ 10*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c
^2)*e^4)*n)*log(c*x^2 + b*x + a) - 60*(c^5*e^4*x^5 + 5*c^5*d*e^3*x^4 + 10*c^5*d^2*e^2*x^3 + 10*c^5*d^3*e*x^2 +
5*c^5*d^4*x)*log(d))/c^5, -1/300*(24*c^5*e^4*n*x^5 + 15*(10*c^5*d*e^3 - b*c^4*e^4)*n*x^4 + 20*(20*c^5*d^2*e^2
- 5*b*c^4*d*e^3 + (b^2*c^3 - 2*a*c^4)*e^4)*n*x^3 + 30*(20*c^5*d^3*e - 10*b*c^4*d^2*e^2 + 5*(b^2*c^3 - 2*a*c^4
)*d*e^3 - (b^3*c^2 - 3*a*b*c^3)*e^4)*n*x^2 - 60*(5*c^4*d^4 - 10*b*c^3*d^3*e + 10*(b^2*c^2 - a*c^3)*d^2*e^2 - 5
*(b^3*c - 2*a*b*c^2)*d*e^3 + (b^4 - 3*a*b^2*c + a^2*c^2)*e^4)*sqrt(-b^2 + 4*a*c)*n*arctan(-sqrt(-b^2 + 4*a*c)*
(2*c*x + b)/(b^2 - 4*a*c)) + 60*(10*c^5*d^4 - 10*b*c^4*d^3*e + 10*(b^2*c^3 - 2*a*c^4)*d^2*e^2 - 5*(b^3*c^2 - 3
*a*b*c^3)*d*e^3 + (b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*e^4)*n*x - 30*(2*c^5*e^4*n*x^5 + 10*c^5*d*e^3*n*x^4 + 20*c
^5*d^2*e^2*n*x^3 + 20*c^5*d^3*e*n*x^2 + 10*c^5*d^4*n*x + (5*b*c^4*d^4 - 10*(b^2*c^3 - 2*a*c^4)*d^3*e + 10*(b^3
*c^2 - 3*a*b*c^3)*d^2*e^2 - 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4)*n
)*log(c*x^2 + b*x + a) - 60*(c^5*e^4*x^5 + 5*c^5*d*e^3*x^4 + 10*c^5*d^2*e^2*x^3 + 10*c^5*d^3*e*x^2 + 5*c^5*d^4
*x)*log(d))/c^5]

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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((e*x+d)**4*ln(d*(c*x**2+b*x+a)**n),x)

[Out]

Timed out

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Giac [B]  time = 1.38521, size = 1338, normalized size = 2.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*log(d*(c*x^2+b*x+a)^n),x, algorithm="giac")

[Out]

1/2*b*d^4*n*log(c*x^2 + b*x + a)/c - (b^2*d^4*n - 4*a*c*d^4*n)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b
^2 + 4*a*c)*c) - (b^2*d^3*n*e - 2*a*c*d^3*n*e)*log(c*x^2 + b*x + a)/c^2 + 2*(b^3*d^3*n*e - 4*a*b*c*d^3*n*e)*ar
ctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2) + (b^3*d^2*n*e^2 - 3*a*b*c*d^2*n*e^2)*log(c*x^2
+ b*x + a)/c^3 - 2*(b^4*d^2*n*e^2 - 5*a*b^2*c*d^2*n*e^2 + 4*a^2*c^2*d^2*n*e^2)*arctan((2*c*x + b)/sqrt(-b^2 +
4*a*c))/(sqrt(-b^2 + 4*a*c)*c^3) - 1/2*(b^4*d*n*e^3 - 4*a*b^2*c*d*n*e^3 + 2*a^2*c^2*d*n*e^3)*log(c*x^2 + b*x +
a)/c^4 + (b^5*d*n*e^3 - 6*a*b^3*c*d*n*e^3 + 8*a^2*b*c^2*d*n*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt
(-b^2 + 4*a*c)*c^4) + 1/300*(60*c^4*n*x^5*e^4*log(c*x^2 + b*x + a) + 300*c^4*d*n*x^4*e^3*log(c*x^2 + b*x + a)
+ 600*c^4*d^2*n*x^3*e^2*log(c*x^2 + b*x + a) + 600*c^4*d^3*n*x^2*e*log(c*x^2 + b*x + a) - 24*c^4*n*x^5*e^4 - 1
50*c^4*d*n*x^4*e^3 - 400*c^4*d^2*n*x^3*e^2 - 600*c^4*d^3*n*x^2*e + 300*c^4*d^4*n*x*log(c*x^2 + b*x + a) + 60*c
^4*x^5*e^4*log(d) + 300*c^4*d*x^4*e^3*log(d) + 600*c^4*d^2*x^3*e^2*log(d) + 600*c^4*d^3*x^2*e*log(d) - 600*c^4
*d^4*n*x + 15*b*c^3*n*x^4*e^4 + 100*b*c^3*d*n*x^3*e^3 + 300*b*c^3*d^2*n*x^2*e^2 + 600*b*c^3*d^3*n*x*e + 300*c^
4*d^4*x*log(d) - 20*b^2*c^2*n*x^3*e^4 + 40*a*c^3*n*x^3*e^4 - 150*b^2*c^2*d*n*x^2*e^3 + 300*a*c^3*d*n*x^2*e^3 -
600*b^2*c^2*d^2*n*x*e^2 + 1200*a*c^3*d^2*n*x*e^2 + 30*b^3*c*n*x^2*e^4 - 90*a*b*c^2*n*x^2*e^4 + 300*b^3*c*d*n*
x*e^3 - 900*a*b*c^2*d*n*x*e^3 - 60*b^4*n*x*e^4 + 240*a*b^2*c*n*x*e^4 - 120*a^2*c^2*n*x*e^4)/c^4 + 1/10*(b^5*n*
e^4 - 5*a*b^3*c*n*e^4 + 5*a^2*b*c^2*n*e^4)*log(c*x^2 + b*x + a)/c^5 - 1/5*(b^6*n*e^4 - 7*a*b^4*c*n*e^4 + 13*a^
2*b^2*c^2*n*e^4 - 4*a^3*c^3*n*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)