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

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

[Out]

-1/5*(10*c^4*d^4+b^4*e^4-10*c^3*d^2*e*(2*a*e+b*d)-b^2*c*e^3*(4*a*e+5*b*d)+c^2*e^2*(2*a^2*e^2+15*a*b*d*e+10*b^2
*d^2))*n*x/c^4-1/10*e*(20*c^3*d^3-b^3*e^3-10*c^2*d*e*(a*e+b*d)+b*c*e^2*(3*a*e+5*b*d))*n*x^2/c^3-1/15*e^2*(20*c
^2*d^2+b^2*e^2-c*e*(2*a*e+5*b*d))*n*x^3/c^2-1/20*e^3*(-b*e+10*c*d)*n*x^4/c-2/25*e^4*n*x^5-1/10*(-b*e+2*c*d)*(c
^4*d^4+b^4*e^4-2*c^3*d^2*e*(5*a*e+b*d)-b^2*c*e^3*(5*a*e+3*b*d)+c^2*e^2*(5*a^2*e^2+10*a*b*d*e+4*b^2*d^2))*n*ln(
c*x^2+b*x+a)/c^5/e+1/5*(e*x+d)^5*ln(d*(c*x^2+b*x+a)^n)/e+1/5*(5*c^4*d^4+b^4*e^4-10*c^3*d^2*e*(a*e+b*d)-b^2*c*e
^3*(3*a*e+5*b*d)+c^2*e^2*(a^2*e^2+10*a*b*d*e+10*b^2*d^2))*n*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))*(-4*a*c+b^2)
^(1/2)/c^5

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Rubi [A]  time = 2.06, antiderivative size = 485, 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} \[ -\frac {n (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{10 c^5 e}-\frac {n x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )}{5 c^4}+\frac {n \sqrt {b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{5 c^5}-\frac {e^2 n x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )}{15 c^2}-\frac {e n x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )}{10 c^3}+\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac {e^3 n x^4 (10 c d-b e)}{20 c}-\frac {2}{25} e^4 n x^5 \]

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 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 (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*}

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Mathematica [A]  time = 1.97, size = 468, normalized size = 0.96 \[ \frac {(d+e x)^5 \log \left (d (a+x (b+c x))^n\right )-\frac {n \left (60 c e x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )+30 (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log (a+x (b+c x))-60 e \sqrt {b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+30 c^2 e^2 x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )+20 c^3 e^3 x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )+15 c^4 e^4 x^4 (10 c d-b e)+24 c^5 e^5 x^5\right )}{60 c^5}}{5 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/60*(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)]))/c^5 + (d + e*x)^5*Log[d*(a + x*(b + c*x))^n])/(5*e)

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fricas [A]  time = 0.57, size = 1270, normalized size = 2.62 \[ \text {result too large to display} \]

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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giac [A]  time = 0.26, size = 817, normalized size = 1.68 \[ \frac {60 \, c^{4} n x^{5} e^{4} \log \left (c x^{2} + b x + a\right ) + 300 \, c^{4} d n x^{4} e^{3} \log \left (c x^{2} + b x + a\right ) + 600 \, c^{4} d^{2} n x^{3} e^{2} \log \left (c x^{2} + b x + a\right ) + 600 \, c^{4} d^{3} n x^{2} e \log \left (c x^{2} + b x + a\right ) - 24 \, c^{4} n x^{5} e^{4} - 150 \, 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 \left (c x^{2} + b x + a\right ) + 60 \, c^{4} x^{5} e^{4} \log \relax (d) + 300 \, c^{4} d x^{4} e^{3} \log \relax (d) + 600 \, c^{4} d^{2} x^{3} e^{2} \log \relax (d) + 600 \, c^{4} d^{3} x^{2} e \log \relax (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 \relax (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}}{300 \, c^{4}} + \frac {{\left (5 \, b c^{4} d^{4} n - 10 \, b^{2} c^{3} d^{3} n e + 20 \, a c^{4} d^{3} n e + 10 \, b^{3} c^{2} d^{2} n e^{2} - 30 \, a b c^{3} d^{2} n e^{2} - 5 \, b^{4} c d n e^{3} + 20 \, a b^{2} c^{2} d n e^{3} - 10 \, a^{2} c^{3} d n e^{3} + b^{5} n e^{4} - 5 \, a b^{3} c n e^{4} + 5 \, a^{2} b c^{2} n e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{10 \, c^{5}} - \frac {{\left (5 \, b^{2} c^{4} d^{4} n - 20 \, a c^{5} d^{4} n - 10 \, b^{3} c^{3} d^{3} n e + 40 \, a b c^{4} d^{3} n e + 10 \, b^{4} c^{2} d^{2} n e^{2} - 50 \, a b^{2} c^{3} d^{2} n e^{2} + 40 \, a^{2} c^{4} d^{2} n e^{2} - 5 \, b^{5} c d n e^{3} + 30 \, a b^{3} c^{2} d n e^{3} - 40 \, a^{2} b c^{3} d n e^{3} + 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}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{5 \, \sqrt {-b^{2} + 4 \, a c} c^{5}} \]

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]

1/300*(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 - 150*c^4*d*n*x^4*e^3 - 4
00*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) + 30
0*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*(5*b*c^4*d^4*n - 10*b^2*c^3*d
^3*n*e + 20*a*c^4*d^3*n*e + 10*b^3*c^2*d^2*n*e^2 - 30*a*b*c^3*d^2*n*e^2 - 5*b^4*c*d*n*e^3 + 20*a*b^2*c^2*d*n*e
^3 - 10*a^2*c^3*d*n*e^3 + 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*(5*b
^2*c^4*d^4*n - 20*a*c^5*d^4*n - 10*b^3*c^3*d^3*n*e + 40*a*b*c^4*d^3*n*e + 10*b^4*c^2*d^2*n*e^2 - 50*a*b^2*c^3*
d^2*n*e^2 + 40*a^2*c^4*d^2*n*e^2 - 5*b^5*c*d*n*e^3 + 30*a*b^3*c^2*d*n*e^3 - 40*a^2*b*c^3*d*n*e^3 + 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)

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maple [C]  time = 1.09, size = 31895, normalized size = 65.76 \[ \text {output too large to display} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> 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?

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mupad [B]  time = 1.03, size = 1240, normalized size = 2.56 \[ x^3\,\left (\frac {b\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{3\,c}+\frac {2\,a\,e^4\,n}{15\,c}-\frac {d\,e^2\,n\,\left (b\,e+4\,c\,d\right )}{3\,c}\right )-x\,\left (\frac {a\,\left (\frac {b\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{c}+\frac {2\,a\,e^4\,n}{5\,c}-\frac {d\,e^2\,n\,\left (b\,e+4\,c\,d\right )}{c}\right )}{c}-\frac {b\,\left (\frac {b\,\left (\frac {b\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{c}+\frac {2\,a\,e^4\,n}{5\,c}-\frac {d\,e^2\,n\,\left (b\,e+4\,c\,d\right )}{c}\right )}{c}-\frac {a\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{c}+\frac {2\,d^2\,e\,n\,\left (b\,e+2\,c\,d\right )}{c}\right )}{c}+\frac {2\,d^3\,n\,\left (b\,e+c\,d\right )}{c}\right )-x^2\,\left (\frac {b\,\left (\frac {b\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{c}+\frac {2\,a\,e^4\,n}{5\,c}-\frac {d\,e^2\,n\,\left (b\,e+4\,c\,d\right )}{c}\right )}{2\,c}-\frac {a\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{2\,c}+\frac {d^2\,e\,n\,\left (b\,e+2\,c\,d\right )}{c}\right )-x^4\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{20\,c}-\frac {b\,e^4\,n}{10\,c}\right )+\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )\,\left (d^4\,x+2\,d^3\,e\,x^2+2\,d^2\,e^2\,x^3+d\,e^3\,x^4+\frac {e^4\,x^5}{5}\right )+\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^5\,e^4\,n+5\,b\,c^4\,d^4\,n+b^4\,e^4\,n\,\sqrt {b^2-4\,a\,c}+5\,c^4\,d^4\,n\,\sqrt {b^2-4\,a\,c}-5\,a\,b^3\,c\,e^4\,n+20\,a\,c^4\,d^3\,e\,n-5\,b^4\,c\,d\,e^3\,n+5\,a^2\,b\,c^2\,e^4\,n-10\,a^2\,c^3\,d\,e^3\,n-10\,b^2\,c^3\,d^3\,e\,n+a^2\,c^2\,e^4\,n\,\sqrt {b^2-4\,a\,c}+10\,b^3\,c^2\,d^2\,e^2\,n-10\,a\,c^3\,d^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}+10\,b^2\,c^2\,d^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}-3\,a\,b^2\,c\,e^4\,n\,\sqrt {b^2-4\,a\,c}-10\,b\,c^3\,d^3\,e\,n\,\sqrt {b^2-4\,a\,c}-5\,b^3\,c\,d\,e^3\,n\,\sqrt {b^2-4\,a\,c}-30\,a\,b\,c^3\,d^2\,e^2\,n+20\,a\,b^2\,c^2\,d\,e^3\,n+10\,a\,b\,c^2\,d\,e^3\,n\,\sqrt {b^2-4\,a\,c}\right )}{10\,c^5}-\frac {2\,e^4\,n\,x^5}{25}+\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^5\,e^4\,n+5\,b\,c^4\,d^4\,n-b^4\,e^4\,n\,\sqrt {b^2-4\,a\,c}-5\,c^4\,d^4\,n\,\sqrt {b^2-4\,a\,c}-5\,a\,b^3\,c\,e^4\,n+20\,a\,c^4\,d^3\,e\,n-5\,b^4\,c\,d\,e^3\,n+5\,a^2\,b\,c^2\,e^4\,n-10\,a^2\,c^3\,d\,e^3\,n-10\,b^2\,c^3\,d^3\,e\,n-a^2\,c^2\,e^4\,n\,\sqrt {b^2-4\,a\,c}+10\,b^3\,c^2\,d^2\,e^2\,n+10\,a\,c^3\,d^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}-10\,b^2\,c^2\,d^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}+3\,a\,b^2\,c\,e^4\,n\,\sqrt {b^2-4\,a\,c}+10\,b\,c^3\,d^3\,e\,n\,\sqrt {b^2-4\,a\,c}+5\,b^3\,c\,d\,e^3\,n\,\sqrt {b^2-4\,a\,c}-30\,a\,b\,c^3\,d^2\,e^2\,n+20\,a\,b^2\,c^2\,d\,e^3\,n-10\,a\,b\,c^2\,d\,e^3\,n\,\sqrt {b^2-4\,a\,c}\right )}{10\,c^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((b*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/(3*c) + (2*a*e^4*n)/(15*c) - (d*e^2*n*(b*e + 4*c*d
))/(3*c)) - x*((a*((b*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/c + (2*a*e^4*n)/(5*c) - (d*e^2*n*(b*
e + 4*c*d))/c))/c - (b*((b*((b*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/c + (2*a*e^4*n)/(5*c) - (d*
e^2*n*(b*e + 4*c*d))/c))/c - (a*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/c + (2*d^2*e*n*(b*e + 2*c*
d))/c))/c + (2*d^3*n*(b*e + c*d))/c) - x^2*((b*((b*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/c + (2*
a*e^4*n)/(5*c) - (d*e^2*n*(b*e + 4*c*d))/c))/(2*c) - (a*((e^3*n*(b*e + 10*c*d))/(5*c) - (2*b*e^4*n)/(5*c)))/(2
*c) + (d^2*e*n*(b*e + 2*c*d))/c) - x^4*((e^3*n*(b*e + 10*c*d))/(20*c) - (b*e^4*n)/(10*c)) + log(d*(a + b*x + c
*x^2)^n)*(d^4*x + (e^4*x^5)/5 + 2*d^3*e*x^2 + d*e^3*x^4 + 2*d^2*e^2*x^3) + (log(b*(b^2 - 4*a*c)^(1/2) - 4*a*c
+ b^2 + 2*c*x*(b^2 - 4*a*c)^(1/2))*(b^5*e^4*n + 5*b*c^4*d^4*n + b^4*e^4*n*(b^2 - 4*a*c)^(1/2) + 5*c^4*d^4*n*(b
^2 - 4*a*c)^(1/2) - 5*a*b^3*c*e^4*n + 20*a*c^4*d^3*e*n - 5*b^4*c*d*e^3*n + 5*a^2*b*c^2*e^4*n - 10*a^2*c^3*d*e^
3*n - 10*b^2*c^3*d^3*e*n + a^2*c^2*e^4*n*(b^2 - 4*a*c)^(1/2) + 10*b^3*c^2*d^2*e^2*n - 10*a*c^3*d^2*e^2*n*(b^2
- 4*a*c)^(1/2) + 10*b^2*c^2*d^2*e^2*n*(b^2 - 4*a*c)^(1/2) - 3*a*b^2*c*e^4*n*(b^2 - 4*a*c)^(1/2) - 10*b*c^3*d^3
*e*n*(b^2 - 4*a*c)^(1/2) - 5*b^3*c*d*e^3*n*(b^2 - 4*a*c)^(1/2) - 30*a*b*c^3*d^2*e^2*n + 20*a*b^2*c^2*d*e^3*n +
10*a*b*c^2*d*e^3*n*(b^2 - 4*a*c)^(1/2)))/(10*c^5) - (2*e^4*n*x^5)/25 + (log(4*a*c + b*(b^2 - 4*a*c)^(1/2) - b
^2 + 2*c*x*(b^2 - 4*a*c)^(1/2))*(b^5*e^4*n + 5*b*c^4*d^4*n - b^4*e^4*n*(b^2 - 4*a*c)^(1/2) - 5*c^4*d^4*n*(b^2
- 4*a*c)^(1/2) - 5*a*b^3*c*e^4*n + 20*a*c^4*d^3*e*n - 5*b^4*c*d*e^3*n + 5*a^2*b*c^2*e^4*n - 10*a^2*c^3*d*e^3*n
- 10*b^2*c^3*d^3*e*n - a^2*c^2*e^4*n*(b^2 - 4*a*c)^(1/2) + 10*b^3*c^2*d^2*e^2*n + 10*a*c^3*d^2*e^2*n*(b^2 - 4
*a*c)^(1/2) - 10*b^2*c^2*d^2*e^2*n*(b^2 - 4*a*c)^(1/2) + 3*a*b^2*c*e^4*n*(b^2 - 4*a*c)^(1/2) + 10*b*c^3*d^3*e*
n*(b^2 - 4*a*c)^(1/2) + 5*b^3*c*d*e^3*n*(b^2 - 4*a*c)^(1/2) - 30*a*b*c^3*d^2*e^2*n + 20*a*b^2*c^2*d*e^3*n - 10
*a*b*c^2*d*e^3*n*(b^2 - 4*a*c)^(1/2)))/(10*c^5)

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

[Out]

Timed out

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