### 3.90 $$\int \frac{\log (d (a+b x+c x^2)^n)}{(d+e x)^4} \, dx$$

Optimal. Leaf size=356 $\frac{n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{6 e \left (a e^2-b d e+c d^2\right )^3}+\frac{n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{3 e (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{n (2 c d-b e) \log (d+e x) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{3 e \left (a e^2-b d e+c d^2\right )^3}+\frac{n \sqrt{b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 \left (a e^2-b d e+c d^2\right )^3}+\frac{n (2 c d-b e)}{6 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3}$

[Out]

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

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Rubi [A]  time = 0.620757, antiderivative size = 356, 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} $\frac{n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{6 e \left (a e^2-b d e+c d^2\right )^3}+\frac{n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{3 e (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{n (2 c d-b e) \log (d+e x) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{3 e \left (a e^2-b d e+c d^2\right )^3}+\frac{n \sqrt{b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 \left (a e^2-b d e+c d^2\right )^3}+\frac{n (2 c d-b e)}{6 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 \frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^4} \, dx &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3}+\frac{n \int \frac{b+2 c x}{(d+e x)^3 \left (a+b x+c x^2\right )} \, dx}{3 e}\\ &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3}+\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)^3}+\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)^2}+\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)}+\frac{3 b^3 c d e^2-b^4 e^3+b c^2 d \left (c d^2-9 a e^2\right )-b^2 c e \left (3 c d^2-4 a e^2\right )+2 a c^2 e \left (3 c d^2-a e^2\right )+c (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}\right ) \, dx}{3 e}\\ &=\frac{(2 c d-b e) n}{6 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac{\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{3 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log (d+e x)}{3 e \left (c d^2-b d e+a e^2\right )^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3}+\frac{n \int \frac{3 b^3 c d e^2-b^4 e^3+b c^2 d \left (c d^2-9 a e^2\right )-b^2 c e \left (3 c d^2-4 a e^2\right )+2 a c^2 e \left (3 c d^2-a e^2\right )+c (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{a+b x+c x^2} \, dx}{3 e \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{(2 c d-b e) n}{6 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac{\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{3 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log (d+e x)}{3 e \left (c d^2-b d e+a e^2\right )^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3}-\frac{\left (\left (b^2-4 a c\right ) \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{6 \left (c d^2-b d e+a e^2\right )^3}+\frac{\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{6 e \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{(2 c d-b e) n}{6 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac{\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{3 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log (d+e x)}{3 e \left (c d^2-b d e+a e^2\right )^3}+\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log \left (a+b x+c x^2\right )}{6 e \left (c d^2-b d e+a e^2\right )^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3}+\frac{\left (\left (b^2-4 a c\right ) \left (3 c^2 d^2+b^2 e^2-c e (3 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 )}{3 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{(2 c d-b e) n}{6 e \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac{\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n}{3 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\sqrt{b^2-4 a c} \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 \left (c d^2-b d e+a e^2\right )^3}-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log (d+e x)}{3 e \left (c d^2-b d e+a e^2\right )^3}+\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log \left (a+b x+c x^2\right )}{6 e \left (c d^2-b d e+a e^2\right )^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 1.19815, size = 310, normalized size = 0.87 $\frac{\frac{n (d+e x) \left (2 (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )-2 (d+e x)^2 (2 c d-b e) \log (d+e x) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )+(d+e x)^2 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log (a+x (b+c x))+2 e \sqrt{b^2-4 a c} (d+e x)^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+(2 c d-b e) \left (e (a e-b d)+c d^2\right )^2\right )}{\left (e (a e-b d)+c d^2\right )^3}-2 \log \left (d (a+x (b+c x))^n\right )}{6 e (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((n*(d + e*x)*((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2 + 2*(c*d^2 + e*(-(b*d) + a*e))*(2*c^2*d^2 + b^2*e^2
- 2*c*e*(b*d + a*e))*(d + e*x) + 2*Sqrt[b^2 - 4*a*c]*e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*(d + e*x)^2*A
rcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] - 2*(2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(d + e*x)^2*Lo
g[d + e*x] + (2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(d + e*x)^2*Log[a + x*(b + c*x)]))/(c*d^2 +
e*(-(b*d) + a*e))^3 - 2*Log[d*(a + x*(b + c*x))^n])/(6*e*(d + e*x)^3)

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Maple [C]  time = 0.303, size = 306209, normalized size = 860.1 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 125.094, size = 6159, normalized size = 17.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(2*(2*c^3*d^4*e^2 - 4*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^4 - b^3*d*e^5 + (a*b^2 - 2*a^2*c)*e^6)*n*x^2 + (10*c^
3*d^5*e - 21*b*c^2*d^4*e^2 - a^2*b*e^6 + 4*(4*b^2*c + a*c^2)*d^3*e^3 - (5*b^3 + 6*a*b*c)*d^2*e^4 + 6*(a*b^2 -
a^2*c)*d*e^5)*n*x - ((3*c^2*d^2*e^4 - 3*b*c*d*e^5 + (b^2 - a*c)*e^6)*n*x^3 + 3*(3*c^2*d^3*e^3 - 3*b*c*d^2*e^4
+ (b^2 - a*c)*d*e^5)*n*x^2 + 3*(3*c^2*d^4*e^2 - 3*b*c*d^3*e^3 + (b^2 - a*c)*d^2*e^4)*n*x + (3*c^2*d^5*e - 3*b*
c*d^4*e^2 + (b^2 - a*c)*d^3*e^3)*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)) + (6*c^3*d^6 - 13*b*c^2*d^5*e - a^2*b*d*e^5 + 2*(5*b^2*c + 2*a*c^2)*d^4*e^2
- 3*(b^3 + 2*a*b*c)*d^3*e^3 + 2*(2*a*b^2 - a^2*c)*d^2*e^4)*n + ((2*c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*(b^2*c -
2*a*c^2)*d*e^5 - (b^3 - 3*a*b*c)*e^6)*n*x^3 + 3*(2*c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*(b^2*c - 2*a*c^2)*d^2*e^
4 - (b^3 - 3*a*b*c)*d*e^5)*n*x^2 + 3*(2*c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*(b^2*c - 2*a*c^2)*d^3*e^3 - (b^3 - 3*a
*b*c)*d^2*e^4)*n*x + (3*b*c^2*d^5*e + 6*a^2*b*d*e^5 - 2*a^3*e^6 - 3*(b^2*c + 4*a*c^2)*d^4*e^2 + (b^3 + 15*a*b*
c)*d^3*e^3 - 6*(a*b^2 + a^2*c)*d^2*e^4)*n)*log(c*x^2 + b*x + a) - 2*((2*c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*(b^2
*c - 2*a*c^2)*d*e^5 - (b^3 - 3*a*b*c)*e^6)*n*x^3 + 3*(2*c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*(b^2*c - 2*a*c^2)*d^
2*e^4 - (b^3 - 3*a*b*c)*d*e^5)*n*x^2 + 3*(2*c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*(b^2*c - 2*a*c^2)*d^3*e^3 - (b^3 -
3*a*b*c)*d^2*e^4)*n*x + (2*c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c - 2*a*c^2)*d^4*e^2 - (b^3 - 3*a*b*c)*d^3*e^3)*n
)*log(e*x + d) - 2*(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a
*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*log(d))/(c^3*d^9*e - 3*b*c^2*d^8*e^2 - 3*a^2*b*d^4*e^6 + a^3*d^3*e^
7 + 3*(b^2*c + a*c^2)*d^7*e^3 - (b^3 + 6*a*b*c)*d^6*e^4 + 3*(a*b^2 + a^2*c)*d^5*e^5 + (c^3*d^6*e^4 - 3*b*c^2*d
^5*e^5 - 3*a^2*b*d*e^9 + a^3*e^10 + 3*(b^2*c + a*c^2)*d^4*e^6 - (b^3 + 6*a*b*c)*d^3*e^7 + 3*(a*b^2 + a^2*c)*d^
2*e^8)*x^3 + 3*(c^3*d^7*e^3 - 3*b*c^2*d^6*e^4 - 3*a^2*b*d^2*e^8 + a^3*d*e^9 + 3*(b^2*c + a*c^2)*d^5*e^5 - (b^3
+ 6*a*b*c)*d^4*e^6 + 3*(a*b^2 + a^2*c)*d^3*e^7)*x^2 + 3*(c^3*d^8*e^2 - 3*b*c^2*d^7*e^3 - 3*a^2*b*d^3*e^7 + a^
3*d^2*e^8 + 3*(b^2*c + a*c^2)*d^6*e^4 - (b^3 + 6*a*b*c)*d^5*e^5 + 3*(a*b^2 + a^2*c)*d^4*e^6)*x), 1/6*(2*(2*c^3
*d^4*e^2 - 4*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^4 - b^3*d*e^5 + (a*b^2 - 2*a^2*c)*e^6)*n*x^2 + (10*c^3*d^5*e - 21*b
*c^2*d^4*e^2 - a^2*b*e^6 + 4*(4*b^2*c + a*c^2)*d^3*e^3 - (5*b^3 + 6*a*b*c)*d^2*e^4 + 6*(a*b^2 - a^2*c)*d*e^5)*
n*x + 2*((3*c^2*d^2*e^4 - 3*b*c*d*e^5 + (b^2 - a*c)*e^6)*n*x^3 + 3*(3*c^2*d^3*e^3 - 3*b*c*d^2*e^4 + (b^2 - a*c
)*d*e^5)*n*x^2 + 3*(3*c^2*d^4*e^2 - 3*b*c*d^3*e^3 + (b^2 - a*c)*d^2*e^4)*n*x + (3*c^2*d^5*e - 3*b*c*d^4*e^2 +
(b^2 - a*c)*d^3*e^3)*n)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (6*c^3*d^6
- 13*b*c^2*d^5*e - a^2*b*d*e^5 + 2*(5*b^2*c + 2*a*c^2)*d^4*e^2 - 3*(b^3 + 2*a*b*c)*d^3*e^3 + 2*(2*a*b^2 - a^2*
c)*d^2*e^4)*n + ((2*c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*(b^2*c - 2*a*c^2)*d*e^5 - (b^3 - 3*a*b*c)*e^6)*n*x^3 + 3
*(2*c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*(b^2*c - 2*a*c^2)*d^2*e^4 - (b^3 - 3*a*b*c)*d*e^5)*n*x^2 + 3*(2*c^3*d^5*
e - 3*b*c^2*d^4*e^2 + 3*(b^2*c - 2*a*c^2)*d^3*e^3 - (b^3 - 3*a*b*c)*d^2*e^4)*n*x + (3*b*c^2*d^5*e + 6*a^2*b*d*
e^5 - 2*a^3*e^6 - 3*(b^2*c + 4*a*c^2)*d^4*e^2 + (b^3 + 15*a*b*c)*d^3*e^3 - 6*(a*b^2 + a^2*c)*d^2*e^4)*n)*log(c
*x^2 + b*x + a) - 2*((2*c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*(b^2*c - 2*a*c^2)*d*e^5 - (b^3 - 3*a*b*c)*e^6)*n*x^3
+ 3*(2*c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*(b^2*c - 2*a*c^2)*d^2*e^4 - (b^3 - 3*a*b*c)*d*e^5)*n*x^2 + 3*(2*c^3*
d^5*e - 3*b*c^2*d^4*e^2 + 3*(b^2*c - 2*a*c^2)*d^3*e^3 - (b^3 - 3*a*b*c)*d^2*e^4)*n*x + (2*c^3*d^6 - 3*b*c^2*d^
5*e + 3*(b^2*c - 2*a*c^2)*d^4*e^2 - (b^3 - 3*a*b*c)*d^3*e^3)*n)*log(e*x + d) - 2*(c^3*d^6 - 3*b*c^2*d^5*e - 3*
a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*log(d
))/(c^3*d^9*e - 3*b*c^2*d^8*e^2 - 3*a^2*b*d^4*e^6 + a^3*d^3*e^7 + 3*(b^2*c + a*c^2)*d^7*e^3 - (b^3 + 6*a*b*c)*
d^6*e^4 + 3*(a*b^2 + a^2*c)*d^5*e^5 + (c^3*d^6*e^4 - 3*b*c^2*d^5*e^5 - 3*a^2*b*d*e^9 + a^3*e^10 + 3*(b^2*c + a
*c^2)*d^4*e^6 - (b^3 + 6*a*b*c)*d^3*e^7 + 3*(a*b^2 + a^2*c)*d^2*e^8)*x^3 + 3*(c^3*d^7*e^3 - 3*b*c^2*d^6*e^4 -
3*a^2*b*d^2*e^8 + a^3*d*e^9 + 3*(b^2*c + a*c^2)*d^5*e^5 - (b^3 + 6*a*b*c)*d^4*e^6 + 3*(a*b^2 + a^2*c)*d^3*e^7)
*x^2 + 3*(c^3*d^8*e^2 - 3*b*c^2*d^7*e^3 - 3*a^2*b*d^3*e^7 + a^3*d^2*e^8 + 3*(b^2*c + a*c^2)*d^6*e^4 - (b^3 + 6
*a*b*c)*d^5*e^5 + 3*(a*b^2 + a^2*c)*d^4*e^6)*x)]

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

[Out]

Timed out

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Giac [B]  time = 1.51773, size = 2650, normalized size = 7.44 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*c^3*d^3*n - 3*b*c^2*d^2*n*e + 3*b^2*c*d*n*e^2 - 6*a*c^2*d*n*e^2 - b^3*n*e^3 + 3*a*b*c*n*e^3)*log(c*x^2
+ b*x + a)/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 +
3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7) - 1/3*(3*b^2*c^2*d^2*n - 12*a*c^3*d^2*n - 3*b^3*c
*d*n*e + 12*a*b*c^2*d*n*e + b^4*n*e^2 - 5*a*b^2*c*n*e^2 + 4*a^2*c^2*n*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*
c))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^
2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-b^2 + 4*a*c)) - 1/6*(4*c^3*d^3*n*x^3*e^3*log(x*e + d)
+ 12*c^3*d^4*n*x^2*e^2*log(x*e + d) + 12*c^3*d^5*n*x*e*log(x*e + d) - 4*c^3*d^4*n*x^2*e^2 - 10*c^3*d^5*n*x*e
+ 2*c^3*d^6*n*log(c*x^2 + b*x + a) - 6*b*c^2*d^5*n*e*log(c*x^2 + b*x + a) + 4*c^3*d^6*n*log(x*e + d) - 6*b*c^2
*d^2*n*x^3*e^4*log(x*e + d) - 18*b*c^2*d^3*n*x^2*e^3*log(x*e + d) - 18*b*c^2*d^4*n*x*e^2*log(x*e + d) - 6*b*c^
2*d^5*n*e*log(x*e + d) - 6*c^3*d^6*n + 8*b*c^2*d^3*n*x^2*e^3 + 21*b*c^2*d^4*n*x*e^2 + 13*b*c^2*d^5*n*e + 6*b^2
*c*d^4*n*e^2*log(c*x^2 + b*x + a) + 6*a*c^2*d^4*n*e^2*log(c*x^2 + b*x + a) + 6*b^2*c*d*n*x^3*e^5*log(x*e + d)
- 12*a*c^2*d*n*x^3*e^5*log(x*e + d) + 18*b^2*c*d^2*n*x^2*e^4*log(x*e + d) - 36*a*c^2*d^2*n*x^2*e^4*log(x*e + d
) + 18*b^2*c*d^3*n*x*e^3*log(x*e + d) - 36*a*c^2*d^3*n*x*e^3*log(x*e + d) + 6*b^2*c*d^4*n*e^2*log(x*e + d) - 1
2*a*c^2*d^4*n*e^2*log(x*e + d) + 2*c^3*d^6*log(d) - 6*b*c^2*d^5*e*log(d) - 6*b^2*c*d^2*n*x^2*e^4 - 16*b^2*c*d^
3*n*x*e^3 - 4*a*c^2*d^3*n*x*e^3 - 10*b^2*c*d^4*n*e^2 - 4*a*c^2*d^4*n*e^2 - 2*b^3*d^3*n*e^3*log(c*x^2 + b*x + a
) - 12*a*b*c*d^3*n*e^3*log(c*x^2 + b*x + a) - 2*b^3*n*x^3*e^6*log(x*e + d) + 6*a*b*c*n*x^3*e^6*log(x*e + d) -
6*b^3*d*n*x^2*e^5*log(x*e + d) + 18*a*b*c*d*n*x^2*e^5*log(x*e + d) - 6*b^3*d^2*n*x*e^4*log(x*e + d) + 18*a*b*c
*d^2*n*x*e^4*log(x*e + d) - 2*b^3*d^3*n*e^3*log(x*e + d) + 6*a*b*c*d^3*n*e^3*log(x*e + d) + 6*b^2*c*d^4*e^2*lo
g(d) + 6*a*c^2*d^4*e^2*log(d) + 2*b^3*d*n*x^2*e^5 + 5*b^3*d^2*n*x*e^4 + 6*a*b*c*d^2*n*x*e^4 + 3*b^3*d^3*n*e^3
+ 6*a*b*c*d^3*n*e^3 + 6*a*b^2*d^2*n*e^4*log(c*x^2 + b*x + a) + 6*a^2*c*d^2*n*e^4*log(c*x^2 + b*x + a) - 2*b^3*
d^3*e^3*log(d) - 12*a*b*c*d^3*e^3*log(d) - 2*a*b^2*n*x^2*e^6 + 4*a^2*c*n*x^2*e^6 - 6*a*b^2*d*n*x*e^5 + 6*a^2*c
*d*n*x*e^5 - 4*a*b^2*d^2*n*e^4 + 2*a^2*c*d^2*n*e^4 - 6*a^2*b*d*n*e^5*log(c*x^2 + b*x + a) + 6*a*b^2*d^2*e^4*lo
g(d) + 6*a^2*c*d^2*e^4*log(d) + a^2*b*n*x*e^6 + a^2*b*d*n*e^5 + 2*a^3*n*e^6*log(c*x^2 + b*x + a) - 6*a^2*b*d*e
^5*log(d) + 2*a^3*e^6*log(d))/(c^3*d^6*x^3*e^4 + 3*c^3*d^7*x^2*e^3 + 3*c^3*d^8*x*e^2 + c^3*d^9*e - 3*b*c^2*d^5
*x^3*e^5 - 9*b*c^2*d^6*x^2*e^4 - 9*b*c^2*d^7*x*e^3 - 3*b*c^2*d^8*e^2 + 3*b^2*c*d^4*x^3*e^6 + 3*a*c^2*d^4*x^3*e
^6 + 9*b^2*c*d^5*x^2*e^5 + 9*a*c^2*d^5*x^2*e^5 + 9*b^2*c*d^6*x*e^4 + 9*a*c^2*d^6*x*e^4 + 3*b^2*c*d^7*e^3 + 3*a
*c^2*d^7*e^3 - b^3*d^3*x^3*e^7 - 6*a*b*c*d^3*x^3*e^7 - 3*b^3*d^4*x^2*e^6 - 18*a*b*c*d^4*x^2*e^6 - 3*b^3*d^5*x*
e^5 - 18*a*b*c*d^5*x*e^5 - b^3*d^6*e^4 - 6*a*b*c*d^6*e^4 + 3*a*b^2*d^2*x^3*e^8 + 3*a^2*c*d^2*x^3*e^8 + 9*a*b^2
*d^3*x^2*e^7 + 9*a^2*c*d^3*x^2*e^7 + 9*a*b^2*d^4*x*e^6 + 9*a^2*c*d^4*x*e^6 + 3*a*b^2*d^5*e^5 + 3*a^2*c*d^5*e^5
- 3*a^2*b*d*x^3*e^9 - 9*a^2*b*d^2*x^2*e^8 - 9*a^2*b*d^3*x*e^7 - 3*a^2*b*d^4*e^6 + a^3*x^3*e^10 + 3*a^3*d*x^2*
e^9 + 3*a^3*d^2*x*e^8 + a^3*d^3*e^7)