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]

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

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Rubi [A]  time = 0.62, antiderivative size = 356, 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} \[ \frac {n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{6 e \left (a e^2-b d e+c d^2\right )^3}+\frac {n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{3 e (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {n (2 c d-b e) \log (d+e x) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{3 e \left (a e^2-b d e+c d^2\right )^3}+\frac {n \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{3 \left (a e^2-b d e+c d^2\right )^3}+\frac {n (2 c d-b e)}{6 e (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e (d+e x)^3} \]

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 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)^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*}

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Mathematica [A]  time = 1.21, 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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fricas [B]  time = 15.09, size = 3013, normalized size = 8.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^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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giac [B]  time = 0.61, size = 1963, normalized size = 5.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/(e*x+d)^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)

________________________________________________________________________________________

maple [C]  time = 1.04, size = 306209, normalized size = 860.14 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(c*x^2+b*x+a)^n)/(e*x+d)^4,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)^4,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 = 11.30, size = 2707, normalized size = 7.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(d + e*x)*(e^3*(b^3*n - 3*a*b*c*n) + e^2*(6*a*c^2*d*n - 3*b^2*c*d*n) - 2*c^3*d^3*n + 3*b*c^2*d^2*e*n))/(3*
a^3*e^7 + 3*c^3*d^6*e - 3*b^3*d^3*e^4 + 9*a*b^2*d^2*e^5 + 9*a*c^2*d^4*e^3 + 9*a^2*c*d^2*e^5 - 9*b*c^2*d^5*e^2
+ 9*b^2*c*d^4*e^3 - 9*a^2*b*d*e^6 - 18*a*b*c*d^3*e^4) - (log(32*a*b^5*e^5 - 2*a*e^5*(b^2 - 4*a*c)^(5/2) - 192*
a*c^5*d^5 + 32*b^6*e^5*x + 48*b^2*c^4*d^5 - 18*b^3*e^5*x*(b^2 - 4*a*c)^(3/2) - 3*b^5*e^5*x*(b^2 - 4*a*c)^(1/2)
+ 96*c^5*d^5*x*(b^2 - 4*a*c)^(1/2) - 208*a^2*b^3*c*e^5 + 320*a^3*b*c^2*e^5 - 704*a^3*c^3*d*e^4 - 48*b^3*c^3*d
^4*e - 16*b^5*c*d^2*e^3 - 64*a^3*c^3*e^5*x + 1152*a^2*c^4*d^3*e^2 + 48*b^4*c^2*d^3*e^2 - 33*b*d*e^4*(b^2 - 4*a
*c)^(5/2) - 11*b*e^5*x*(b^2 - 4*a*c)^(5/2) - 24*a*b^2*e^5*(b^2 - 4*a*c)^(3/2) - 6*a*b^4*e^5*(b^2 - 4*a*c)^(1/2
) + 48*b*c^4*d^5*(b^2 - 4*a*c)^(1/2) + 18*b^3*d*e^4*(b^2 - 4*a*c)^(3/2) + 15*b^5*d*e^4*(b^2 - 4*a*c)^(1/2) + 4
4*c*d^2*e^3*(b^2 - 4*a*c)^(5/2) + 72*c^3*d^4*e*(b^2 - 4*a*c)^(3/2) + 22*c*d*e^4*x*(b^2 - 4*a*c)^(5/2) + 192*a*
b*c^4*d^4*e - 128*a*b^4*c*d*e^4 + 120*b^3*c^2*d^3*e^2*(b^2 - 4*a*c)^(1/2) - 224*a*b^4*c*e^5*x - 576*a*c^5*d^4*
e*x - 160*b^5*c*d*e^4*x + 144*b^2*c^4*d^4*e*x - 72*b*c^2*d^3*e^2*(b^2 - 4*a*c)^(3/2) - 120*b^2*c^3*d^4*e*(b^2
- 4*a*c)^(1/2) - 60*b^4*c*d^2*e^3*(b^2 - 4*a*c)^(1/2) + 144*c^3*d^3*e^2*x*(b^2 - 4*a*c)^(3/2) - 480*a*b^2*c^3*
d^3*e^2 + 320*a*b^3*c^2*d^2*e^3 - 1024*a^2*b*c^3*d^2*e^3 + 688*a^2*b^2*c^2*d*e^4 + 400*a^2*b^2*c^2*e^5*x + 140
8*a^2*c^4*d^2*e^3*x - 288*b^3*c^3*d^3*e^2*x + 304*b^4*c^2*d^2*e^3*x - 216*b*c^2*d^2*e^3*x*(b^2 - 4*a*c)^(3/2)
- 1568*a*b^2*c^3*d^2*e^3*x + 240*b^2*c^3*d^3*e^2*x*(b^2 - 4*a*c)^(1/2) - 120*b^3*c^2*d^2*e^3*x*(b^2 - 4*a*c)^(
1/2) - 240*b*c^4*d^4*e*x*(b^2 - 4*a*c)^(1/2) + 108*b^2*c*d*e^4*x*(b^2 - 4*a*c)^(3/2) + 30*b^4*c*d*e^4*x*(b^2 -
4*a*c)^(1/2) + 1152*a*b*c^4*d^3*e^2*x + 992*a*b^3*c^2*d*e^4*x - 1408*a^2*b*c^3*d*e^4*x)*(e^3*((b^3*n)/6 - (b^
2*n*(b^2 - 4*a*c)^(1/2))/6 + (a*c*n*(b^2 - 4*a*c)^(1/2))/6 - (a*b*c*n)/2) + e^2*(a*c^2*d*n - (b^2*c*d*n)/2 + (
b*c*d*n*(b^2 - 4*a*c)^(1/2))/2) + e*((b*c^2*d^2*n)/2 - (c^2*d^2*n*(b^2 - 4*a*c)^(1/2))/2) - (c^3*d^3*n)/3))/(a
^3*e^7 + c^3*d^6*e - b^3*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - 3*b*c^2*d^5*e^2 + 3*b
^2*c*d^4*e^3 - 3*a^2*b*d*e^6 - 6*a*b*c*d^3*e^4) - (log(2*a*e^5*(b^2 - 4*a*c)^(5/2) + 32*a*b^5*e^5 - 192*a*c^5*
d^5 + 32*b^6*e^5*x + 48*b^2*c^4*d^5 + 18*b^3*e^5*x*(b^2 - 4*a*c)^(3/2) + 3*b^5*e^5*x*(b^2 - 4*a*c)^(1/2) - 96*
c^5*d^5*x*(b^2 - 4*a*c)^(1/2) - 208*a^2*b^3*c*e^5 + 320*a^3*b*c^2*e^5 - 704*a^3*c^3*d*e^4 - 48*b^3*c^3*d^4*e -
16*b^5*c*d^2*e^3 - 64*a^3*c^3*e^5*x + 1152*a^2*c^4*d^3*e^2 + 48*b^4*c^2*d^3*e^2 + 33*b*d*e^4*(b^2 - 4*a*c)^(5
/2) + 11*b*e^5*x*(b^2 - 4*a*c)^(5/2) + 24*a*b^2*e^5*(b^2 - 4*a*c)^(3/2) + 6*a*b^4*e^5*(b^2 - 4*a*c)^(1/2) - 48
*b*c^4*d^5*(b^2 - 4*a*c)^(1/2) - 18*b^3*d*e^4*(b^2 - 4*a*c)^(3/2) - 15*b^5*d*e^4*(b^2 - 4*a*c)^(1/2) - 44*c*d^
2*e^3*(b^2 - 4*a*c)^(5/2) - 72*c^3*d^4*e*(b^2 - 4*a*c)^(3/2) - 22*c*d*e^4*x*(b^2 - 4*a*c)^(5/2) + 192*a*b*c^4*
d^4*e - 128*a*b^4*c*d*e^4 - 120*b^3*c^2*d^3*e^2*(b^2 - 4*a*c)^(1/2) - 224*a*b^4*c*e^5*x - 576*a*c^5*d^4*e*x -
160*b^5*c*d*e^4*x + 144*b^2*c^4*d^4*e*x + 72*b*c^2*d^3*e^2*(b^2 - 4*a*c)^(3/2) + 120*b^2*c^3*d^4*e*(b^2 - 4*a*
c)^(1/2) + 60*b^4*c*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 144*c^3*d^3*e^2*x*(b^2 - 4*a*c)^(3/2) - 480*a*b^2*c^3*d^3*e^
2 + 320*a*b^3*c^2*d^2*e^3 - 1024*a^2*b*c^3*d^2*e^3 + 688*a^2*b^2*c^2*d*e^4 + 400*a^2*b^2*c^2*e^5*x + 1408*a^2*
c^4*d^2*e^3*x - 288*b^3*c^3*d^3*e^2*x + 304*b^4*c^2*d^2*e^3*x + 216*b*c^2*d^2*e^3*x*(b^2 - 4*a*c)^(3/2) - 1568
*a*b^2*c^3*d^2*e^3*x - 240*b^2*c^3*d^3*e^2*x*(b^2 - 4*a*c)^(1/2) + 120*b^3*c^2*d^2*e^3*x*(b^2 - 4*a*c)^(1/2) +
240*b*c^4*d^4*e*x*(b^2 - 4*a*c)^(1/2) - 108*b^2*c*d*e^4*x*(b^2 - 4*a*c)^(3/2) - 30*b^4*c*d*e^4*x*(b^2 - 4*a*c
)^(1/2) + 1152*a*b*c^4*d^3*e^2*x + 992*a*b^3*c^2*d*e^4*x - 1408*a^2*b*c^3*d*e^4*x)*(e^3*((b^3*n)/6 + (b^2*n*(b
^2 - 4*a*c)^(1/2))/6 - (a*c*n*(b^2 - 4*a*c)^(1/2))/6 - (a*b*c*n)/2) - e^2*((b^2*c*d*n)/2 - a*c^2*d*n + (b*c*d*
n*(b^2 - 4*a*c)^(1/2))/2) + e*((b*c^2*d^2*n)/2 + (c^2*d^2*n*(b^2 - 4*a*c)^(1/2))/2) - (c^3*d^3*n)/3))/(a^3*e^7
+ c^3*d^6*e - b^3*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - 3*b*c^2*d^5*e^2 + 3*b^2*c*d
^4*e^3 - 3*a^2*b*d*e^6 - 6*a*b*c*d^3*e^4) - ((a*b*e^3*n - 6*c^2*d^3*n - 3*b^2*d*e^2*n + 2*a*c*d*e^2*n + 7*b*c*
d^2*e*n)/(2*(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2)) - (n*x*(b^2*e^3 + 2
*c^2*d^2*e - 2*a*c*e^3 - 2*b*c*d*e^2))/(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^
2*e^2))/(3*d^2*e + 3*e^3*x^2 + 6*d*e^2*x) - log(d*(a + b*x + c*x^2)^n)/(3*e*(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d
^2*e*x))

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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)**4,x)

[Out]

Timed out

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