### 3.89 $$\int \frac{\log (d (a+b x+c x^2)^n)}{(d+e x)^3} \, dx$$

Optimal. Leaf size=259 $\frac{n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{4 e \left (a e^2-b d e+c d^2\right )^2}-\frac{n \log (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{2 e \left (a e^2-b d e+c d^2\right )^2}+\frac{n \sqrt{b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac{n (2 c d-b e)}{2 e (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}$

[Out]

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

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Rubi [A]  time = 0.405627, antiderivative size = 259, 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} $\frac{n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{4 e \left (a e^2-b d e+c d^2\right )^2}-\frac{n \log (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{2 e \left (a e^2-b d e+c d^2\right )^2}+\frac{n \sqrt{b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac{n (2 c d-b e)}{2 e (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.59451, size = 215, normalized size = 0.83 $\frac{\frac{n (d+e x) \left (-2 (d+e x) \log (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )+(d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log (a+x (b+c x))-2 e \sqrt{b^2-4 a c} (d+e x) (b e-2 c d) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+2 (2 c d-b e) \left (e (a e-b d)+c d^2\right )\right )}{\left (e (a e-b d)+c d^2\right )^2}-2 \log \left (d (a+x (b+c x))^n\right )}{4 e (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^3,x]

[Out]

((n*(d + e*x)*(2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)) - 2*Sqrt[b^2 - 4*a*c]*e*(-2*c*d + b*e)*(d + e*x)*Arc
Tanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] - 2*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*(d + e*x)*Log[d + e*x] + (2*
c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*(d + e*x)*Log[a + x*(b + c*x)]))/(c*d^2 + e*(-(b*d) + a*e))^2 - 2*Log[d
*(a + x*(b + c*x))^n])/(4*e*(d + e*x)^2)

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Maple [C]  time = 0.237, size = 55216, normalized size = 213.2 \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)^3,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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 17.1336, size = 2867, normalized size = 11.07 \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)^3,x, algorithm="fricas")

[Out]

[1/4*(2*(2*c^2*d^3*e - 3*b*c*d^2*e^2 - a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*n*x - ((2*c*d*e^3 - b*e^4)*n*x^2 + 2*(2*
c*d^2*e^2 - b*d*e^3)*n*x + (2*c*d^3*e - b*d^2*e^2)*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)) + 2*(2*c^2*d^4 - 3*b*c*d^3*e - a*b*d*e^3 + (b^2 + 2*a*c)*
d^2*e^2)*n + ((2*c^2*d^2*e^2 - 2*b*c*d*e^3 + (b^2 - 2*a*c)*e^4)*n*x^2 + 2*(2*c^2*d^3*e - 2*b*c*d^2*e^2 + (b^2
- 2*a*c)*d*e^3)*n*x + (2*b*c*d^3*e + 4*a*b*d*e^3 - 2*a^2*e^4 - (b^2 + 6*a*c)*d^2*e^2)*n)*log(c*x^2 + b*x + a)
- 2*((2*c^2*d^2*e^2 - 2*b*c*d*e^3 + (b^2 - 2*a*c)*e^4)*n*x^2 + 2*(2*c^2*d^3*e - 2*b*c*d^2*e^2 + (b^2 - 2*a*c)*
d*e^3)*n*x + (2*c^2*d^4 - 2*b*c*d^3*e + (b^2 - 2*a*c)*d^2*e^2)*n)*log(e*x + d) - 2*(c^2*d^4 - 2*b*c*d^3*e - 2*
a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*log(d))/(c^2*d^6*e - 2*b*c*d^5*e^2 - 2*a*b*d^3*e^4 + a^2*d^2*e^5
+ (b^2 + 2*a*c)*d^4*e^3 + (c^2*d^4*e^3 - 2*b*c*d^3*e^4 - 2*a*b*d*e^6 + a^2*e^7 + (b^2 + 2*a*c)*d^2*e^5)*x^2 +
2*(c^2*d^5*e^2 - 2*b*c*d^4*e^3 - 2*a*b*d^2*e^5 + a^2*d*e^6 + (b^2 + 2*a*c)*d^3*e^4)*x), 1/4*(2*(2*c^2*d^3*e -
3*b*c*d^2*e^2 - a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*n*x + 2*((2*c*d*e^3 - b*e^4)*n*x^2 + 2*(2*c*d^2*e^2 - b*d*e^3)*
n*x + (2*c*d^3*e - b*d^2*e^2)*n)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*
(2*c^2*d^4 - 3*b*c*d^3*e - a*b*d*e^3 + (b^2 + 2*a*c)*d^2*e^2)*n + ((2*c^2*d^2*e^2 - 2*b*c*d*e^3 + (b^2 - 2*a*c
)*e^4)*n*x^2 + 2*(2*c^2*d^3*e - 2*b*c*d^2*e^2 + (b^2 - 2*a*c)*d*e^3)*n*x + (2*b*c*d^3*e + 4*a*b*d*e^3 - 2*a^2*
e^4 - (b^2 + 6*a*c)*d^2*e^2)*n)*log(c*x^2 + b*x + a) - 2*((2*c^2*d^2*e^2 - 2*b*c*d*e^3 + (b^2 - 2*a*c)*e^4)*n*
x^2 + 2*(2*c^2*d^3*e - 2*b*c*d^2*e^2 + (b^2 - 2*a*c)*d*e^3)*n*x + (2*c^2*d^4 - 2*b*c*d^3*e + (b^2 - 2*a*c)*d^2
*e^2)*n)*log(e*x + d) - 2*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*log(d))/(c^2
*d^6*e - 2*b*c*d^5*e^2 - 2*a*b*d^3*e^4 + a^2*d^2*e^5 + (b^2 + 2*a*c)*d^4*e^3 + (c^2*d^4*e^3 - 2*b*c*d^3*e^4 -
2*a*b*d*e^6 + a^2*e^7 + (b^2 + 2*a*c)*d^2*e^5)*x^2 + 2*(c^2*d^5*e^2 - 2*b*c*d^4*e^3 - 2*a*b*d^2*e^5 + a^2*d*e^
6 + (b^2 + 2*a*c)*d^3*e^4)*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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.32779, size = 1197, normalized size = 4.62 \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)^3,x, algorithm="giac")

[Out]

1/4*(2*c^2*d^2*n - 2*b*c*d*n*e + b^2*n*e^2 - 2*a*c*n*e^2)*log(c*x^2 + b*x + a)/(c^2*d^4*e - 2*b*c*d^3*e^2 + b^
2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5) - 1/2*(2*b^2*c*d*n - 8*a*c^2*d*n - b^3*n*e + 4*a*b*c*n*e)*a
rctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^
2*e^4)*sqrt(-b^2 + 4*a*c)) - 1/2*(2*c^2*d^2*n*x^2*e^2*log(x*e + d) + 4*c^2*d^3*n*x*e*log(x*e + d) - 2*c^2*d^3*
n*x*e + c^2*d^4*n*log(c*x^2 + b*x + a) - 2*b*c*d^3*n*e*log(c*x^2 + b*x + a) + 2*c^2*d^4*n*log(x*e + d) - 2*b*c
*d*n*x^2*e^3*log(x*e + d) - 4*b*c*d^2*n*x*e^2*log(x*e + d) - 2*b*c*d^3*n*e*log(x*e + d) - 2*c^2*d^4*n + 3*b*c*
d^2*n*x*e^2 + 3*b*c*d^3*n*e + b^2*d^2*n*e^2*log(c*x^2 + b*x + a) + 2*a*c*d^2*n*e^2*log(c*x^2 + b*x + a) + b^2*
n*x^2*e^4*log(x*e + d) - 2*a*c*n*x^2*e^4*log(x*e + d) + 2*b^2*d*n*x*e^3*log(x*e + d) - 4*a*c*d*n*x*e^3*log(x*e
+ d) + b^2*d^2*n*e^2*log(x*e + d) - 2*a*c*d^2*n*e^2*log(x*e + d) + c^2*d^4*log(d) - 2*b*c*d^3*e*log(d) - b^2*
d*n*x*e^3 - 2*a*c*d*n*x*e^3 - b^2*d^2*n*e^2 - 2*a*c*d^2*n*e^2 - 2*a*b*d*n*e^3*log(c*x^2 + b*x + a) + b^2*d^2*e
^2*log(d) + 2*a*c*d^2*e^2*log(d) + a*b*n*x*e^4 + a*b*d*n*e^3 + a^2*n*e^4*log(c*x^2 + b*x + a) - 2*a*b*d*e^3*lo
g(d) + a^2*e^4*log(d))/(c^2*d^4*x^2*e^3 + 2*c^2*d^5*x*e^2 + c^2*d^6*e - 2*b*c*d^3*x^2*e^4 - 4*b*c*d^4*x*e^3 -
2*b*c*d^5*e^2 + b^2*d^2*x^2*e^5 + 2*a*c*d^2*x^2*e^5 + 2*b^2*d^3*x*e^4 + 4*a*c*d^3*x*e^4 + b^2*d^4*e^3 + 2*a*c*
d^4*e^3 - 2*a*b*d*x^2*e^6 - 4*a*b*d^2*x*e^5 - 2*a*b*d^3*e^4 + a^2*x^2*e^7 + 2*a^2*d*x*e^6 + a^2*d^2*e^5)