3.2204 \(\int \frac{(d+e x)^2}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=198 \[ -\frac{-x \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-3 b \left (a e^2+c d^2\right )+4 a c d e+2 b^2 d e}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

-((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (2*b^2*d*e + 4*a*c*d*e -
3*b*(c*d^2 + a*e^2) - (6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*x)/((b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (2*
(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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Rubi [A]  time = 0.21828, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {738, 638, 618, 206} \[ -\frac{-x \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-3 b \left (a e^2+c d^2\right )+4 a c d e+2 b^2 d e}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (2*b^2*d*e + 4*a*c*d*e -
3*b*(c*d^2 + a*e^2) - (6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*x)/((b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (2*
(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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])

Rubi steps

\begin{align*} \int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{2 \left (3 c d^2-e (2 b d-a e)\right )+2 e (2 c d-b e) x}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{(d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{2 b^2 d e+4 a c d e-3 b \left (c d^2+a e^2\right )-\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac{(d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{2 b^2 d e+4 a c d e-3 b \left (c d^2+a e^2\right )-\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{(d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{2 b^2 d e+4 a c d e-3 b \left (c d^2+a e^2\right )-\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.309917, size = 203, normalized size = 1.03 \[ \frac{1}{2} \left (\frac{(b+2 c x) \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x}{c \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{4 \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (a*b*e^2 +
2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (
4*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2

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Maple [B]  time = 0.159, size = 508, normalized size = 2.6 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ({\frac{c \left ( 2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2} \right ){x}^{3}}{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}}+{\frac{3\,b \left ( 2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2} \right ){x}^{2}}{32\,{a}^{2}{c}^{2}-16\,ac{b}^{2}+2\,{b}^{4}}}-{\frac{ \left ( 2\,{a}^{2}c{e}^{2}-5\,a{b}^{2}{e}^{2}+10\,abcde-10\,a{c}^{2}{d}^{2}+2\,{b}^{3}de-2\,c{b}^{2}{d}^{2} \right ) x}{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}}+{\frac{6\,{a}^{2}b{e}^{2}-16\,{a}^{2}cde-2\,a{b}^{2}de+10\,abc{d}^{2}-{b}^{3}{d}^{2}}{32\,{a}^{2}{c}^{2}-16\,ac{b}^{2}+2\,{b}^{4}}} \right ) }+4\,{\frac{ac{e}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{b}^{2}{e}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-12\,{\frac{bcde}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{c}^{2}{d}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2/(c*x^2+b*x+a)^3,x)

[Out]

(c*(2*a*c*e^2+b^2*e^2-6*b*c*d*e+6*c^2*d^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+3/2*b*(2*a*c*e^2+b^2*e^2-6*b*c*d*e+6
*c^2*d^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-(2*a^2*c*e^2-5*a*b^2*e^2+10*a*b*c*d*e-10*a*c^2*d^2+2*b^3*d*e-2*b^2*c*
d^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x+1/2*(6*a^2*b*e^2-16*a^2*c*d*e-2*a*b^2*d*e+10*a*b*c*d^2-b^3*d^2)/(16*a^2*c^2-
8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1
/2))*a*c*e^2+2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*e^2-12/(16
*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*d*e+12/(16*a^2*c^2-8*a*b^2*c
+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^2*d^2

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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)^2/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.15503, size = 3270, normalized size = 16.52 \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)^2/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[1/2*(2*(6*(b^2*c^3 - 4*a*c^4)*d^2 - 6*(b^3*c^2 - 4*a*b*c^3)*d*e + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*e^2)*x^3
- (b^5 - 14*a*b^3*c + 40*a^2*b*c^2)*d^2 - 2*(a*b^4 + 4*a^2*b^2*c - 32*a^3*c^2)*d*e + 6*(a^2*b^3 - 4*a^3*b*c)*e
^2 + 3*(6*(b^3*c^2 - 4*a*b*c^3)*d^2 - 6*(b^4*c - 4*a*b^2*c^2)*d*e + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*e^2)*x^2 +
2*(6*a^2*c^2*d^2 - 6*a^2*b*c*d*e + (6*c^4*d^2 - 6*b*c^3*d*e + (b^2*c^2 + 2*a*c^3)*e^2)*x^4 + 2*(6*b*c^3*d^2 -
6*b^2*c^2*d*e + (b^3*c + 2*a*b*c^2)*e^2)*x^3 + (a^2*b^2 + 2*a^3*c)*e^2 + (6*(b^2*c^2 + 2*a*c^3)*d^2 - 6*(b^3*
c + 2*a*b*c^2)*d*e + (b^4 + 4*a*b^2*c + 4*a^2*c^2)*e^2)*x^2 + 2*(6*a*b*c^2*d^2 - 6*a*b^2*c*d*e + (a*b^3 + 2*a^
2*b*c)*e^2)*x)*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*(b^4*c + a*b^2*c^2 - 20*a^2*c^3)*d^2 - 2*(b^5 + a*b^3*c - 20*a^2*b*c^2)*d*e + (5*a*b^4 -
22*a^2*b^2*c + 8*a^3*c^2)*e^2)*x)/(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + (b^6*c^2 - 12*a*b^4*
c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^3 + (b^8 -
10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 -
64*a^4*b*c^3)*x), 1/2*(2*(6*(b^2*c^3 - 4*a*c^4)*d^2 - 6*(b^3*c^2 - 4*a*b*c^3)*d*e + (b^4*c - 2*a*b^2*c^2 - 8*a
^2*c^3)*e^2)*x^3 - (b^5 - 14*a*b^3*c + 40*a^2*b*c^2)*d^2 - 2*(a*b^4 + 4*a^2*b^2*c - 32*a^3*c^2)*d*e + 6*(a^2*b
^3 - 4*a^3*b*c)*e^2 + 3*(6*(b^3*c^2 - 4*a*b*c^3)*d^2 - 6*(b^4*c - 4*a*b^2*c^2)*d*e + (b^5 - 2*a*b^3*c - 8*a^2*
b*c^2)*e^2)*x^2 - 4*(6*a^2*c^2*d^2 - 6*a^2*b*c*d*e + (6*c^4*d^2 - 6*b*c^3*d*e + (b^2*c^2 + 2*a*c^3)*e^2)*x^4 +
2*(6*b*c^3*d^2 - 6*b^2*c^2*d*e + (b^3*c + 2*a*b*c^2)*e^2)*x^3 + (a^2*b^2 + 2*a^3*c)*e^2 + (6*(b^2*c^2 + 2*a*c
^3)*d^2 - 6*(b^3*c + 2*a*b*c^2)*d*e + (b^4 + 4*a*b^2*c + 4*a^2*c^2)*e^2)*x^2 + 2*(6*a*b*c^2*d^2 - 6*a*b^2*c*d*
e + (a*b^3 + 2*a^2*b*c)*e^2)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(
2*(b^4*c + a*b^2*c^2 - 20*a^2*c^3)*d^2 - 2*(b^5 + a*b^3*c - 20*a^2*b*c^2)*d*e + (5*a*b^4 - 22*a^2*b^2*c + 8*a^
3*c^2)*e^2)*x)/(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + (b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^
4 - 64*a^3*c^5)*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^3 + (b^8 - 10*a*b^6*c + 24*a^
2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*x)]

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Sympy [B]  time = 3.86347, size = 1052, normalized size = 5.31 \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)**2/(c*x**2+b*x+a)**3,x)

[Out]

-sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)*log(x + (-64*a**3*c**3*sqrt(-1/
(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + 48*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**
2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) - 12*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2
+ b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + 2*a*b*c*e**2 + b**6*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e*
*2 - 6*b*c*d*e + 6*c**2*d**2) + b**3*e**2 - 6*b**2*c*d*e + 6*b*c**2*d**2)/(4*a*c**2*e**2 + 2*b**2*c*e**2 - 12*
b*c**2*d*e + 12*c**3*d**2)) + sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)*lo
g(x + (64*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) - 48*a**2*b*
*2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + 12*a*b**4*c*sqrt(-1/(4
*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + 2*a*b*c*e**2 - b**6*sqrt(-1/(4*a*c - b**
2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + b**3*e**2 - 6*b**2*c*d*e + 6*b*c**2*d**2)/(4*a*c**
2*e**2 + 2*b**2*c*e**2 - 12*b*c**2*d*e + 12*c**3*d**2)) + (6*a**2*b*e**2 - 16*a**2*c*d*e - 2*a*b**2*d*e + 10*a
*b*c*d**2 - b**3*d**2 + x**3*(4*a*c**2*e**2 + 2*b**2*c*e**2 - 12*b*c**2*d*e + 12*c**3*d**2) + x**2*(6*a*b*c*e*
*2 + 3*b**3*e**2 - 18*b**2*c*d*e + 18*b*c**2*d**2) + x*(-4*a**2*c*e**2 + 10*a*b**2*e**2 - 20*a*b*c*d*e + 20*a*
c**2*d**2 - 4*b**3*d*e + 4*b**2*c*d**2))/(32*a**4*c**2 - 16*a**3*b**2*c + 2*a**2*b**4 + x**4*(32*a**2*c**4 - 1
6*a*b**2*c**3 + 2*b**4*c**2) + x**3*(64*a**2*b*c**3 - 32*a*b**3*c**2 + 4*b**5*c) + x**2*(64*a**3*c**3 - 12*a*b
**4*c + 2*b**6) + x*(64*a**3*b*c**2 - 32*a**2*b**3*c + 4*a*b**5))

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Giac [A]  time = 1.10699, size = 414, normalized size = 2.09 \begin{align*} \frac{2 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{3} d^{2} x^{3} - 12 \, b c^{2} d x^{3} e + 18 \, b c^{2} d^{2} x^{2} + 2 \, b^{2} c x^{3} e^{2} + 4 \, a c^{2} x^{3} e^{2} - 18 \, b^{2} c d x^{2} e + 4 \, b^{2} c d^{2} x + 20 \, a c^{2} d^{2} x + 3 \, b^{3} x^{2} e^{2} + 6 \, a b c x^{2} e^{2} - 4 \, b^{3} d x e - 20 \, a b c d x e - b^{3} d^{2} + 10 \, a b c d^{2} + 10 \, a b^{2} x e^{2} - 4 \, a^{2} c x e^{2} - 2 \, a b^{2} d e - 16 \, a^{2} c d e + 6 \, a^{2} b e^{2}}{2 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*
a^2*c^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(12*c^3*d^2*x^3 - 12*b*c^2*d*x^3*e + 18*b*c^2*d^2*x^2 + 2*b^2*c*x^3*e^2 + 4
*a*c^2*x^3*e^2 - 18*b^2*c*d*x^2*e + 4*b^2*c*d^2*x + 20*a*c^2*d^2*x + 3*b^3*x^2*e^2 + 6*a*b*c*x^2*e^2 - 4*b^3*d
*x*e - 20*a*b*c*d*x*e - b^3*d^2 + 10*a*b*c*d^2 + 10*a*b^2*x*e^2 - 4*a^2*c*x*e^2 - 2*a*b^2*d*e - 16*a^2*c*d*e +
6*a^2*b*e^2)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*(c*x^2 + b*x + a)^2)