3.2216 $$\int \frac{(d+e x)^2}{(a+b x+c x^2)^4} \, dx$$

Optimal. Leaf size=260 $-\frac{2 (b+2 c x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{-2 x \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+8 a c d e+3 b^2 d e}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{8 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}$

[Out]

-((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (3*b^2*d*e + 8*a*c*d*e -
5*b*(c*d^2 + a*e^2) - 2*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*x)/(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) -
(2*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (8*c*(5*c^2*d
^2 + b^2*e^2 - c*e*(5*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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Rubi [A]  time = 0.261681, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {738, 638, 614, 618, 206} $-\frac{2 (b+2 c x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{-2 x \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+8 a c d e+3 b^2 d e}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{8 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (3*b^2*d*e + 8*a*c*d*e -
5*b*(c*d^2 + a*e^2) - 2*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*x)/(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) -
(2*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (8*c*(5*c^2*d
^2 + b^2*e^2 - c*e*(5*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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 614

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

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 )^4} \, dx &=-\frac{(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac{\int \frac{2 \left (5 c d^2-e (3 b d-a e)\right )+4 e (2 c d-b e) x}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac{3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{\left (2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^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)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac{3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{\left (4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3}\\ &=-\frac{(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac{3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{\left (8 c \left (5 c^2 d^2+b^2 e^2-c e (5 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 )^3}\\ &=-\frac{(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac{3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{8 c \left (5 c^2 d^2+b^2 e^2-c e (5 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 )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.562601, size = 258, normalized size = 0.99 $\frac{1}{3} \left (-\frac{6 (b+2 c x) \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac{(b+2 c x) \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac{a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x}{c \left (4 a c-b^2\right ) (a+x (b+c x))^3}+\frac{24 c \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(((5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (6*(5*c^2*
d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^3*(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))^3) + (24*c*(5*c^2*d
^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2))/3

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Maple [B]  time = 0.161, size = 850, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*c^3*(a*c*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5+10*c^2*(a*c*e^2+b^
2*e^2-5*b*c*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*b*x^4+2/3*(16*a*c+11*b^2)*c*(a*c*e^2+b^2
*e^2-5*b*c*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3+b*(16*a*c+b^2)*(a*c*e^2+b^2*e^2-5*b*c
*d*e+5*c^2*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-(4*a^3*c^2*e^2-22*a^2*b^2*c*e^2+44*a^2*b*c^2*d*
e-44*a^2*c^3*d^2-a*b^4*e^2+18*a*b^3*c*d*e-18*a*b^2*c^2*d^2-b^5*d*e+b^4*c*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*
b^4*c-b^6)*x+1/3*(26*a^3*b*c*e^2-64*a^3*c^2*d*e+a^2*b^3*e^2-18*a^2*b^2*c*d*e+66*a^2*b*c^2*d^2+a*b^4*d*e-13*a*b
^3*c*d^2+b^5*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3+8*c^2/(64*a^3*c^3-48*a^2*b^2*c^2
+12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*e^2+8*c/(64*a^3*c^3-48*a^2*b^2*c^2+12
*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*e^2-40*c^2/(64*a^3*c^3-48*a^2*b^2*c^2+
12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d*e+40*c^3/(64*a^3*c^3-48*a^2*b^2*c^2+
12*a*b^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.44419, size = 5802, normalized size = 22.32 \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)^4,x, algorithm="fricas")

[Out]

[-1/3*(12*(5*(b^2*c^5 - 4*a*c^6)*d^2 - 5*(b^3*c^4 - 4*a*b*c^5)*d*e + (b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*e^2)*
x^5 + 30*(5*(b^3*c^4 - 4*a*b*c^5)*d^2 - 5*(b^4*c^3 - 4*a*b^2*c^4)*d*e + (b^5*c^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*
e^2)*x^4 + 2*(5*(11*b^4*c^3 - 28*a*b^2*c^4 - 64*a^2*c^5)*d^2 - 5*(11*b^5*c^2 - 28*a*b^3*c^3 - 64*a^2*b*c^4)*d*
e + (11*b^6*c - 17*a*b^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*e^2)*x^3 + (b^7 - 17*a*b^5*c + 118*a^2*b^3*c^2 - 2
64*a^3*b*c^3)*d^2 + (a*b^6 - 22*a^2*b^4*c + 8*a^3*b^2*c^2 + 256*a^4*c^3)*d*e + (a^2*b^5 + 22*a^3*b^3*c - 104*a
^4*b*c^2)*e^2 + 3*(5*(b^5*c^2 + 12*a*b^3*c^3 - 64*a^2*b*c^4)*d^2 - 5*(b^6*c + 12*a*b^4*c^2 - 64*a^2*b^2*c^3)*d
*e + (b^7 + 13*a*b^5*c - 52*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^2)*x^2 + 12*(5*a^3*c^3*d^2 - 5*a^3*b*c^2*d*e + (5*c^
6*d^2 - 5*b*c^5*d*e + (b^2*c^4 + a*c^5)*e^2)*x^6 + 3*(5*b*c^5*d^2 - 5*b^2*c^4*d*e + (b^3*c^3 + a*b*c^4)*e^2)*x
^5 + 3*(5*(b^2*c^4 + a*c^5)*d^2 - 5*(b^3*c^3 + a*b*c^4)*d*e + (b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)*e^2)*x^4 + (5*
(b^3*c^3 + 6*a*b*c^4)*d^2 - 5*(b^4*c^2 + 6*a*b^2*c^3)*d*e + (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*e^2)*x^3 + (a^
3*b^2*c + a^4*c^2)*e^2 + 3*(5*(a*b^2*c^3 + a^2*c^4)*d^2 - 5*(a*b^3*c^2 + a^2*b*c^3)*d*e + (a*b^4*c + 2*a^2*b^2
*c^2 + a^3*c^3)*e^2)*x^2 + 3*(5*a^2*b*c^3*d^2 - 5*a^2*b^2*c^2*d*e + (a^2*b^3*c + a^3*b*c^2)*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)) - 3*((b^6*c
- 22*a*b^4*c^2 + 28*a^2*b^2*c^3 + 176*a^3*c^4)*d^2 - (b^7 - 22*a*b^5*c + 28*a^2*b^3*c^2 + 176*a^3*b*c^3)*d*e
- (a*b^6 + 18*a^2*b^4*c - 92*a^3*b^2*c^2 + 16*a^4*c^3)*e^2)*x)/(a^3*b^8 - 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*
a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 256*a^3*b^2*c^6 + 256*a^4*c^7)*x^6 + 3*
(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 + 256*a^4*b*c^6)*x^5 + 3*(b^10*c - 15*a*b^8*c^2 + 8
0*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^11 - 10*a*b^9*c + 320*a^3*b^5*c^3 - 1280*a^4*b^3*c^4 +
1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6*c^2 - 160*a^4*b^4*c^3 + 256*a^6*c^5)*x^2 + 3*(a^2
*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 + 256*a^6*b*c^4)*x), -1/3*(12*(5*(b^2*c^5 - 4*a*c^6)*d^
2 - 5*(b^3*c^4 - 4*a*b*c^5)*d*e + (b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*e^2)*x^5 + 30*(5*(b^3*c^4 - 4*a*b*c^5)*d
^2 - 5*(b^4*c^3 - 4*a*b^2*c^4)*d*e + (b^5*c^2 - 3*a*b^3*c^3 - 4*a^2*b*c^4)*e^2)*x^4 + 2*(5*(11*b^4*c^3 - 28*a*
b^2*c^4 - 64*a^2*c^5)*d^2 - 5*(11*b^5*c^2 - 28*a*b^3*c^3 - 64*a^2*b*c^4)*d*e + (11*b^6*c - 17*a*b^4*c^2 - 92*a
^2*b^2*c^3 - 64*a^3*c^4)*e^2)*x^3 + (b^7 - 17*a*b^5*c + 118*a^2*b^3*c^2 - 264*a^3*b*c^3)*d^2 + (a*b^6 - 22*a^2
*b^4*c + 8*a^3*b^2*c^2 + 256*a^4*c^3)*d*e + (a^2*b^5 + 22*a^3*b^3*c - 104*a^4*b*c^2)*e^2 + 3*(5*(b^5*c^2 + 12*
a*b^3*c^3 - 64*a^2*b*c^4)*d^2 - 5*(b^6*c + 12*a*b^4*c^2 - 64*a^2*b^2*c^3)*d*e + (b^7 + 13*a*b^5*c - 52*a^2*b^3
*c^2 - 64*a^3*b*c^3)*e^2)*x^2 - 24*(5*a^3*c^3*d^2 - 5*a^3*b*c^2*d*e + (5*c^6*d^2 - 5*b*c^5*d*e + (b^2*c^4 + a*
c^5)*e^2)*x^6 + 3*(5*b*c^5*d^2 - 5*b^2*c^4*d*e + (b^3*c^3 + a*b*c^4)*e^2)*x^5 + 3*(5*(b^2*c^4 + a*c^5)*d^2 - 5
*(b^3*c^3 + a*b*c^4)*d*e + (b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)*e^2)*x^4 + (5*(b^3*c^3 + 6*a*b*c^4)*d^2 - 5*(b^4*
c^2 + 6*a*b^2*c^3)*d*e + (b^5*c + 7*a*b^3*c^2 + 6*a^2*b*c^3)*e^2)*x^3 + (a^3*b^2*c + a^4*c^2)*e^2 + 3*(5*(a*b^
2*c^3 + a^2*c^4)*d^2 - 5*(a*b^3*c^2 + a^2*b*c^3)*d*e + (a*b^4*c + 2*a^2*b^2*c^2 + a^3*c^3)*e^2)*x^2 + 3*(5*a^2
*b*c^3*d^2 - 5*a^2*b^2*c^2*d*e + (a^2*b^3*c + a^3*b*c^2)*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)) - 3*((b^6*c - 22*a*b^4*c^2 + 28*a^2*b^2*c^3 + 176*a^3*c^4)*d^2 - (b^7 - 22*a*b^5*c
+ 28*a^2*b^3*c^2 + 176*a^3*b*c^3)*d*e - (a*b^6 + 18*a^2*b^4*c - 92*a^3*b^2*c^2 + 16*a^4*c^3)*e^2)*x)/(a^3*b^8
- 16*a^4*b^6*c + 96*a^5*b^4*c^2 - 256*a^6*b^2*c^3 + 256*a^7*c^4 + (b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 -
256*a^3*b^2*c^6 + 256*a^4*c^7)*x^6 + 3*(b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 - 256*a^3*b^3*c^5 + 256*a^4*b*
c^6)*x^5 + 3*(b^10*c - 15*a*b^8*c^2 + 80*a^2*b^6*c^3 - 160*a^3*b^4*c^4 + 256*a^5*c^6)*x^4 + (b^11 - 10*a*b^9*c
+ 320*a^3*b^5*c^3 - 1280*a^4*b^3*c^4 + 1536*a^5*b*c^5)*x^3 + 3*(a*b^10 - 15*a^2*b^8*c + 80*a^3*b^6*c^2 - 160*
a^4*b^4*c^3 + 256*a^6*c^5)*x^2 + 3*(a^2*b^9 - 16*a^3*b^7*c + 96*a^4*b^5*c^2 - 256*a^5*b^3*c^3 + 256*a^6*b*c^4)
*x)]

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Sympy [B]  time = 7.96913, size = 1635, normalized size = 6.29 \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)**4,x)

[Out]

-4*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(x + (-1024*a**4*c**5*sqrt
(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 1024*a**3*b**2*c**4*sqrt(-1/(4*a*c -
b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 384*a**2*b**4*c**3*sqrt(-1/(4*a*c - b**2)**7)*(a
*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 64*a*b**6*c**2*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e*
*2 - 5*b*c*d*e + 5*c**2*d**2) + 4*a*b*c**2*e**2 - 4*b**8*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 -
5*b*c*d*e + 5*c**2*d**2) + 4*b**3*c*e**2 - 20*b**2*c**2*d*e + 20*b*c**3*d**2)/(8*a*c**3*e**2 + 8*b**2*c**2*e**
2 - 40*b*c**3*d*e + 40*c**4*d**2)) + 4*c*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2
*d**2)*log(x + (1024*a**4*c**5*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 1
024*a**3*b**2*c**4*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 384*a**2*b**4
*c**3*sqrt(-1/(4*a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) - 64*a*b**6*c**2*sqrt(-1/(4*
a*c - b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 4*a*b*c**2*e**2 + 4*b**8*c*sqrt(-1/(4*a*c -
b**2)**7)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2) + 4*b**3*c*e**2 - 20*b**2*c**2*d*e + 20*b*c**3*d**
2)/(8*a*c**3*e**2 + 8*b**2*c**2*e**2 - 40*b*c**3*d*e + 40*c**4*d**2)) + (26*a**3*b*c*e**2 - 64*a**3*c**2*d*e +
a**2*b**3*e**2 - 18*a**2*b**2*c*d*e + 66*a**2*b*c**2*d**2 + a*b**4*d*e - 13*a*b**3*c*d**2 + b**5*d**2 + x**5*
(12*a*c**4*e**2 + 12*b**2*c**3*e**2 - 60*b*c**4*d*e + 60*c**5*d**2) + x**4*(30*a*b*c**3*e**2 + 30*b**3*c**2*e*
*2 - 150*b**2*c**3*d*e + 150*b*c**4*d**2) + x**3*(32*a**2*c**3*e**2 + 54*a*b**2*c**2*e**2 - 160*a*b*c**3*d*e +
160*a*c**4*d**2 + 22*b**4*c*e**2 - 110*b**3*c**2*d*e + 110*b**2*c**3*d**2) + x**2*(48*a**2*b*c**2*e**2 + 51*a
*b**3*c*e**2 - 240*a*b**2*c**2*d*e + 240*a*b*c**3*d**2 + 3*b**5*e**2 - 15*b**4*c*d*e + 15*b**3*c**2*d**2) + x*
(-12*a**3*c**2*e**2 + 66*a**2*b**2*c*e**2 - 132*a**2*b*c**2*d*e + 132*a**2*c**3*d**2 + 3*a*b**4*e**2 - 54*a*b*
*3*c*d*e + 54*a*b**2*c**2*d**2 + 3*b**5*d*e - 3*b**4*c*d**2))/(192*a**6*c**3 - 144*a**5*b**2*c**2 + 36*a**4*b*
*4*c - 3*a**3*b**6 + x**6*(192*a**3*c**6 - 144*a**2*b**2*c**5 + 36*a*b**4*c**4 - 3*b**6*c**3) + x**5*(576*a**3
*b*c**5 - 432*a**2*b**3*c**4 + 108*a*b**5*c**3 - 9*b**7*c**2) + x**4*(576*a**4*c**5 + 144*a**3*b**2*c**4 - 324
*a**2*b**4*c**3 + 99*a*b**6*c**2 - 9*b**8*c) + x**3*(1152*a**4*b*c**4 - 672*a**3*b**3*c**3 + 72*a**2*b**5*c**2
+ 18*a*b**7*c - 3*b**9) + x**2*(576*a**5*c**4 + 144*a**4*b**2*c**3 - 324*a**3*b**4*c**2 + 99*a**2*b**6*c - 9*
a*b**8) + x*(576*a**5*b*c**3 - 432*a**4*b**3*c**2 + 108*a**3*b**5*c - 9*a**2*b**7))

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Giac [B]  time = 1.12732, size = 814, normalized size = 3.13 \begin{align*} -\frac{8 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{60 \, c^{5} d^{2} x^{5} - 60 \, b c^{4} d x^{5} e + 150 \, b c^{4} d^{2} x^{4} + 12 \, b^{2} c^{3} x^{5} e^{2} + 12 \, a c^{4} x^{5} e^{2} - 150 \, b^{2} c^{3} d x^{4} e + 110 \, b^{2} c^{3} d^{2} x^{3} + 160 \, a c^{4} d^{2} x^{3} + 30 \, b^{3} c^{2} x^{4} e^{2} + 30 \, a b c^{3} x^{4} e^{2} - 110 \, b^{3} c^{2} d x^{3} e - 160 \, a b c^{3} d x^{3} e + 15 \, b^{3} c^{2} d^{2} x^{2} + 240 \, a b c^{3} d^{2} x^{2} + 22 \, b^{4} c x^{3} e^{2} + 54 \, a b^{2} c^{2} x^{3} e^{2} + 32 \, a^{2} c^{3} x^{3} e^{2} - 15 \, b^{4} c d x^{2} e - 240 \, a b^{2} c^{2} d x^{2} e - 3 \, b^{4} c d^{2} x + 54 \, a b^{2} c^{2} d^{2} x + 132 \, a^{2} c^{3} d^{2} x + 3 \, b^{5} x^{2} e^{2} + 51 \, a b^{3} c x^{2} e^{2} + 48 \, a^{2} b c^{2} x^{2} e^{2} + 3 \, b^{5} d x e - 54 \, a b^{3} c d x e - 132 \, a^{2} b c^{2} d x e + b^{5} d^{2} - 13 \, a b^{3} c d^{2} + 66 \, a^{2} b c^{2} d^{2} + 3 \, a b^{4} x e^{2} + 66 \, a^{2} b^{2} c x e^{2} - 12 \, a^{3} c^{2} x e^{2} + a b^{4} d e - 18 \, a^{2} b^{2} c d e - 64 \, a^{3} c^{2} d e + a^{2} b^{3} e^{2} + 26 \, a^{3} b c e^{2}}{3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}{\left (c x^{2} + b x + a\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2 + a*c^2*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c
+ 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/3*(60*c^5*d^2*x^5 - 60*b*c^4*d*x^5*e + 150*b*c^4*d^2*x
^4 + 12*b^2*c^3*x^5*e^2 + 12*a*c^4*x^5*e^2 - 150*b^2*c^3*d*x^4*e + 110*b^2*c^3*d^2*x^3 + 160*a*c^4*d^2*x^3 + 3
0*b^3*c^2*x^4*e^2 + 30*a*b*c^3*x^4*e^2 - 110*b^3*c^2*d*x^3*e - 160*a*b*c^3*d*x^3*e + 15*b^3*c^2*d^2*x^2 + 240*
a*b*c^3*d^2*x^2 + 22*b^4*c*x^3*e^2 + 54*a*b^2*c^2*x^3*e^2 + 32*a^2*c^3*x^3*e^2 - 15*b^4*c*d*x^2*e - 240*a*b^2*
c^2*d*x^2*e - 3*b^4*c*d^2*x + 54*a*b^2*c^2*d^2*x + 132*a^2*c^3*d^2*x + 3*b^5*x^2*e^2 + 51*a*b^3*c*x^2*e^2 + 48
*a^2*b*c^2*x^2*e^2 + 3*b^5*d*x*e - 54*a*b^3*c*d*x*e - 132*a^2*b*c^2*d*x*e + b^5*d^2 - 13*a*b^3*c*d^2 + 66*a^2*
b*c^2*d^2 + 3*a*b^4*x*e^2 + 66*a^2*b^2*c*x*e^2 - 12*a^3*c^2*x*e^2 + a*b^4*d*e - 18*a^2*b^2*c*d*e - 64*a^3*c^2*
d*e + a^2*b^3*e^2 + 26*a^3*b*c*e^2)/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c*x^2 + b*x + a)^3)