3.270 \(\int \frac{(d+e x^2)^3}{(a+b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=563 \[ -\frac{\left (-b^2 \left (a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (e \sqrt{b^2-4 a c}+2 c d\right )-b c \left (c d^2 \left (d \sqrt{b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}+8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (-b^2 \left (-a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt{b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}-8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (c \left (-\frac{a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

(x*(c*(b^2*d^3 - 2*a*d*(c*d^2 - 3*a*e^2) - (a*b*e*(3*c*d^2 + a*e^2))/c) - (a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^
2) - b*c*d*(c*d^2 + 3*a*e^2))*x^2))/(2*a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((a*b^3*e^3 + 6*a*c*(2*c*d + S
qrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) - b^2*(c^2*d^3 - 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) - b*c*(a*e^2*(3*Sq
rt[b^2 - 4*a*c]*d + 8*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b
^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^3*e^3 + 6*a*c*(2*
c*d - Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) - b^2*(c^2*d^3 - 3*a*c*d*e^2 - a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*d^
2*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 8*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b +
Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

________________________________________________________________________________________

Rubi [A]  time = 3.51858, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1205, 1166, 205} \[ -\frac{\left (-b^2 \left (a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (e \sqrt{b^2-4 a c}+2 c d\right )-b c \left (c d^2 \left (d \sqrt{b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}+8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (-b^2 \left (-a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt{b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}-8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (c \left (-\frac{a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x]

[Out]

(x*(c*(b^2*d^3 - 2*a*d*(c*d^2 - 3*a*e^2) - (a*b*e*(3*c*d^2 + a*e^2))/c) - (a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^
2) - b*c*d*(c*d^2 + 3*a*e^2))*x^2))/(2*a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((a*b^3*e^3 + 6*a*c*(2*c*d + S
qrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) - b^2*(c^2*d^3 - 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) - b*c*(a*e^2*(3*Sq
rt[b^2 - 4*a*c]*d + 8*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b
^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((a*b^3*e^3 + 6*a*c*(2*
c*d - Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) - b^2*(c^2*d^3 - 3*a*c*d*e^2 - a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*d^
2*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 8*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b +
Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 1205

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(
2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], 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] && IGtQ[q, 1] && LtQ[p, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac{x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac{a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-b^2 d^3+6 a d \left (c d^2+a e^2\right )-\frac{a b e \left (3 c d^2+a e^2\right )}{c}+\left (-\frac{a b^2 e^3}{c}+6 a e \left (c d^2+a e^2\right )-b \left (c d^3+3 a d e^2\right )\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac{a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (a b^3 e^3+6 a c \left (2 c d-\sqrt{b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt{b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt{b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt{b^2-4 a c} d-8 a e\right )\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a c \left (b^2-4 a c\right )^{3/2}}-\frac{\left (a b^3 e^3+6 a c \left (2 c d+\sqrt{b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt{b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt{b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt{b^2-4 a c} d+12 a e\right )\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a c \left (b^2-4 a c\right )^{3/2}}\\ &=\frac{x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac{a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (a b^3 e^3+6 a c \left (2 c d+\sqrt{b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt{b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt{b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt{b^2-4 a c} d+12 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (a b^3 e^3+6 a c \left (2 c d-\sqrt{b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt{b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt{b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt{b^2-4 a c} d-8 a e\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 1.71249, size = 540, normalized size = 0.96 \[ \frac{\frac{2 \sqrt{c} x \left (b \left (-a^2 e^3-3 a c d e \left (d-e x^2\right )+c^2 d^3 x^2\right )+b^2 \left (c d^3-a e^3 x^2\right )+2 a c \left (a e^2 \left (3 d+e x^2\right )-c d^2 \left (d+3 e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \left (b^2 \left (a e^3 \sqrt{b^2-4 a c}-3 a c d e^2+c^2 d^3\right )-6 a c \left (a e^2+c d^2\right ) \left (e \sqrt{b^2-4 a c}+2 c d\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}+12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}+8 a e\right )\right )-a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (b^2 \left (a e^3 \sqrt{b^2-4 a c}+3 a c d e^2-c^2 d^3\right )+6 a c \left (a e^2+c d^2\right ) \left (2 c d-e \sqrt{b^2-4 a c}\right )+b c \left (c d^2 \left (d \sqrt{b^2-4 a c}-12 a e\right )+a e^2 \left (3 d \sqrt{b^2-4 a c}-8 a e\right )\right )+a b^3 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 a c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^3/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*Sqrt[c]*x*(b^2*(c*d^3 - a*e^3*x^2) + b*(-(a^2*e^3) + c^2*d^3*x^2 - 3*a*c*d*e*(d - e*x^2)) + 2*a*c*(a*e^2*(
3*d + e*x^2) - c*d^2*(d + 3*e*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(a*b^3*e^3) - 6*a*c*(2*
c*d + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(c^2*d^3 - 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(a*e^
2*(3*Sqrt[b^2 - 4*a*c]*d + 8*a*e) + c*d^2*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b -
Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(a*b^3*e^3 + 6*a*c*(2*c*d -
Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(-(c^2*d^3) + 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*d^2*
(Sqrt[b^2 - 4*a*c]*d - 12*a*e) + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 8*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + S
qrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*a*c^(3/2))

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Maple [B]  time = 0.047, size = 1846, 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^2+d)^3/(c*x^4+b*x^2+a)^2,x)

[Out]

-3*a/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b
^2)^(1/2)-b)*c)^(1/2))*d*e^2+3/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arcta
nh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b*d^2*e+1/4/a/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*
a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^2*d^3-3*a/(4*a*c-b^2)*c/(-4
*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*
d*e^2+3/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a
*c+b^2)^(1/2))*c)^(1/2))*b*d^2*e+1/4/a/(4*a*c-b^2)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/
2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*d^3+(-1/2*(2*a^2*c*e^3-a*b^2*e^3+3*a*b*c*d*e^2-6*a
*c^2*d^2*e+b*c^2*d^3)/a/c/(4*a*c-b^2)*x^3+1/2/c*(a^2*b*e^3-6*a^2*c*d*e^2+3*a*b*c*d^2*e+2*a*c^2*d^3-b^2*c*d^3)/
(4*a*c-b^2)/a*x)/(c*x^4+b*x^2+a)+1/4/a/(4*a*c-b^2)*c*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1
/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b*d^3+2*a/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)
*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b*e^3-1/4/(4*a*c-b^2)/c/(-4*a*c+b^2)^(1/2)*2^(
1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*b^3*e^3-3/4/(4*a*c
-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*
c)^(1/2))*b^2*d*e^2-1/4/a/(4*a*c-b^2)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*
c+b^2)^(1/2))*c)^(1/2))*b*d^3+2*a/(4*a*c-b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
an(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*e^3-1/4/(4*a*c-b^2)/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*
c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*e^3-3/4/(4*a*c-b^2)/(-4*a*c+b^
2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*d*e
^2-3/2*a/(4*a*c-b^2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(
1/2))*e^3+3/2*a/(4*a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2))*e^3+1/4/(4*a*c-b^2)/c*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^
(1/2)-b)*c)^(1/2))*b^2*e^3+3/4/(4*a*c-b^2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*
a*c+b^2)^(1/2)-b)*c)^(1/2))*b*d*e^2-3/2/(4*a*c-b^2)*c*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(
1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*d^2*e-3/(4*a*c-b^2)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/(((-4*a*c+b^2)^(1/2)
-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))*d^3-1/4/(4*a*c-b^2)/c*2^(1/2)/((b+(-4*a*c+b
^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^2*e^3-3/4/(4*a*c-b^2)*2^(1/2)/((b+(
-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b*d*e^2+3/2/(4*a*c-b^2)*c*2^(
1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*d^2*e-3/(4*a*c-b^2)
*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)
^(1/2))*d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} -{\left (a b^{2} - 2 \, a^{2} c\right )} e^{3}\right )} x^{3} -{\left (3 \, a b c d^{2} e - 6 \, a^{2} c d e^{2} + a^{2} b e^{3} -{\left (b^{2} c - 2 \, a c^{2}\right )} d^{3}\right )} x}{2 \,{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2} +{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x^{4} +{\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} x^{2}\right )}} - \frac{-\int \frac{3 \, a b c d^{2} e - 6 \, a^{2} c d e^{2} + a^{2} b e^{3} +{\left (b^{2} c - 6 \, a c^{2}\right )} d^{3} +{\left (b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} +{\left (a b^{2} - 6 \, a^{2} c\right )} e^{3}\right )} x^{2}}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*((b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3)*x^3 - (3*a*b*c*d^2*e - 6*a^2*c*d*e^2
+ a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3)*x)/(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^
2*b*c^2)*x^2) - 1/2*integrate(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 + (b^2*c - 6*a*c^2)*d^3 + (b*c^2*d^3
- 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 + (a*b^2 - 6*a^2*c)*e^3)*x^2)/(c*x^4 + b*x^2 + a), x)/(a*b^2*c - 4*a^2*c^2)

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Fricas [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((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

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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((e*x**2+d)**3/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3/(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError