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

Optimal. Leaf size=363 \[ -\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{7/4}}+\frac{x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (c d^2-3 a e^2\right )\right )}{4 a c \left (a+c x^4\right )}-\frac{e^3 x^3}{c \left (a+c x^4\right )} \]

[Out]

-((e^3*x^3)/(c*(a + c*x^4))) + (x*(d*(c*d^2 - 3*a*e^2) + 3*e*(c*d^2 + a*e^2)*x^2))/(4*a*c*(a + c*x^4)) - (3*(S
qrt[c]*d + Sqrt[a]*e)*(c*d^2 + a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c^(7/4)) + (
3*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 + a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c^(7/4))
- (3*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt
[2]*a^(7/4)*c^(7/4)) + (3*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*c^(7/4))

________________________________________________________________________________________

Rubi [A]  time = 0.410194, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {1207, 1858, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{7/4}}+\frac{x \left (3 e x^2 \left (a e^2+c d^2\right )+d \left (c d^2-3 a e^2\right )\right )}{4 a c \left (a+c x^4\right )}-\frac{e^3 x^3}{c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3/(a + c*x^4)^2,x]

[Out]

-((e^3*x^3)/(c*(a + c*x^4))) + (x*(d*(c*d^2 - 3*a*e^2) + 3*e*(c*d^2 + a*e^2)*x^2))/(4*a*c*(a + c*x^4)) - (3*(S
qrt[c]*d + Sqrt[a]*e)*(c*d^2 + a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c^(7/4)) + (
3*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 + a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*c^(7/4))
- (3*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt
[2]*a^(7/4)*c^(7/4)) + (3*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*c^(7/4))

Rule 1207

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

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1168

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

Rule 1162

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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{\left (d+e x^2\right )^3}{\left (a+c x^4\right )^2} \, dx &=-\frac{e^3 x^3}{c \left (a+c x^4\right )}-\frac{\int \frac{-c d^3-3 e \left (c d^2+a e^2\right ) x^2-3 c d e^2 x^4}{\left (a+c x^4\right )^2} \, dx}{c}\\ &=-\frac{e^3 x^3}{c \left (a+c x^4\right )}+\frac{x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{\int \frac{3 c d \left (c d^2+a e^2\right )+3 c e \left (c d^2+a e^2\right ) x^2}{a+c x^4} \, dx}{4 a c^2}\\ &=-\frac{e^3 x^3}{c \left (a+c x^4\right )}+\frac{x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}+\frac{\left (3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^2}+\frac{\left (3 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^2}\\ &=-\frac{e^3 x^3}{c \left (a+c x^4\right )}+\frac{x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac{\left (3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt{2} a^{7/4} c^{7/4}}-\frac{\left (3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt{2} a^{7/4} c^{7/4}}+\frac{\left (3 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^2}+\frac{\left (3 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (c d^2+a e^2\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^2}\\ &=-\frac{e^3 x^3}{c \left (a+c x^4\right )}+\frac{x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{7/4}}+\frac{\left (3 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (c d^2+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{7/4}}-\frac{\left (3 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (c d^2+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{7/4}}\\ &=-\frac{e^3 x^3}{c \left (a+c x^4\right )}+\frac{x \left (d \left (c d^2-3 a e^2\right )+3 e \left (c d^2+a e^2\right ) x^2\right )}{4 a c \left (a+c x^4\right )}-\frac{3 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{c} d+\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{c} d-\sqrt{a} e\right ) \left (c d^2+a e^2\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{7/4}}\\ \end{align*}

Mathematica [A]  time = 0.269247, size = 371, normalized size = 1.02 \[ \frac{-\frac{8 a^{3/4} c^{3/4} \left (a e^2 x \left (3 d+e x^2\right )-c d^2 x \left (d+3 e x^2\right )\right )}{a+c x^4}+3 \sqrt{2} \left (a^{3/2} e^3+\sqrt{a} c d^2 e-a \sqrt{c} d e^2-c^{3/2} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+3 \sqrt{2} \left (-a^{3/2} e^3-\sqrt{a} c d^2 e+a \sqrt{c} d e^2+c^{3/2} d^3\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-6 \sqrt{2} \left (a^{3/2} e^3+\sqrt{a} c d^2 e+a \sqrt{c} d e^2+c^{3/2} d^3\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt{2} \left (a^{3/2} e^3+\sqrt{a} c d^2 e+a \sqrt{c} d e^2+c^{3/2} d^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{32 a^{7/4} c^{7/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*a^(3/4)*c^(3/4)*(a*e^2*x*(3*d + e*x^2) - c*d^2*x*(d + 3*e*x^2)))/(a + c*x^4) - 6*Sqrt[2]*(c^(3/2)*d^3 + S
qrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 6*Sqrt[2]*(c^(3/2)*d
^3 + Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 3*Sqrt[2]*(-(c
^(3/2)*d^3) + Sqrt[a]*c*d^2*e - a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[
c]*x^2] + 3*Sqrt[2]*(c^(3/2)*d^3 - Sqrt[a]*c*d^2*e + a*Sqrt[c]*d*e^2 - a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*c^(1/4)*x + Sqrt[c]*x^2])/(32*a^(7/4)*c^(7/4))

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Maple [B]  time = 0.058, size = 624, normalized size = 1.7 \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+a)^2,x)

[Out]

(-1/4*e*(a*e^2-3*c*d^2)/a/c*x^3-1/4*d*(3*a*e^2-c*d^2)/a/c*x)/(c*x^4+a)+3/32/a/c*d*(a/c)^(1/4)*2^(1/2)*ln((x^2+
(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*e^2+3/32/a^2*d^3*(a/c)^(1/4)*2^(1/
2)*ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+3/16/a/c*d*(a/c)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*e^2+3/16/a^2*d^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+3
/16/a/c*d*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*e^2+3/16/a^2*d^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(a/c)^(1/4)*x-1)+3/32/c^2*e^3/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1
/4)*x*2^(1/2)+(a/c)^(1/2)))+3/32/a/c*e/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/
c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))*d^2+3/16/c^2*e^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+3/16/a/c
*e/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^2+3/16/c^2*e^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/
c)^(1/4)*x-1)+3/16/a/c*e/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*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^2+d)^3/(c*x^4+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.41217, size = 4196, normalized size = 11.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*(3*c*d^2*e - a*e^3)*x^3 - 3*(a*c^2*x^4 + a^2*c)*sqrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 + a^3
*c^3*sqrt(-(c^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*
e^10 + a^6*e^12)/(a^7*c^7)))/(a^3*c^3))*log(-27*(c^5*d^10 + 3*a*c^4*d^8*e^2 + 2*a^2*c^3*d^6*e^4 - 2*a^3*c^2*d^
4*e^6 - 3*a^4*c*d^2*e^8 - a^5*e^10)*x + 27*(a^2*c^5*d^7 + a^3*c^4*d^5*e^2 - a^4*c^3*d^3*e^4 - a^5*c^2*d*e^6 +
a^6*c^5*e*sqrt(-(c^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c
*d^2*e^10 + a^6*e^12)/(a^7*c^7)))*sqrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 + a^3*c^3*sqrt(-(c^6*d^12 +
2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*
c^7)))/(a^3*c^3))) + 3*(a*c^2*x^4 + a^2*c)*sqrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 + a^3*c^3*sqrt(-(c
^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^
12)/(a^7*c^7)))/(a^3*c^3))*log(-27*(c^5*d^10 + 3*a*c^4*d^8*e^2 + 2*a^2*c^3*d^6*e^4 - 2*a^3*c^2*d^4*e^6 - 3*a^4
*c*d^2*e^8 - a^5*e^10)*x - 27*(a^2*c^5*d^7 + a^3*c^4*d^5*e^2 - a^4*c^3*d^3*e^4 - a^5*c^2*d*e^6 + a^6*c^5*e*sqr
t(-(c^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a
^6*e^12)/(a^7*c^7)))*sqrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 + a^3*c^3*sqrt(-(c^6*d^12 + 2*a*c^5*d^10
*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)))/(a^3*c
^3))) - 3*(a*c^2*x^4 + a^2*c)*sqrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 - a^3*c^3*sqrt(-(c^6*d^12 + 2*a
*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)
))/(a^3*c^3))*log(-27*(c^5*d^10 + 3*a*c^4*d^8*e^2 + 2*a^2*c^3*d^6*e^4 - 2*a^3*c^2*d^4*e^6 - 3*a^4*c*d^2*e^8 -
a^5*e^10)*x + 27*(a^2*c^5*d^7 + a^3*c^4*d^5*e^2 - a^4*c^3*d^3*e^4 - a^5*c^2*d*e^6 - a^6*c^5*e*sqrt(-(c^6*d^12
+ 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7
*c^7)))*sqrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 - a^3*c^3*sqrt(-(c^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^
4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)))/(a^3*c^3))) + 3*(a*
c^2*x^4 + a^2*c)*sqrt(-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 - a^3*c^3*sqrt(-(c^6*d^12 + 2*a*c^5*d^10*e^2
- a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)))/(a^3*c^3))
*log(-27*(c^5*d^10 + 3*a*c^4*d^8*e^2 + 2*a^2*c^3*d^6*e^4 - 2*a^3*c^2*d^4*e^6 - 3*a^4*c*d^2*e^8 - a^5*e^10)*x -
27*(a^2*c^5*d^7 + a^3*c^4*d^5*e^2 - a^4*c^3*d^3*e^4 - a^5*c^2*d*e^6 - a^6*c^5*e*sqrt(-(c^6*d^12 + 2*a*c^5*d^1
0*e^2 - a^2*c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)))*sqrt(
-(2*c^2*d^5*e + 4*a*c*d^3*e^3 + 2*a^2*d*e^5 - a^3*c^3*sqrt(-(c^6*d^12 + 2*a*c^5*d^10*e^2 - a^2*c^4*d^8*e^4 - 4
*a^3*c^3*d^6*e^6 - a^4*c^2*d^4*e^8 + 2*a^5*c*d^2*e^10 + a^6*e^12)/(a^7*c^7)))/(a^3*c^3))) + 4*(c*d^3 - 3*a*d*e
^2)*x)/(a*c^2*x^4 + a^2*c)

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Sympy [A]  time = 3.67759, size = 352, normalized size = 0.97 \begin{align*} \operatorname{RootSum}{\left (65536 t^{4} a^{7} c^{7} + t^{2} \left (9216 a^{6} c^{4} d e^{5} + 18432 a^{5} c^{5} d^{3} e^{3} + 9216 a^{4} c^{6} d^{5} e\right ) + 81 a^{6} e^{12} + 486 a^{5} c d^{2} e^{10} + 1215 a^{4} c^{2} d^{4} e^{8} + 1620 a^{3} c^{3} d^{6} e^{6} + 1215 a^{2} c^{4} d^{8} e^{4} + 486 a c^{5} d^{10} e^{2} + 81 c^{6} d^{12}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{6} c^{5} e + 432 t a^{5} c^{2} d e^{6} + 720 t a^{4} c^{3} d^{3} e^{4} + 144 t a^{3} c^{4} d^{5} e^{2} - 144 t a^{2} c^{5} d^{7}}{27 a^{5} e^{10} + 81 a^{4} c d^{2} e^{8} + 54 a^{3} c^{2} d^{4} e^{6} - 54 a^{2} c^{3} d^{6} e^{4} - 81 a c^{4} d^{8} e^{2} - 27 c^{5} d^{10}} \right )} \right )\right )} - \frac{x^{3} \left (a e^{3} - 3 c d^{2} e\right ) + x \left (3 a d e^{2} - c d^{3}\right )}{4 a^{2} c + 4 a c^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(65536*_t**4*a**7*c**7 + _t**2*(9216*a**6*c**4*d*e**5 + 18432*a**5*c**5*d**3*e**3 + 9216*a**4*c**6*d**5
*e) + 81*a**6*e**12 + 486*a**5*c*d**2*e**10 + 1215*a**4*c**2*d**4*e**8 + 1620*a**3*c**3*d**6*e**6 + 1215*a**2*
c**4*d**8*e**4 + 486*a*c**5*d**10*e**2 + 81*c**6*d**12, Lambda(_t, _t*log(x + (4096*_t**3*a**6*c**5*e + 432*_t
*a**5*c**2*d*e**6 + 720*_t*a**4*c**3*d**3*e**4 + 144*_t*a**3*c**4*d**5*e**2 - 144*_t*a**2*c**5*d**7)/(27*a**5*
e**10 + 81*a**4*c*d**2*e**8 + 54*a**3*c**2*d**4*e**6 - 54*a**2*c**3*d**6*e**4 - 81*a*c**4*d**8*e**2 - 27*c**5*
d**10)))) - (x**3*(a*e**3 - 3*c*d**2*e) + x*(3*a*d*e**2 - c*d**3))/(4*a**2*c + 4*a*c**2*x**4)

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Giac [A]  time = 1.18259, size = 574, normalized size = 1.58 \begin{align*} \frac{3 \, c d^{2} x^{3} e + c d^{3} x - a x^{3} e^{3} - 3 \, a d x e^{2}}{4 \,{\left (c x^{4} + a\right )} a c} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac{3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac{3}{4}} a e^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} + \left (a c^{3}\right )^{\frac{3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac{3}{4}} a e^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} c^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac{3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac{3}{4}} a e^{3}\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{4}} - \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} - \left (a c^{3}\right )^{\frac{3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac{3}{4}} a e^{3}\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a^{2} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*c*d^2*x^3*e + c*d^3*x - a*x^3*e^3 - 3*a*d*x*e^2)/((c*x^4 + a)*a*c) + 3/16*sqrt(2)*((a*c^3)^(1/4)*c^3*d^
3 + (a*c^3)^(1/4)*a*c^2*d*e^2 + (a*c^3)^(3/4)*c*d^2*e + (a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)
*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^4) + 3/16*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4)*a*c^2*d*e^2 + (a*c^
3)^(3/4)*c*d^2*e + (a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^4)
+ 3/32*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4)*a*c^2*d*e^2 - (a*c^3)^(3/4)*c*d^2*e - (a*c^3)^(3/4)*a*e^
3)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4) - 3/32*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 + (a*c^3)^(1/4
)*a*c^2*d*e^2 - (a*c^3)^(3/4)*c*d^2*e - (a*c^3)^(3/4)*a*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2
*c^4)