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

Optimal. Leaf size=370 \[ -\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )-\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )-\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )+\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )+\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^3}{3 c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.501238, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {1171, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )-\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )-\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )+\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )+\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^3}{3 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3/(a + c*x^4),x]

[Out]

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

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

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

Mathematica [A]  time = 0.280898, size = 360, normalized size = 0.97 \[ \frac{-3 \sqrt{2} \left (a^{3/2} e^3-3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+3 \sqrt{2} \left (a^{3/2} e^3-3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+6 \sqrt{2} \left (a^{3/2} e^3-3 \sqrt{a} c d^2 e+3 a \sqrt{c} d e^2-c^{3/2} d^3\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt{2} \left (-a^{3/2} e^3+3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+72 a^{3/4} c^{3/4} d e^2 x+8 a^{3/4} c^{3/4} e^3 x^3}{24 a^{3/4} c^{7/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(72*a^(3/4)*c^(3/4)*d*e^2*x + 8*a^(3/4)*c^(3/4)*e^3*x^3 + 6*Sqrt[2]*(-(c^(3/2)*d^3) - 3*Sqrt[a]*c*d^2*e + 3*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 + 3*Sqrt[a]*c*d^
2*e - 3*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 - 3*Sq
rt[a]*c*d^2*e - 3*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*Sq
rt[2]*(c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - 3*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])/(24*a^(3/4)*c^(7/4))

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Maple [A]  time = 0.049, size = 572, normalized size = 1.6 \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),x)

[Out]

1/3*e^3*x^3/c+3*d*e^2*x/c-3/4/c*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d*e^2+1/4*(a/c)^(1/4)/a*2^
(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^3-3/8/c*(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*e^2+1/8*(a/c)^(1/4)/a*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^3-3/4/c*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*
d*e^2+1/4*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^3-1/8/c^2/(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)))*a*e^3+3/8/c/(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*e-1/4/c^2/(a/c)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(a/c)^(1/4)*x-1)*a*e^3+3/4/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^2*e-1/4/c^2/
(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*a*e^3+3/4/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)
*x+1)*d^2*e

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 8.33023, size = 4462, normalized size = 12.06 \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),x, algorithm="fricas")

[Out]

1/12*(4*e^3*x^3 + 36*d*e^2*x - 3*c*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 + a*c^3*sqrt(-(c^6*d^12 -
30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6
*e^12)/(a^3*c^7)))/(a*c^3))*log(-(c^6*d^12 - 12*a*c^5*d^10*e^2 - 27*a^2*c^4*d^8*e^4 + 27*a^4*c^2*d^4*e^8 + 12*
a^5*c*d^2*e^10 - a^6*e^12)*x + (a*c^6*d^9 - 18*a^2*c^5*d^7*e^2 + 60*a^3*c^4*d^5*e^4 - 46*a^4*c^3*d^3*e^6 + 3*a
^5*c^2*d*e^8 + (3*a^3*c^6*d^2*e - a^4*c^5*e^3)*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452
*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))*sqrt(-(6*c^2*d^5*e - 20*a*c
*d^3*e^3 + 6*a^2*d*e^5 + a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6
+ 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3))) + 3*c*sqrt(-(6*c^2*d^5*e - 20*a*c
*d^3*e^3 + 6*a^2*d*e^5 + a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6
+ 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3))*log(-(c^6*d^12 - 12*a*c^5*d^10*e^2
- 27*a^2*c^4*d^8*e^4 + 27*a^4*c^2*d^4*e^8 + 12*a^5*c*d^2*e^10 - a^6*e^12)*x - (a*c^6*d^9 - 18*a^2*c^5*d^7*e^2
+ 60*a^3*c^4*d^5*e^4 - 46*a^4*c^3*d^3*e^6 + 3*a^5*c^2*d*e^8 + (3*a^3*c^6*d^2*e - a^4*c^5*e^3)*sqrt(-(c^6*d^12
- 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a
^6*e^12)/(a^3*c^7)))*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 + a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10
*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^
7)))/(a*c^3))) - 3*c*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 - a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10
*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^
7)))/(a*c^3))*log(-(c^6*d^12 - 12*a*c^5*d^10*e^2 - 27*a^2*c^4*d^8*e^4 + 27*a^4*c^2*d^4*e^8 + 12*a^5*c*d^2*e^10
- a^6*e^12)*x + (a*c^6*d^9 - 18*a^2*c^5*d^7*e^2 + 60*a^3*c^4*d^5*e^4 - 46*a^4*c^3*d^3*e^6 + 3*a^5*c^2*d*e^8 -
(3*a^3*c^6*d^2*e - a^4*c^5*e^3)*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e
^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a
^2*d*e^5 - a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2
*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3))) + 3*c*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a
^2*d*e^5 - a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2
*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3))*log(-(c^6*d^12 - 12*a*c^5*d^10*e^2 - 27*a^2*c^4*
d^8*e^4 + 27*a^4*c^2*d^4*e^8 + 12*a^5*c*d^2*e^10 - a^6*e^12)*x - (a*c^6*d^9 - 18*a^2*c^5*d^7*e^2 + 60*a^3*c^4*
d^5*e^4 - 46*a^4*c^3*d^3*e^6 + 3*a^5*c^2*d*e^8 - (3*a^3*c^6*d^2*e - a^4*c^5*e^3)*sqrt(-(c^6*d^12 - 30*a*c^5*d^
10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*
c^7)))*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 - a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2
*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3)))
)/c

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Sympy [A]  time = 2.36805, size = 350, normalized size = 0.95 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{3} c^{7} + t^{2} \left (192 a^{4} c^{4} d e^{5} - 640 a^{3} c^{5} d^{3} e^{3} + 192 a^{2} c^{6} d^{5} e\right ) + a^{6} e^{12} + 6 a^{5} c d^{2} e^{10} + 15 a^{4} c^{2} d^{4} e^{8} + 20 a^{3} c^{3} d^{6} e^{6} + 15 a^{2} c^{4} d^{8} e^{4} + 6 a c^{5} d^{10} e^{2} + c^{6} d^{12}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{4} c^{5} e^{3} + 192 t^{3} a^{3} c^{6} d^{2} e - 36 t a^{5} c^{2} d e^{8} + 336 t a^{4} c^{3} d^{3} e^{6} - 504 t a^{3} c^{4} d^{5} e^{4} + 144 t a^{2} c^{5} d^{7} e^{2} - 4 t a c^{6} d^{9}}{a^{6} e^{12} - 12 a^{5} c d^{2} e^{10} - 27 a^{4} c^{2} d^{4} e^{8} + 27 a^{2} c^{4} d^{8} e^{4} + 12 a c^{5} d^{10} e^{2} - c^{6} d^{12}} \right )} \right )\right )} + \frac{3 d e^{2} x}{c} + \frac{e^{3} x^{3}}{3 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4*a**3*c**7 + _t**2*(192*a**4*c**4*d*e**5 - 640*a**3*c**5*d**3*e**3 + 192*a**2*c**6*d**5*e) +
a**6*e**12 + 6*a**5*c*d**2*e**10 + 15*a**4*c**2*d**4*e**8 + 20*a**3*c**3*d**6*e**6 + 15*a**2*c**4*d**8*e**4 +
6*a*c**5*d**10*e**2 + c**6*d**12, Lambda(_t, _t*log(x + (-64*_t**3*a**4*c**5*e**3 + 192*_t**3*a**3*c**6*d**2*e
- 36*_t*a**5*c**2*d*e**8 + 336*_t*a**4*c**3*d**3*e**6 - 504*_t*a**3*c**4*d**5*e**4 + 144*_t*a**2*c**5*d**7*e*
*2 - 4*_t*a*c**6*d**9)/(a**6*e**12 - 12*a**5*c*d**2*e**10 - 27*a**4*c**2*d**4*e**8 + 27*a**2*c**4*d**8*e**4 +
12*a*c**5*d**10*e**2 - c**6*d**12)))) + 3*d*e**2*x/c + e**3*x**3/(3*c)

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Giac [A]  time = 1.17081, size = 547, normalized size = 1.48 \begin{align*} \frac{c^{2} x^{3} e^{3} + 9 \, c^{2} d x e^{2}}{3 \, c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} + 3 \, \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 )}{4 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} + 3 \, \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 )}{4 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} - 3 \, \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 )}{8 \, a c^{4}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} - 3 \, \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 )}{8 \, a 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),x, algorithm="giac")

[Out]

1/3*(c^2*x^3*e^3 + 9*c^2*d*x*e^2)/c^3 + 1/4*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 - 3*(a*c^3)^(1/4)*a*c^2*d*e^2 + 3*(
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*c^4
) + 1/4*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 - 3*(a*c^3)^(1/4)*a*c^2*d*e^2 + 3*(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*c^4) + 1/8*sqrt(2)*((a*c^3)^(1/4)*c^3*d
^3 - 3*(a*c^3)^(1/4)*a*c^2*d*e^2 - 3*(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*c^4) - 1/8*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 - 3*(a*c^3)^(1/4)*a*c^2*d*e^2 - 3*(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*c^4)