3.303 \(\int \frac{x^8}{(d+e x^2) (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=387 \[ -\frac{\left (-\frac{3 a^2 b c e+2 a^2 c^2 d-4 a b^2 c d-a b^3 e+b^4 d}{\sqrt{b^2-4 a c}}+a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{5/2} \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}-\frac{\left (\frac{3 a^2 b c e+2 a^2 c^2 d-4 a b^2 c d-a b^3 e+b^4 d}{\sqrt{b^2-4 a c}}+a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{5/2} \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac{x (b e+c d)}{c^2 e^2}+\frac{x^3}{3 c e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 4.03166, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1287, 205, 1166} \[ -\frac{\left (-\frac{3 a^2 b c e+2 a^2 c^2 d-4 a b^2 c d-a b^3 e+b^4 d}{\sqrt{b^2-4 a c}}+a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{5/2} \sqrt{b-\sqrt{b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}-\frac{\left (\frac{3 a^2 b c e+2 a^2 c^2 d-4 a b^2 c d-a b^3 e+b^4 d}{\sqrt{b^2-4 a c}}+a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{5/2} \sqrt{\sqrt{b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac{x (b e+c d)}{c^2 e^2}+\frac{x^3}{3 c e} \]

Antiderivative was successfully verified.

[In]

Int[x^8/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

Rule 1287

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.634654, size = 463, normalized size = 1.2 \[ \frac{\left (a^2 c \left (e \sqrt{b^2-4 a c}-2 c d\right )+b^3 \left (d \sqrt{b^2-4 a c}+a e\right )+a b^2 \left (4 c d-e \sqrt{b^2-4 a c}\right )-a b c \left (2 d \sqrt{b^2-4 a c}+3 a e\right )+b^4 (-d)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}} \left (e (b d-a e)-c d^2\right )}+\frac{\left (a^2 c \left (e \sqrt{b^2-4 a c}+2 c d\right )+b^3 \left (d \sqrt{b^2-4 a c}-a e\right )-a b^2 \left (e \sqrt{b^2-4 a c}+4 c d\right )+a b c \left (3 a e-2 d \sqrt{b^2-4 a c}\right )+b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{5/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b} \left (e (b d-a e)-c d^2\right )}+\frac{d^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac{x (b e+c d)}{c^2 e^2}+\frac{x^3}{3 c e} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/((d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Maple [B]  time = 0.039, size = 1449, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(e*x^2+d)/(c*x^4+b*x^2+a),x)

[Out]

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

________________________________________________________________________________________

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(x^8/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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(x^8/(e*x^2+d)/(c*x^4+b*x^2+a),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(x**8/(e*x**2+d)/(c*x**4+b*x**2+a),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(x^8/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError