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

Optimal. Leaf size=166 \[ \frac{x \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right ) \left (7 c d^2-e (a e+3 b d)\right )}{2 d^{3/2} e^{9/2}}-\frac{2 c x^3 (c d-b e)}{3 e^3}+\frac{c^2 x^5}{5 e^2} \]

[Out]

((3*c^2*d^2 + b^2*e^2 - 2*c*e*(2*b*d - a*e))*x)/e^4 - (2*c*(c*d - b*e)*x^3)/(3*e^3) + (c^2*x^5)/(5*e^2) + ((c*
d^2 - b*d*e + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) - ((c*d^2 - b*d*e + a*e^2)*(7*c*d^2 - e*(3*b*d + a*e))*ArcTan[
(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(9/2))

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Rubi [A]  time = 0.297636, antiderivative size = 166, 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 = {1157, 1810, 205} \[ \frac{x \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right ) \left (7 c d^2-e (a e+3 b d)\right )}{2 d^{3/2} e^{9/2}}-\frac{2 c x^3 (c d-b e)}{3 e^3}+\frac{c^2 x^5}{5 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*c^2*d^2 + b^2*e^2 - 2*c*e*(2*b*d - a*e))*x)/e^4 - (2*c*(c*d - b*e)*x^3)/(3*e^3) + (c^2*x^5)/(5*e^2) + ((c*
d^2 - b*d*e + a*e^2)^2*x)/(2*d*e^4*(d + e*x^2)) - ((c*d^2 - b*d*e + a*e^2)*(7*c*d^2 - e*(3*b*d + a*e))*ArcTan[
(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(9/2))

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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 (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx &=\frac{\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac{\int \frac{\frac{c^2 d^4-2 c d^2 e (b d-a e)+e^2 \left (b^2 d^2-2 a b d e-a^2 e^2\right )}{e^4}-\frac{2 d \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^2}{e^3}+\frac{2 c d (c d-2 b e) x^4}{e^2}-\frac{2 c^2 d x^6}{e}}{d+e x^2} \, dx}{2 d}\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac{\int \left (-\frac{2 d \left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right )}{e^4}+\frac{4 c d (c d-b e) x^2}{e^3}-\frac{2 c^2 d x^4}{e^2}+\frac{7 c^2 d^4-10 b c d^3 e+3 b^2 d^2 e^2+6 a c d^2 e^2-2 a b d e^3-a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx}{2 d}\\ &=\frac{\left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right ) x}{e^4}-\frac{2 c (c d-b e) x^3}{3 e^3}+\frac{c^2 x^5}{5 e^2}+\frac{\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac{\left (7 c^2 d^4-10 b c d^3 e+3 b^2 d^2 e^2+6 a c d^2 e^2-2 a b d e^3-a^2 e^4\right ) \int \frac{1}{d+e x^2} \, dx}{2 d e^4}\\ &=\frac{\left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right ) x}{e^4}-\frac{2 c (c d-b e) x^3}{3 e^3}+\frac{c^2 x^5}{5 e^2}+\frac{\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac{\left (7 c d^2-3 b d e-a e^2\right ) \left (c d^2-b d e+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.102643, size = 183, normalized size = 1.1 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-e^2 \left (a^2 e^2+2 a b d e-3 b^2 d^2\right )+2 c d^2 e (3 a e-5 b d)+7 c^2 d^4\right )}{2 d^{3/2} e^{9/2}}+\frac{x \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac{x \left (e (a e-b d)+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}+\frac{2 c x^3 (b e-c d)}{3 e^3}+\frac{c^2 x^5}{5 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*c^2*d^2 + b^2*e^2 + 2*c*e*(-2*b*d + a*e))*x)/e^4 + (2*c*(-(c*d) + b*e)*x^3)/(3*e^3) + (c^2*x^5)/(5*e^2) +
((c*d^2 + e*(-(b*d) + a*e))^2*x)/(2*d*e^4*(d + e*x^2)) - ((7*c^2*d^4 + 2*c*d^2*e*(-5*b*d + 3*a*e) - e^2*(-3*b^
2*d^2 + 2*a*b*d*e + a^2*e^2))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*e^(9/2))

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Maple [B]  time = 0.013, size = 320, normalized size = 1.9 \begin{align*}{\frac{{c}^{2}{x}^{5}}{5\,{e}^{2}}}+{\frac{2\,{x}^{3}bc}{3\,{e}^{2}}}-{\frac{2\,d{c}^{2}{x}^{3}}{3\,{e}^{3}}}+2\,{\frac{acx}{{e}^{2}}}+{\frac{{b}^{2}x}{{e}^{2}}}-4\,{\frac{bcdx}{{e}^{3}}}+3\,{\frac{{c}^{2}{d}^{2}x}{{e}^{4}}}+{\frac{{a}^{2}x}{2\,d \left ( e{x}^{2}+d \right ) }}-{\frac{xab}{e \left ( e{x}^{2}+d \right ) }}+{\frac{adxc}{{e}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{dx{b}^{2}}{2\,{e}^{2} \left ( e{x}^{2}+d \right ) }}-{\frac{{d}^{2}xbc}{{e}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{{d}^{3}x{c}^{2}}{2\,{e}^{4} \left ( e{x}^{2}+d \right ) }}+{\frac{{a}^{2}}{2\,d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{ab}{e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-3\,{\frac{acd}{{e}^{2}\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }-{\frac{3\,{b}^{2}d}{2\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+5\,{\frac{bc{d}^{2}}{{e}^{3}\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }-{\frac{7\,{c}^{2}{d}^{3}}{2\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5/e^2+2/3/e^2*x^3*b*c-2/3*c^2*d*x^3/e^3+2/e^2*a*c*x+1/e^2*b^2*x-4/e^3*b*c*d*x+3/e^4*c^2*d^2*x+1/2/d*
x/(e*x^2+d)*a^2-1/e*x/(e*x^2+d)*a*b+1/e^2*d*x/(e*x^2+d)*a*c+1/2/e^2*d*x/(e*x^2+d)*b^2-1/e^3*d^2*x/(e*x^2+d)*b*
c+1/2/e^4*d^3*x/(e*x^2+d)*c^2+1/2/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a^2+1/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(
1/2))*a*b-3/e^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a*c-3/2/e^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b^2+5/
e^3*d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b*c-7/2/e^4*d^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*c^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((c*x^4+b*x^2+a)^2/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.71495, size = 1272, normalized size = 7.66 \begin{align*} \left [\frac{12 \, c^{2} d^{2} e^{4} x^{7} - 4 \,{\left (7 \, c^{2} d^{3} e^{3} - 10 \, b c d^{2} e^{4}\right )} x^{5} + 20 \,{\left (7 \, c^{2} d^{4} e^{2} - 10 \, b c d^{3} e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{3} + 15 \,{\left (7 \, c^{2} d^{5} - 10 \, b c d^{4} e - 2 \, a b d^{2} e^{3} - a^{2} d e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (7 \, c^{2} d^{4} e - 10 \, b c d^{3} e^{2} - 2 \, a b d e^{4} - a^{2} e^{5} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x^{2}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right ) + 30 \,{\left (7 \, c^{2} d^{5} e - 10 \, b c d^{4} e^{2} - 2 \, a b d^{2} e^{4} + a^{2} d e^{5} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x}{60 \,{\left (d^{2} e^{6} x^{2} + d^{3} e^{5}\right )}}, \frac{6 \, c^{2} d^{2} e^{4} x^{7} - 2 \,{\left (7 \, c^{2} d^{3} e^{3} - 10 \, b c d^{2} e^{4}\right )} x^{5} + 10 \,{\left (7 \, c^{2} d^{4} e^{2} - 10 \, b c d^{3} e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{3} - 15 \,{\left (7 \, c^{2} d^{5} - 10 \, b c d^{4} e - 2 \, a b d^{2} e^{3} - a^{2} d e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (7 \, c^{2} d^{4} e - 10 \, b c d^{3} e^{2} - 2 \, a b d e^{4} - a^{2} e^{5} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) + 15 \,{\left (7 \, c^{2} d^{5} e - 10 \, b c d^{4} e^{2} - 2 \, a b d^{2} e^{4} + a^{2} d e^{5} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x}{30 \,{\left (d^{2} e^{6} x^{2} + d^{3} e^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/60*(12*c^2*d^2*e^4*x^7 - 4*(7*c^2*d^3*e^3 - 10*b*c*d^2*e^4)*x^5 + 20*(7*c^2*d^4*e^2 - 10*b*c*d^3*e^3 + 3*(b
^2 + 2*a*c)*d^2*e^4)*x^3 + 15*(7*c^2*d^5 - 10*b*c*d^4*e - 2*a*b*d^2*e^3 - a^2*d*e^4 + 3*(b^2 + 2*a*c)*d^3*e^2
+ (7*c^2*d^4*e - 10*b*c*d^3*e^2 - 2*a*b*d*e^4 - a^2*e^5 + 3*(b^2 + 2*a*c)*d^2*e^3)*x^2)*sqrt(-d*e)*log((e*x^2
- 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) + 30*(7*c^2*d^5*e - 10*b*c*d^4*e^2 - 2*a*b*d^2*e^4 + a^2*d*e^5 + 3*(b^2 + 2
*a*c)*d^3*e^3)*x)/(d^2*e^6*x^2 + d^3*e^5), 1/30*(6*c^2*d^2*e^4*x^7 - 2*(7*c^2*d^3*e^3 - 10*b*c*d^2*e^4)*x^5 +
10*(7*c^2*d^4*e^2 - 10*b*c*d^3*e^3 + 3*(b^2 + 2*a*c)*d^2*e^4)*x^3 - 15*(7*c^2*d^5 - 10*b*c*d^4*e - 2*a*b*d^2*e
^3 - a^2*d*e^4 + 3*(b^2 + 2*a*c)*d^3*e^2 + (7*c^2*d^4*e - 10*b*c*d^3*e^2 - 2*a*b*d*e^4 - a^2*e^5 + 3*(b^2 + 2*
a*c)*d^2*e^3)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + 15*(7*c^2*d^5*e - 10*b*c*d^4*e^2 - 2*a*b*d^2*e^4 + a^2*d*
e^5 + 3*(b^2 + 2*a*c)*d^3*e^3)*x)/(d^2*e^6*x^2 + d^3*e^5)]

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Sympy [B]  time = 3.27443, size = 479, normalized size = 2.89 \begin{align*} \frac{c^{2} x^{5}}{5 e^{2}} + \frac{x \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{2 d^{2} e^{4} + 2 d e^{5} x^{2}} - \frac{\sqrt{- \frac{1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right ) \log{\left (- \frac{d^{2} e^{4} \sqrt{- \frac{1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right )}{a^{2} e^{4} + 2 a b d e^{3} - 6 a c d^{2} e^{2} - 3 b^{2} d^{2} e^{2} + 10 b c d^{3} e - 7 c^{2} d^{4}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right ) \log{\left (\frac{d^{2} e^{4} \sqrt{- \frac{1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right )}{a^{2} e^{4} + 2 a b d e^{3} - 6 a c d^{2} e^{2} - 3 b^{2} d^{2} e^{2} + 10 b c d^{3} e - 7 c^{2} d^{4}} + x \right )}}{4} + \frac{x^{3} \left (2 b c e - 2 c^{2} d\right )}{3 e^{3}} + \frac{x \left (2 a c e^{2} + b^{2} e^{2} - 4 b c d e + 3 c^{2} d^{2}\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*x**5/(5*e**2) + x*(a**2*e**4 - 2*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 - 2*b*c*d**3*e + c**2*d**4
)/(2*d**2*e**4 + 2*d*e**5*x**2) - sqrt(-1/(d**3*e**9))*(a*e**2 - b*d*e + c*d**2)*(a*e**2 + 3*b*d*e - 7*c*d**2)
*log(-d**2*e**4*sqrt(-1/(d**3*e**9))*(a*e**2 - b*d*e + c*d**2)*(a*e**2 + 3*b*d*e - 7*c*d**2)/(a**2*e**4 + 2*a*
b*d*e**3 - 6*a*c*d**2*e**2 - 3*b**2*d**2*e**2 + 10*b*c*d**3*e - 7*c**2*d**4) + x)/4 + sqrt(-1/(d**3*e**9))*(a*
e**2 - b*d*e + c*d**2)*(a*e**2 + 3*b*d*e - 7*c*d**2)*log(d**2*e**4*sqrt(-1/(d**3*e**9))*(a*e**2 - b*d*e + c*d*
*2)*(a*e**2 + 3*b*d*e - 7*c*d**2)/(a**2*e**4 + 2*a*b*d*e**3 - 6*a*c*d**2*e**2 - 3*b**2*d**2*e**2 + 10*b*c*d**3
*e - 7*c**2*d**4) + x)/4 + x**3*(2*b*c*e - 2*c**2*d)/(3*e**3) + x*(2*a*c*e**2 + b**2*e**2 - 4*b*c*d*e + 3*c**2
*d**2)/e**4

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Giac [A]  time = 1.12264, size = 279, normalized size = 1.68 \begin{align*} \frac{1}{15} \,{\left (3 \, c^{2} x^{5} e^{8} - 10 \, c^{2} d x^{3} e^{7} + 10 \, b c x^{3} e^{8} + 45 \, c^{2} d^{2} x e^{6} - 60 \, b c d x e^{7} + 15 \, b^{2} x e^{8} + 30 \, a c x e^{8}\right )} e^{\left (-10\right )} - \frac{{\left (7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} - 2 \, a b d e^{3} - a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{2 \, d^{\frac{3}{2}}} + \frac{{\left (c^{2} d^{4} x - 2 \, b c d^{3} x e + b^{2} d^{2} x e^{2} + 2 \, a c d^{2} x e^{2} - 2 \, a b d x e^{3} + a^{2} x e^{4}\right )} e^{\left (-4\right )}}{2 \,{\left (x^{2} e + d\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*c^2*x^5*e^8 - 10*c^2*d*x^3*e^7 + 10*b*c*x^3*e^8 + 45*c^2*d^2*x*e^6 - 60*b*c*d*x*e^7 + 15*b^2*x*e^8 + 3
0*a*c*x*e^8)*e^(-10) - 1/2*(7*c^2*d^4 - 10*b*c*d^3*e + 3*b^2*d^2*e^2 + 6*a*c*d^2*e^2 - 2*a*b*d*e^3 - a^2*e^4)*
arctan(x*e^(1/2)/sqrt(d))*e^(-9/2)/d^(3/2) + 1/2*(c^2*d^4*x - 2*b*c*d^3*x*e + b^2*d^2*x*e^2 + 2*a*c*d^2*x*e^2
- 2*a*b*d*x*e^3 + a^2*x*e^4)*e^(-4)/((x^2*e + d)*d)