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

Optimal. Leaf size=66 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}-\frac{x (c d-b e)}{e^2}+\frac{c x^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.0446231, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1153, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}-\frac{x (c d-b e)}{e^2}+\frac{c x^3}{3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, 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[p, 0] && IGtQ[q, -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{a+b x^2+c x^4}{d+e x^2} \, dx &=\int \left (-\frac{c d-b e}{e^2}+\frac{c x^2}{e}+\frac{c d^2-b d e+a e^2}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^3}{3 e}+\frac{\left (c d^2-b d e+a e^2\right ) \int \frac{1}{d+e x^2} \, dx}{e^2}\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^3}{3 e}+\frac{\left (c d^2-b d e+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0512229, size = 65, normalized size = 0.98 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{d} e^{5/2}}+\frac{x (b e-c d)}{e^2}+\frac{c x^3}{3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 84, normalized size = 1.3 \begin{align*}{\frac{c{x}^{3}}{3\,e}}+{\frac{bx}{e}}-{\frac{cdx}{{e}^{2}}}+{a\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{bd}{e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c{d}^{2}}{{e}^{2}}\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)/(e*x^2+d),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.63718, size = 347, normalized size = 5.26 \begin{align*} \left [\frac{2 \, c d e^{2} x^{3} - 3 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right ) - 6 \,{\left (c d^{2} e - b d e^{2}\right )} x}{6 \, d e^{3}}, \frac{c d e^{2} x^{3} + 3 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) - 3 \,{\left (c d^{2} e - b d e^{2}\right )} x}{3 \, d e^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(2*c*d*e^2*x^3 - 3*(c*d^2 - b*d*e + a*e^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 6*(
c*d^2*e - b*d*e^2)*x)/(d*e^3), 1/3*(c*d*e^2*x^3 + 3*(c*d^2 - b*d*e + a*e^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) -
3*(c*d^2*e - b*d*e^2)*x)/(d*e^3)]

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Sympy [B]  time = 0.640437, size = 117, normalized size = 1.77 \begin{align*} \frac{c x^{3}}{3 e} - \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (- d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{d e^{5}}} \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d e^{2} \sqrt{- \frac{1}{d e^{5}}} + x \right )}}{2} + \frac{x \left (b e - c d\right )}{e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x**3/(3*e) - sqrt(-1/(d*e**5))*(a*e**2 - b*d*e + c*d**2)*log(-d*e**2*sqrt(-1/(d*e**5)) + x)/2 + sqrt(-1/(d*e
**5))*(a*e**2 - b*d*e + c*d**2)*log(d*e**2*sqrt(-1/(d*e**5)) + x)/2 + x*(b*e - c*d)/e**2

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Giac [A]  time = 1.26151, size = 76, normalized size = 1.15 \begin{align*} \frac{{\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{5}{2}\right )}}{\sqrt{d}} + \frac{1}{3} \,{\left (c x^{3} e^{2} - 3 \, c d x e + 3 \, b x e^{2}\right )} e^{\left (-3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*d^2 - b*d*e + a*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-5/2)/sqrt(d) + 1/3*(c*x^3*e^2 - 3*c*d*x*e + 3*b*x*e^2)*e
^(-3)