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

Optimal. Leaf size=143 \[ \frac{x^3 \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 e^3}-\frac{x (c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )^2}{\sqrt{d} e^{9/2}}-\frac{c x^5 (c d-2 b e)}{5 e^2}+\frac{c^2 x^7}{7 e} \]

[Out]

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

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Rubi [A]  time = 0.140219, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1153, 205} \[ \frac{x^3 \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 e^3}-\frac{x (c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )^2}{\sqrt{d} e^{9/2}}-\frac{c x^5 (c d-2 b e)}{5 e^2}+\frac{c^2 x^7}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0649803, size = 144, normalized size = 1.01 \[ \frac{x^3 \left (2 a c e^2+b^2 e^2-2 b c d e+c^2 d^2\right )}{3 e^3}+\frac{x (b e-c d) \left (2 a e^2-b d e+c d^2\right )}{e^4}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a e^2-b d e+c d^2\right )^2}{\sqrt{d} e^{9/2}}+\frac{c x^5 (2 b e-c d)}{5 e^2}+\frac{c^2 x^7}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.005, size = 267, normalized size = 1.9 \begin{align*}{\frac{{c}^{2}{x}^{7}}{7\,e}}+{\frac{2\,{x}^{5}bc}{5\,e}}-{\frac{d{c}^{2}{x}^{5}}{5\,{e}^{2}}}+{\frac{2\,{x}^{3}ac}{3\,e}}+{\frac{{x}^{3}{b}^{2}}{3\,e}}-{\frac{2\,{x}^{3}bcd}{3\,{e}^{2}}}+{\frac{{x}^{3}{c}^{2}{d}^{2}}{3\,{e}^{3}}}+2\,{\frac{abx}{e}}-2\,{\frac{acdx}{{e}^{2}}}-{\frac{{b}^{2}dx}{{e}^{2}}}+2\,{\frac{bc{d}^{2}x}{{e}^{3}}}-{\frac{{c}^{2}{d}^{3}x}{{e}^{4}}}+{{a}^{2}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-2\,{\frac{dab}{e\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }+2\,{\frac{ac{d}^{2}}{{e}^{2}\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }+{\frac{{b}^{2}{d}^{2}}{{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-2\,{\frac{bc{d}^{3}}{{e}^{3}\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }+{\frac{{c}^{2}{d}^{4}}{{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),x)

[Out]

1/7*c^2*x^7/e+2/5/e*x^5*b*c-1/5*c^2*d*x^5/e^2+2/3/e*x^3*a*c+1/3/e*x^3*b^2-2/3/e^2*x^3*b*c*d+1/3/e^3*x^3*c^2*d^
2+2/e*a*b*x-2/e^2*a*c*d*x-1/e^2*b^2*d*x+2/e^3*d^2*b*c*x-1/e^4*c^2*d^3*x+1/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*
a^2-2/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a*b*d+2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a*c*d^2+1/e^2/(d*e
)^(1/2)*arctan(x*e/(d*e)^(1/2))*b^2*d^2-2/e^3/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b*c*d^3+1/e^4/(d*e)^(1/2)*ar
ctan(x*e/(d*e)^(1/2))*c^2*d^4

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61883, size = 892, normalized size = 6.24 \begin{align*} \left [\frac{30 \, c^{2} d e^{4} x^{7} - 42 \,{\left (c^{2} d^{2} e^{3} - 2 \, b c d e^{4}\right )} x^{5} + 70 \,{\left (c^{2} d^{3} e^{2} - 2 \, b c d^{2} e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{3} - 105 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right ) - 210 \,{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x}{210 \, d e^{5}}, \frac{15 \, c^{2} d e^{4} x^{7} - 21 \,{\left (c^{2} d^{2} e^{3} - 2 \, b c d e^{4}\right )} x^{5} + 35 \,{\left (c^{2} d^{3} e^{2} - 2 \, b c d^{2} e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{3} + 105 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) - 105 \,{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x}{105 \, d e^{5}}\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),x, algorithm="fricas")

[Out]

[1/210*(30*c^2*d*e^4*x^7 - 42*(c^2*d^2*e^3 - 2*b*c*d*e^4)*x^5 + 70*(c^2*d^3*e^2 - 2*b*c*d^2*e^3 + (b^2 + 2*a*c
)*d*e^4)*x^3 - 105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*sqrt(-d*e)*log((e*x
^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 210*(c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*
x)/(d*e^5), 1/105*(15*c^2*d*e^4*x^7 - 21*(c^2*d^2*e^3 - 2*b*c*d*e^4)*x^5 + 35*(c^2*d^3*e^2 - 2*b*c*d^2*e^3 + (
b^2 + 2*a*c)*d*e^4)*x^3 + 105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*sqrt(d*e
)*arctan(sqrt(d*e)*x/d) - 105*(c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*x)/(d*e^5)]

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Sympy [B]  time = 1.26761, size = 366, normalized size = 2.56 \begin{align*} \frac{c^{2} x^{7}}{7 e} - \frac{\sqrt{- \frac{1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} \log{\left (- \frac{d e^{4} \sqrt{- \frac{1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{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}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} \log{\left (\frac{d e^{4} \sqrt{- \frac{1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{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}} + x \right )}}{2} + \frac{x^{5} \left (2 b c e - c^{2} d\right )}{5 e^{2}} + \frac{x^{3} \left (2 a c e^{2} + b^{2} e^{2} - 2 b c d e + c^{2} d^{2}\right )}{3 e^{3}} + \frac{x \left (2 a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 2 b c d^{2} e - c^{2} d^{3}\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),x)

[Out]

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

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Giac [A]  time = 1.1435, size = 250, normalized size = 1.75 \begin{align*} \frac{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, 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 )}}{\sqrt{d}} + \frac{1}{105} \,{\left (15 \, c^{2} x^{7} e^{6} - 21 \, c^{2} d x^{5} e^{5} + 42 \, b c x^{5} e^{6} + 35 \, c^{2} d^{2} x^{3} e^{4} - 70 \, b c d x^{3} e^{5} - 105 \, c^{2} d^{3} x e^{3} + 35 \, b^{2} x^{3} e^{6} + 70 \, a c x^{3} e^{6} + 210 \, b c d^{2} x e^{4} - 105 \, b^{2} d x e^{5} - 210 \, a c d x e^{5} + 210 \, a b x e^{6}\right )} e^{\left (-7\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),x, algorithm="giac")

[Out]

(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*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)/sqrt(d) + 1/105*(15*c^2*x^7*e^6 - 21*c^2*d*x^5*e^5 + 42*b*c*x^5*e^6 + 35*c^2*d^2*x^3*e^4 - 70*b*c*d*x^3*e^5
- 105*c^2*d^3*x*e^3 + 35*b^2*x^3*e^6 + 70*a*c*x^3*e^6 + 210*b*c*d^2*x*e^4 - 105*b^2*d*x*e^5 - 210*a*c*d*x*e^5
+ 210*a*b*x*e^6)*e^(-7)