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

Optimal. Leaf size=108 \[ \frac{c x^3 \left (2 a e^2+c d^2\right )}{3 e^3}-\frac{c d x \left (2 a e^2+c d^2\right )}{e^4}+\frac{\left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{9/2}}-\frac{c^2 d x^5}{5 e^2}+\frac{c^2 x^7}{7 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0767978, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1154, 205} \[ \frac{c x^3 \left (2 a e^2+c d^2\right )}{3 e^3}-\frac{c d x \left (2 a e^2+c d^2\right )}{e^4}+\frac{\left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{9/2}}-\frac{c^2 d x^5}{5 e^2}+\frac{c^2 x^7}{7 e} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2/(d + e*x^2),x]

[Out]

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

Rule 1154

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a
+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + 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+c x^4\right )^2}{d+e x^2} \, dx &=\int \left (-\frac{c d \left (c d^2+2 a e^2\right )}{e^4}+\frac{c \left (c d^2+2 a e^2\right ) x^2}{e^3}-\frac{c^2 d x^4}{e^2}+\frac{c^2 x^6}{e}+\frac{c^2 d^4+2 a c d^2 e^2+a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac{c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac{c \left (c d^2+2 a e^2\right ) x^3}{3 e^3}-\frac{c^2 d x^5}{5 e^2}+\frac{c^2 x^7}{7 e}+\frac{\left (c d^2+a e^2\right )^2 \int \frac{1}{d+e x^2} \, dx}{e^4}\\ &=-\frac{c d \left (c d^2+2 a e^2\right ) x}{e^4}+\frac{c \left (c d^2+2 a e^2\right ) x^3}{3 e^3}-\frac{c^2 d x^5}{5 e^2}+\frac{c^2 x^7}{7 e}+\frac{\left (c d^2+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 = 0.0798412, size = 97, normalized size = 0.9 \[ \frac{c x \left (70 a e^2 \left (e x^2-3 d\right )+c \left (35 d^2 e x^2-105 d^3-21 d e^2 x^4+15 e^3 x^6\right )\right )}{105 e^4}+\frac{\left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} e^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(70*a*e^2*(-3*d + e*x^2) + c*(-105*d^3 + 35*d^2*e*x^2 - 21*d*e^2*x^4 + 15*e^3*x^6)))/(105*e^4) + ((c*d^2
+ a*e^2)^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(9/2))

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 136, normalized size = 1.3 \begin{align*}{\frac{{c}^{2}{x}^{7}}{7\,e}}-{\frac{{c}^{2}d{x}^{5}}{5\,{e}^{2}}}+{\frac{2\,c{x}^{3}a}{3\,e}}+{\frac{{c}^{2}{d}^{2}{x}^{3}}{3\,{e}^{3}}}-2\,{\frac{acdx}{{e}^{2}}}-{\frac{{c}^{2}{d}^{3}x}{{e}^{4}}}+{{a}^{2}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+2\,{\frac{ac{d}^{2}}{{e}^{2}\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+a)^2/(e*x^2+d),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.8622, size = 585, normalized size = 5.42 \begin{align*} \left [\frac{30 \, c^{2} d e^{4} x^{7} - 42 \, c^{2} d^{2} e^{3} x^{5} + 70 \,{\left (c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{3} - 105 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\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 \, a c d^{2} e^{3}\right )} x}{210 \, d e^{5}}, \frac{15 \, c^{2} d e^{4} x^{7} - 21 \, c^{2} d^{2} e^{3} x^{5} + 35 \,{\left (c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{3} + 105 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) - 105 \,{\left (c^{2} d^{4} e + 2 \, a c 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+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*x^5 + 70*(c^2*d^3*e^2 + 2*a*c*d*e^4)*x^3 - 105*(c^2*d^4 + 2*a*c*d^2*
e^2 + a^2*e^4)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 210*(c^2*d^4*e + 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*x^5 + 35*(c^2*d^3*e^2 + 2*a*c*d*e^4)*x^3 + 105*(c^2*d^4 + 2*a
*c*d^2*e^2 + a^2*e^4)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) - 105*(c^2*d^4*e + 2*a*c*d^2*e^3)*x)/(d*e^5)]

________________________________________________________________________________________

Sympy [B]  time = 0.593639, size = 235, normalized size = 2.18 \begin{align*} - \frac{c^{2} d x^{5}}{5 e^{2}} + \frac{c^{2} x^{7}}{7 e} - \frac{\sqrt{- \frac{1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2} \log{\left (- \frac{d e^{4} \sqrt{- \frac{1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2}}{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2} \log{\left (\frac{d e^{4} \sqrt{- \frac{1}{d e^{9}}} \left (a e^{2} + c d^{2}\right )^{2}}{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}} + x \right )}}{2} + \frac{x^{3} \left (2 a c e^{2} + c^{2} d^{2}\right )}{3 e^{3}} - \frac{x \left (2 a c d e^{2} + c^{2} d^{3}\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.10459, size = 142, normalized size = 1.31 \begin{align*} \frac{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 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} + 35 \, c^{2} d^{2} x^{3} e^{4} - 105 \, c^{2} d^{3} x e^{3} + 70 \, a c x^{3} e^{6} - 210 \, a c d x e^{5}\right )} e^{\left (-7\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*d^4 + 2*a*c*d^2*e^2 + 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 + 35*c^2*d^2*x^3*e^4 - 105*c^2*d^3*x*e^3 + 70*a*c*x^3*e^6 - 210*a*c*d*x*e^5)*e^(-7)