3.142 \(\int \frac{\left (a+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx\)

Optimal. Leaf size=155 \[ \frac{\left (3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{9/2}}-\frac{x \left (13 c d^2-3 a e^2\right ) \left (a e^2+c d^2\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^3}{3 e^3} \]

[Out]

(-3*c^2*d*x)/e^4 + (c^2*x^3)/(3*e^3) + ((c*d^2 + a*e^2)^2*x)/(4*d*e^4*(d + e*x^2
)^2) - ((13*c*d^2 - 3*a*e^2)*(c*d^2 + a*e^2)*x)/(8*d^2*e^4*(d + e*x^2)) + ((35*c
^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(9
/2))

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Rubi [A]  time = 0.398501, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\left (3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{9/2}}-\frac{x \left (13 c d^2-3 a e^2\right ) \left (a e^2+c d^2\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{x \left (a e^2+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac{3 c^2 d x}{e^4}+\frac{c^2 x^3}{3 e^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^2/(d + e*x^2)^3,x]

[Out]

(-3*c^2*d*x)/e^4 + (c^2*x^3)/(3*e^3) + ((c*d^2 + a*e^2)^2*x)/(4*d*e^4*(d + e*x^2
)^2) - ((13*c*d^2 - 3*a*e^2)*(c*d^2 + a*e^2)*x)/(8*d^2*e^4*(d + e*x^2)) + ((35*c
^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^(9
/2))

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Rubi in Sympy [A]  time = 85.9228, size = 177, normalized size = 1.14 \[ - \frac{7 c^{2} d x}{3 e^{4}} + \frac{c^{2} x^{7}}{3 e \left (d + e x^{2}\right )^{2}} + \frac{x \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 7 c^{2} d^{4}\right )}{12 d e^{4} \left (d + e x^{2}\right )^{2}} + \frac{x \left (3 a^{2} e^{4} - 10 a c d^{2} e^{2} - 21 c^{2} d^{4}\right )}{8 d^{2} e^{4} \left (d + e x^{2}\right )} + \frac{\left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 d^{\frac{5}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**2/(e*x**2+d)**3,x)

[Out]

-7*c**2*d*x/(3*e**4) + c**2*x**7/(3*e*(d + e*x**2)**2) + x*(3*a**2*e**4 + 6*a*c*
d**2*e**2 + 7*c**2*d**4)/(12*d*e**4*(d + e*x**2)**2) + x*(3*a**2*e**4 - 10*a*c*d
**2*e**2 - 21*c**2*d**4)/(8*d**2*e**4*(d + e*x**2)) + (3*a**2*e**4 + 6*a*c*d**2*
e**2 + 35*c**2*d**4)*atan(sqrt(e)*x/sqrt(d))/(8*d**(5/2)*e**(9/2))

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Mathematica [A]  time = 0.187493, size = 154, normalized size = 0.99 \[ \frac{x \left (3 a^2 e^4 \left (5 d+3 e x^2\right )-6 a c d^2 e^2 \left (3 d+5 e x^2\right )-c^2 d^2 \left (105 d^3+175 d^2 e x^2+56 d e^2 x^4-8 e^3 x^6\right )\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{\left (3 a^2 e^4+6 a c d^2 e^2+35 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^4)^2/(d + e*x^2)^3,x]

[Out]

(x*(3*a^2*e^4*(5*d + 3*e*x^2) - 6*a*c*d^2*e^2*(3*d + 5*e*x^2) - c^2*d^2*(105*d^3
+ 175*d^2*e*x^2 + 56*d*e^2*x^4 - 8*e^3*x^6)))/(24*d^2*e^4*(d + e*x^2)^2) + ((35
*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*e^
(9/2))

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Maple [A]  time = 0.015, size = 211, normalized size = 1.4 \[{\frac{{c}^{2}{x}^{3}}{3\,{e}^{3}}}-3\,{\frac{{c}^{2}dx}{{e}^{4}}}+{\frac{3\,{a}^{2}e{x}^{3}}{8\, \left ( e{x}^{2}+d \right ) ^{2}{d}^{2}}}-{\frac{5\,{x}^{3}ac}{4\,e \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{13\,{d}^{2}{x}^{3}{c}^{2}}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{5\,{a}^{2}x}{8\, \left ( e{x}^{2}+d \right ) ^{2}d}}-{\frac{3\,adxc}{4\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{11\,{d}^{3}x{c}^{2}}{8\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{3\,{a}^{2}}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,ac}{4\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{c}^{2}{d}^{2}}{8\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^2/(e*x^2+d)^3,x)

[Out]

1/3*c^2*x^3/e^3-3*c^2*d*x/e^4+3/8*e/(e*x^2+d)^2/d^2*x^3*a^2-5/4/e/(e*x^2+d)^2*x^
3*a*c-13/8/e^3/(e*x^2+d)^2*d^2*x^3*c^2+5/8/(e*x^2+d)^2/d*x*a^2-3/4/e^2/(e*x^2+d)
^2*d*x*a*c-11/8/e^4/(e*x^2+d)^2*d^3*x*c^2+3/8/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(
1/2))*a^2+3/4/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a*c+35/8/e^4*d^2/(d*e)^(1/
2)*arctan(x*e/(d*e)^(1/2))*c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^2/(e*x^2 + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285642, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (35 \, c^{2} d^{6} + 6 \, a c d^{4} e^{2} + 3 \, a^{2} d^{2} e^{4} +{\left (35 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x^{4} + 2 \,{\left (35 \, c^{2} d^{5} e + 6 \, a c d^{3} e^{3} + 3 \, a^{2} d e^{5}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + 2 \,{\left (8 \, c^{2} d^{2} e^{3} x^{7} - 56 \, c^{2} d^{3} e^{2} x^{5} -{\left (175 \, c^{2} d^{4} e + 30 \, a c d^{2} e^{3} - 9 \, a^{2} e^{5}\right )} x^{3} - 3 \,{\left (35 \, c^{2} d^{5} + 6 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}\right )} x\right )} \sqrt{-d e}}{48 \,{\left (d^{2} e^{6} x^{4} + 2 \, d^{3} e^{5} x^{2} + d^{4} e^{4}\right )} \sqrt{-d e}}, \frac{3 \,{\left (35 \, c^{2} d^{6} + 6 \, a c d^{4} e^{2} + 3 \, a^{2} d^{2} e^{4} +{\left (35 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x^{4} + 2 \,{\left (35 \, c^{2} d^{5} e + 6 \, a c d^{3} e^{3} + 3 \, a^{2} d e^{5}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (8 \, c^{2} d^{2} e^{3} x^{7} - 56 \, c^{2} d^{3} e^{2} x^{5} -{\left (175 \, c^{2} d^{4} e + 30 \, a c d^{2} e^{3} - 9 \, a^{2} e^{5}\right )} x^{3} - 3 \,{\left (35 \, c^{2} d^{5} + 6 \, a c d^{3} e^{2} - 5 \, a^{2} d e^{4}\right )} x\right )} \sqrt{d e}}{24 \,{\left (d^{2} e^{6} x^{4} + 2 \, d^{3} e^{5} x^{2} + d^{4} e^{4}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^2/(e*x^2 + d)^3,x, algorithm="fricas")

[Out]

[1/48*(3*(35*c^2*d^6 + 6*a*c*d^4*e^2 + 3*a^2*d^2*e^4 + (35*c^2*d^4*e^2 + 6*a*c*d
^2*e^4 + 3*a^2*e^6)*x^4 + 2*(35*c^2*d^5*e + 6*a*c*d^3*e^3 + 3*a^2*d*e^5)*x^2)*lo
g((2*d*e*x + (e*x^2 - d)*sqrt(-d*e))/(e*x^2 + d)) + 2*(8*c^2*d^2*e^3*x^7 - 56*c^
2*d^3*e^2*x^5 - (175*c^2*d^4*e + 30*a*c*d^2*e^3 - 9*a^2*e^5)*x^3 - 3*(35*c^2*d^5
+ 6*a*c*d^3*e^2 - 5*a^2*d*e^4)*x)*sqrt(-d*e))/((d^2*e^6*x^4 + 2*d^3*e^5*x^2 + d
^4*e^4)*sqrt(-d*e)), 1/24*(3*(35*c^2*d^6 + 6*a*c*d^4*e^2 + 3*a^2*d^2*e^4 + (35*c
^2*d^4*e^2 + 6*a*c*d^2*e^4 + 3*a^2*e^6)*x^4 + 2*(35*c^2*d^5*e + 6*a*c*d^3*e^3 +
3*a^2*d*e^5)*x^2)*arctan(sqrt(d*e)*x/d) + (8*c^2*d^2*e^3*x^7 - 56*c^2*d^3*e^2*x^
5 - (175*c^2*d^4*e + 30*a*c*d^2*e^3 - 9*a^2*e^5)*x^3 - 3*(35*c^2*d^5 + 6*a*c*d^3
*e^2 - 5*a^2*d*e^4)*x)*sqrt(d*e))/((d^2*e^6*x^4 + 2*d^3*e^5*x^2 + d^4*e^4)*sqrt(
d*e))]

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Sympy [A]  time = 6.49274, size = 257, normalized size = 1.66 \[ - \frac{3 c^{2} d x}{e^{4}} + \frac{c^{2} x^{3}}{3 e^{3}} - \frac{\sqrt{- \frac{1}{d^{5} e^{9}}} \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log{\left (- d^{3} e^{4} \sqrt{- \frac{1}{d^{5} e^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{5} e^{9}}} \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 35 c^{2} d^{4}\right ) \log{\left (d^{3} e^{4} \sqrt{- \frac{1}{d^{5} e^{9}}} + x \right )}}{16} + \frac{x^{3} \left (3 a^{2} e^{5} - 10 a c d^{2} e^{3} - 13 c^{2} d^{4} e\right ) + x \left (5 a^{2} d e^{4} - 6 a c d^{3} e^{2} - 11 c^{2} d^{5}\right )}{8 d^{4} e^{4} + 16 d^{3} e^{5} x^{2} + 8 d^{2} e^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**2/(e*x**2+d)**3,x)

[Out]

-3*c**2*d*x/e**4 + c**2*x**3/(3*e**3) - sqrt(-1/(d**5*e**9))*(3*a**2*e**4 + 6*a*
c*d**2*e**2 + 35*c**2*d**4)*log(-d**3*e**4*sqrt(-1/(d**5*e**9)) + x)/16 + sqrt(-
1/(d**5*e**9))*(3*a**2*e**4 + 6*a*c*d**2*e**2 + 35*c**2*d**4)*log(d**3*e**4*sqrt
(-1/(d**5*e**9)) + x)/16 + (x**3*(3*a**2*e**5 - 10*a*c*d**2*e**3 - 13*c**2*d**4*
e) + x*(5*a**2*d*e**4 - 6*a*c*d**3*e**2 - 11*c**2*d**5))/(8*d**4*e**4 + 16*d**3*
e**5*x**2 + 8*d**2*e**6*x**4)

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GIAC/XCAS [A]  time = 0.273537, size = 196, normalized size = 1.26 \[ \frac{1}{3} \,{\left (c^{2} x^{3} e^{6} - 9 \, c^{2} d x e^{5}\right )} e^{\left (-9\right )} + \frac{{\left (35 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{8 \, d^{\frac{5}{2}}} - \frac{{\left (13 \, c^{2} d^{4} x^{3} e + 11 \, c^{2} d^{5} x + 10 \, a c d^{2} x^{3} e^{3} + 6 \, a c d^{3} x e^{2} - 3 \, a^{2} x^{3} e^{5} - 5 \, a^{2} d x e^{4}\right )} e^{\left (-4\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + a)^2/(e*x^2 + d)^3,x, algorithm="giac")

[Out]

1/3*(c^2*x^3*e^6 - 9*c^2*d*x*e^5)*e^(-9) + 1/8*(35*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a
^2*e^4)*arctan(x*e^(1/2)/sqrt(d))*e^(-9/2)/d^(5/2) - 1/8*(13*c^2*d^4*x^3*e + 11*
c^2*d^5*x + 10*a*c*d^2*x^3*e^3 + 6*a*c*d^3*x*e^2 - 3*a^2*x^3*e^5 - 5*a^2*d*x*e^4
)*e^(-4)/((x^2*e + d)^2*d^2)