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

Optimal. Leaf size=86 \[ -\frac{(2 c d-b e)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{5/2} \sqrt{e} \sqrt{c d-b e}}+\frac{x (3 c d-b e)}{c^2}+\frac{e x^3}{3 c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.183219, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{(2 c d-b e)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{c^{5/2} \sqrt{e} \sqrt{c d-b e}}+\frac{x (3 c d-b e)}{c^2}+\frac{e x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \left (b e - 3 c d\right ) \int \frac{1}{c^{2}}\, dx + \frac{e x^{3}}{3 c} + \frac{\left (b e - 2 c d\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e - c d}} \right )}}{c^{\frac{5}{2}} \sqrt{e} \sqrt{b e - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*e - 3*c*d)*Integral(c**(-2), x) + e*x**3/(3*c) + (b*e - 2*c*d)**2*atan(sqrt(
c)*sqrt(e)*x/sqrt(b*e - c*d))/(c**(5/2)*sqrt(e)*sqrt(b*e - c*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.073463, size = 84, normalized size = 0.98 \[ \frac{(b e-2 c d)^2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{c^{5/2} \sqrt{e} \sqrt{b e-c d}}-\frac{x (b e-3 c d)}{c^2}+\frac{e x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 142, normalized size = 1.7 \[{\frac{e{x}^{3}}{3\,c}}-{\frac{bex}{{c}^{2}}}+3\,{\frac{dx}{c}}+{\frac{{b}^{2}{e}^{2}}{{c}^{2}}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}}-4\,{\frac{bde}{c\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) }+4\,{\frac{{d}^{2}}{\sqrt{ \left ( be-cd \right ) ce}}\arctan \left ({\frac{cex}{\sqrt{ \left ( be-cd \right ) ce}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^3/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.287878, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \log \left (-\frac{2 \,{\left (c^{2} d e - b c e^{2}\right )} x - \sqrt{c^{2} d e - b c e^{2}}{\left (c e x^{2} + c d - b e\right )}}{c e x^{2} - c d + b e}\right ) + 2 \,{\left (c e x^{3} + 3 \,{\left (3 \, c d - b e\right )} x\right )} \sqrt{c^{2} d e - b c e^{2}}}{6 \, \sqrt{c^{2} d e - b c e^{2}} c^{2}}, \frac{3 \,{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \arctan \left (-\frac{\sqrt{-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) +{\left (c e x^{3} + 3 \,{\left (3 \, c d - b e\right )} x\right )} \sqrt{-c^{2} d e + b c e^{2}}}{3 \, \sqrt{-c^{2} d e + b c e^{2}} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*log(-(2*(c^2*d*e - b*c*e^2)*x - sqrt(c
^2*d*e - b*c*e^2)*(c*e*x^2 + c*d - b*e))/(c*e*x^2 - c*d + b*e)) + 2*(c*e*x^3 + 3
*(3*c*d - b*e)*x)*sqrt(c^2*d*e - b*c*e^2))/(sqrt(c^2*d*e - b*c*e^2)*c^2), 1/3*(3
*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*arctan(-sqrt(-c^2*d*e + b*c*e^2)*x/(c*d - b*e
)) + (c*e*x^3 + 3*(3*c*d - b*e)*x)*sqrt(-c^2*d*e + b*c*e^2))/(sqrt(-c^2*d*e + b*
c*e^2)*c^2)]

_______________________________________________________________________________________

Sympy [A]  time = 2.57207, size = 275, normalized size = 3.2 \[ - \frac{\sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} \log{\left (x + \frac{- b c^{2} e \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} + c^{3} d \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2}}{b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )}}{2} + \frac{\sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} \log{\left (x + \frac{b c^{2} e \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2} - c^{3} d \sqrt{- \frac{1}{c^{5} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{2}}{b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}} \right )}}{2} + \frac{e x^{3}}{3 c} - \frac{x \left (b e - 3 c d\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out