3.222 \(\int \frac{1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx\)
Optimal. Leaf size=136 \[ -\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^2}-\frac{(4 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \]
[Out]
-x/(2*d*(2*c*d - b*e)*(d + e*x^2)) - ((4*c*d - b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])
/(2*d^(3/2)*Sqrt[e]*(2*c*d - b*e)^2) - (c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt
[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^2)
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Rubi [A] time = 0.39953, antiderivative size = 136, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128
\[ -\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^2}-\frac{(4 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]
[Out]
-x/(2*d*(2*c*d - b*e)*(d + e*x^2)) - ((4*c*d - b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])
/(2*d^(3/2)*Sqrt[e]*(2*c*d - b*e)^2) - (c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt
[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^2)
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Rubi in Sympy [A] time = 87.4403, size = 117, normalized size = 0.86 \[ \frac{c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e - c d}} \right )}}{\sqrt{e} \left (b e - 2 c d\right )^{2} \sqrt{b e - c d}} + \frac{x}{2 d \left (d + e x^{2}\right ) \left (b e - 2 c d\right )} + \frac{\left (b e - 4 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 d^{\frac{3}{2}} \sqrt{e} \left (b e - 2 c d\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x**2+d)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
c**(3/2)*atan(sqrt(c)*sqrt(e)*x/sqrt(b*e - c*d))/(sqrt(e)*(b*e - 2*c*d)**2*sqrt(
b*e - c*d)) + x/(2*d*(d + e*x**2)*(b*e - 2*c*d)) + (b*e - 4*c*d)*atan(sqrt(e)*x/
sqrt(d))/(2*d**(3/2)*sqrt(e)*(b*e - 2*c*d)**2)
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Mathematica [A] time = 0.388379, size = 133, normalized size = 0.98 \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{\sqrt{e} (b e-2 c d)^2 \sqrt{b e-c d}}+\frac{(b e-4 c d) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]
[Out]
-x/(2*d*(2*c*d - b*e)*(d + e*x^2)) + ((-4*c*d + b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
)/(2*d^(3/2)*Sqrt[e]*(2*c*d - b*e)^2) + (c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt
[-(c*d) + b*e]])/(Sqrt[e]*(-2*c*d + b*e)^2*Sqrt[-(c*d) + b*e])
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Maple [A] time = 0.024, size = 155, normalized size = 1.1 \[{\frac{{c}^{2}}{ \left ( be-2\,cd \right ) ^{2}}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}}+{\frac{bxe}{2\, \left ( be-2\,cd \right ) ^{2}d \left ( e{x}^{2}+d \right ) }}-{\frac{cx}{ \left ( be-2\,cd \right ) ^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{be}{2\, \left ( be-2\,cd \right ) ^{2}d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-2\,{\frac{c}{ \left ( be-2\,cd \right ) ^{2}\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)
[Out]
c^2/(b*e-2*c*d)^2/((b*e-c*d)*c*e)^(1/2)*arctan(x*c*e/((b*e-c*d)*c*e)^(1/2))+1/2/
(b*e-2*c*d)^2/d*x/(e*x^2+d)*b*e-1/(b*e-2*c*d)^2*x/(e*x^2+d)*c+1/2/(b*e-2*c*d)^2/
d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b*e-2/(b*e-2*c*d)^2/(d*e)^(1/2)*arctan(x*e
/(d*e)^(1/2))*c
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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(1/((c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e)*(e*x^2 + d)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.419647, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e)*(e*x^2 + d)),x, algorithm="fricas")
[Out]
[1/4*(2*(c*d*e*x^2 + c*d^2)*sqrt(-d*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 - 2*
(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2 - c*d + b*e)) -
2*(2*c*d - b*e)*sqrt(-d*e)*x - (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*log((2*
d*e*x + (e*x^2 - d)*sqrt(-d*e))/(e*x^2 + d)))/((4*c^2*d^4 - 4*b*c*d^3*e + b^2*d^
2*e^2 + (4*c^2*d^3*e - 4*b*c*d^2*e^2 + b^2*d*e^3)*x^2)*sqrt(-d*e)), 1/2*((c*d*e*
x^2 + c*d^2)*sqrt(d*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 - 2*(c*d*e - b*e^2)*
x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2 - c*d + b*e)) - (2*c*d - b*e)*sq
rt(d*e)*x - (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*arctan(sqrt(d*e)*x/d))/((4
*c^2*d^4 - 4*b*c*d^3*e + b^2*d^2*e^2 + (4*c^2*d^3*e - 4*b*c*d^2*e^2 + b^2*d*e^3)
*x^2)*sqrt(d*e)), 1/4*(4*(c*d*e*x^2 + c*d^2)*sqrt(-d*e)*sqrt(-c/(c*d*e - b*e^2))
*arctan(-c*x/((c*d - b*e)*sqrt(-c/(c*d*e - b*e^2)))) - 2*(2*c*d - b*e)*sqrt(-d*e
)*x - (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*log((2*d*e*x + (e*x^2 - d)*sqrt(
-d*e))/(e*x^2 + d)))/((4*c^2*d^4 - 4*b*c*d^3*e + b^2*d^2*e^2 + (4*c^2*d^3*e - 4*
b*c*d^2*e^2 + b^2*d*e^3)*x^2)*sqrt(-d*e)), 1/2*(2*(c*d*e*x^2 + c*d^2)*sqrt(d*e)*
sqrt(-c/(c*d*e - b*e^2))*arctan(-c*x/((c*d - b*e)*sqrt(-c/(c*d*e - b*e^2)))) - (
2*c*d - b*e)*sqrt(d*e)*x - (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*arctan(sqrt
(d*e)*x/d))/((4*c^2*d^4 - 4*b*c*d^3*e + b^2*d^2*e^2 + (4*c^2*d^3*e - 4*b*c*d^2*e
^2 + b^2*d*e^3)*x^2)*sqrt(d*e))]
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Sympy [A] time = 60.3676, size = 2664, normalized size = 19.59 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x**2+d)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
x/(2*b*d**2*e - 4*c*d**3 + x**2*(2*b*d*e**2 - 4*c*d**2*e)) - sqrt(-1/(d**3*e))*(
b*e - 4*c*d)*log(x + (-b**7*d**3*e**8*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(2*(
b*e - 2*c*d)**6) + 7*b**6*c*d**4*e**7*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e
- 2*c*d)**6 - 79*b**5*c**2*d**5*e**6*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(2*(
b*e - 2*c*d)**6) - b**5*e**5*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(2*(b*e - 2*c*d)**2
) + 117*b**4*c**3*d**6*e**5*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)*
*6 + 7*b**4*c*d*e**4*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2 - 196*b**3
*c**4*d**7*e**4*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 73*b**3
*c**2*d**2*e**3*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(2*(b*e - 2*c*d)**2) + 184*b**2*
c**5*d**8*e**3*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 + 86*b**2*
c**3*d**3*e**2*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2 - 88*b*c**6*d**9
*e**2*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 88*b*c**4*d**4*e*
sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2 + 16*c**7*d**10*e*(-1/(d**3*e))
**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 + 36*c**5*d**5*sqrt(-1/(d**3*e))*(b*e
- 4*c*d)/(b*e - 2*c*d)**2)/(b**2*c**2*e**2 - 9*b*c**3*d*e + 20*c**4*d**2))/(4*(b
*e - 2*c*d)**2) + sqrt(-1/(d**3*e))*(b*e - 4*c*d)*log(x + (b**7*d**3*e**8*(-1/(d
**3*e))**(3/2)*(b*e - 4*c*d)**3/(2*(b*e - 2*c*d)**6) - 7*b**6*c*d**4*e**7*(-1/(d
**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 + 79*b**5*c**2*d**5*e**6*(-1/(d
**3*e))**(3/2)*(b*e - 4*c*d)**3/(2*(b*e - 2*c*d)**6) + b**5*e**5*sqrt(-1/(d**3*e
))*(b*e - 4*c*d)/(2*(b*e - 2*c*d)**2) - 117*b**4*c**3*d**6*e**5*(-1/(d**3*e))**(
3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 7*b**4*c*d*e**4*sqrt(-1/(d**3*e))*(b*e
- 4*c*d)/(b*e - 2*c*d)**2 + 196*b**3*c**4*d**7*e**4*(-1/(d**3*e))**(3/2)*(b*e -
4*c*d)**3/(b*e - 2*c*d)**6 + 73*b**3*c**2*d**2*e**3*sqrt(-1/(d**3*e))*(b*e - 4*c
*d)/(2*(b*e - 2*c*d)**2) - 184*b**2*c**5*d**8*e**3*(-1/(d**3*e))**(3/2)*(b*e - 4
*c*d)**3/(b*e - 2*c*d)**6 - 86*b**2*c**3*d**3*e**2*sqrt(-1/(d**3*e))*(b*e - 4*c*
d)/(b*e - 2*c*d)**2 + 88*b*c**6*d**9*e**2*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/
(b*e - 2*c*d)**6 + 88*b*c**4*d**4*e*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d
)**2 - 16*c**7*d**10*e*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 -
36*c**5*d**5*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2)/(b**2*c**2*e**2 -
9*b*c**3*d*e + 20*c**4*d**2))/(4*(b*e - 2*c*d)**2) - sqrt(-c**3/(e*(b*e - c*d))
)*log(x + (-4*b**7*d**3*e**8*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 + 5
6*b**6*c*d**4*e**7*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - 316*b**5*c*
*2*d**5*e**6*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - b**5*e**5*sqrt(-c
**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 + 936*b**4*c**3*d**6*e**5*(-c**3/(e*(b*e -
c*d)))**(3/2)/(b*e - 2*c*d)**6 + 14*b**4*c*d*e**4*sqrt(-c**3/(e*(b*e - c*d)))/(
b*e - 2*c*d)**2 - 1568*b**3*c**4*d**7*e**4*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e -
2*c*d)**6 - 73*b**3*c**2*d**2*e**3*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2
+ 1472*b**2*c**5*d**8*e**3*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 + 17
2*b**2*c**3*d**3*e**2*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 - 704*b*c**6*
d**9*e**2*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - 176*b*c**4*d**4*e*sq
rt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 + 128*c**7*d**10*e*(-c**3/(e*(b*e - c
*d)))**(3/2)/(b*e - 2*c*d)**6 + 72*c**5*d**5*sqrt(-c**3/(e*(b*e - c*d)))/(b*e -
2*c*d)**2)/(b**2*c**2*e**2 - 9*b*c**3*d*e + 20*c**4*d**2))/(2*(b*e - 2*c*d)**2)
+ sqrt(-c**3/(e*(b*e - c*d)))*log(x + (4*b**7*d**3*e**8*(-c**3/(e*(b*e - c*d)))*
*(3/2)/(b*e - 2*c*d)**6 - 56*b**6*c*d**4*e**7*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*
e - 2*c*d)**6 + 316*b**5*c**2*d**5*e**6*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*
c*d)**6 + b**5*e**5*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 - 936*b**4*c**3
*d**6*e**5*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - 14*b**4*c*d*e**4*sq
rt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 + 1568*b**3*c**4*d**7*e**4*(-c**3/(e*
(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 + 73*b**3*c**2*d**2*e**3*sqrt(-c**3/(e*(b*
e - c*d)))/(b*e - 2*c*d)**2 - 1472*b**2*c**5*d**8*e**3*(-c**3/(e*(b*e - c*d)))**
(3/2)/(b*e - 2*c*d)**6 - 172*b**2*c**3*d**3*e**2*sqrt(-c**3/(e*(b*e - c*d)))/(b*
e - 2*c*d)**2 + 704*b*c**6*d**9*e**2*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d
)**6 + 176*b*c**4*d**4*e*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 - 128*c**7
*d**10*e*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - 72*c**5*d**5*sqrt(-c*
*3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2)/(b**2*c**2*e**2 - 9*b*c**3*d*e + 20*c**4*d
**2))/(2*(b*e - 2*c*d)**2)
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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(1/((c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e)*(e*x^2 + d)),x, algorithm="giac")
[Out]
Timed out