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

Optimal. Leaf size=130 \[ \frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}+2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right )}{\sqrt{c} \sqrt{b e+2 c d}}-\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}-2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right )}{\sqrt{c} \sqrt{b e+2 c d}} \]

[Out]

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

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Rubi [A]  time = 0.25932, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}+2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right )}{\sqrt{c} \sqrt{b e+2 c d}}-\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}-2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right )}{\sqrt{c} \sqrt{b e+2 c d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.1256, size = 121, normalized size = 0.93 \[ \frac{e^{\frac{3}{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{e} x - \sqrt{- b e + 2 c d}}{\sqrt{b e + 2 c d}} \right )}}{\sqrt{c} \sqrt{b e + 2 c d}} + \frac{e^{\frac{3}{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{e} x + \sqrt{- b e + 2 c d}}{\sqrt{b e + 2 c d}} \right )}}{\sqrt{c} \sqrt{b e + 2 c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**(3/2)*atan((2*sqrt(c)*sqrt(e)*x - sqrt(-b*e + 2*c*d))/sqrt(b*e + 2*c*d))/(sqr
t(c)*sqrt(b*e + 2*c*d)) + e**(3/2)*atan((2*sqrt(c)*sqrt(e)*x + sqrt(-b*e + 2*c*d
))/sqrt(b*e + 2*c*d))/(sqrt(c)*sqrt(b*e + 2*c*d))

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Mathematica [A]  time = 0.0714055, size = 248, normalized size = 1.91 \[ \frac{e^{3/2} \left (\frac{\left (\sqrt{b^2 e^2-4 c^2 d^2}-b e+2 c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}\right )}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}+\frac{\left (\sqrt{b^2 e^2-4 c^2 d^2}+b e-2 c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2 e^2-4 c^2 d^2}} \]

Antiderivative was successfully verified.

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

[Out]

(e^(3/2)*(((2*c*d - b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sq
rt[e]*x)/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b
^2*e^2]] + ((-2*c*d + b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*
Sqrt[e]*x)/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e + Sqrt[-4*c^2*d^2 +
b^2*e^2]]))/(Sqrt[2]*Sqrt[c]*Sqrt[-4*c^2*d^2 + b^2*e^2])

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Maple [B]  time = 0.003, size = 582, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{2} + d}{b x^{2} +{\left (x^{4} + \frac{d^{2}}{e^{2}}\right )} c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.286317, size = 1, normalized size = 0.01 \[ \left [\frac{1}{2} \, e \sqrt{-\frac{e}{2 \, c^{2} d + b c e}} \log \left (\frac{c e^{2} x^{4} + c d^{2} -{\left (4 \, c d e + b e^{2}\right )} x^{2} + 2 \,{\left ({\left (2 \, c^{2} d e + b c e^{2}\right )} x^{3} -{\left (2 \, c^{2} d^{2} + b c d e\right )} x\right )} \sqrt{-\frac{e}{2 \, c^{2} d + b c e}}}{c e^{2} x^{4} + b e^{2} x^{2} + c d^{2}}\right ), e \sqrt{\frac{e}{2 \, c^{2} d + b c e}} \arctan \left (\frac{e x}{{\left (2 \, c d + b e\right )} \sqrt{\frac{e}{2 \, c^{2} d + b c e}}}\right ) + e \sqrt{\frac{e}{2 \, c^{2} d + b c e}} \arctan \left (\frac{c e^{2} x^{3} +{\left (c d e + b e^{2}\right )} x}{{\left (2 \, c^{2} d^{2} + b c d e\right )} \sqrt{\frac{e}{2 \, c^{2} d + b c e}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.84113, size = 160, normalized size = 1.23 \[ - \frac{\sqrt{- \frac{e^{3}}{c \left (b e + 2 c d\right )}} \log{\left (- \frac{d}{e} + x^{2} + \frac{x \left (- b e \sqrt{- \frac{e^{3}}{c \left (b e + 2 c d\right )}} - 2 c d \sqrt{- \frac{e^{3}}{c \left (b e + 2 c d\right )}}\right )}{e^{2}} \right )}}{2} + \frac{\sqrt{- \frac{e^{3}}{c \left (b e + 2 c d\right )}} \log{\left (- \frac{d}{e} + x^{2} + \frac{x \left (b e \sqrt{- \frac{e^{3}}{c \left (b e + 2 c d\right )}} + 2 c d \sqrt{- \frac{e^{3}}{c \left (b e + 2 c d\right )}}\right )}{e^{2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.705802, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done