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

Optimal. Leaf size=156 \[ \frac{x (a e+3 b d)}{8 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x (b d-a e)}{4 a b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (a e+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

((3*b*d + a*e)*x)/(8*a^2*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((b*d - a*e)*x)/(4
*a*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((3*b*d + a*e)*(a + b*x^2)*A
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.201955, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x (a e+3 b d)}{8 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x (b d-a e)}{4 a b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (a e+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

((3*b*d + a*e)*x)/(8*a^2*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((b*d - a*e)*x)/(4
*a*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((3*b*d + a*e)*(a + b*x^2)*A
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.0971276, size = 108, normalized size = 0.69 \[ \frac{\sqrt{a} \sqrt{b} x \left (a^2 (-e)+a b \left (5 d+e x^2\right )+3 b^2 d x^2\right )+\left (a+b x^2\right )^2 (a e+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*Sqrt[b]*x*(-(a^2*e) + 3*b^2*d*x^2 + a*b*(5*d + e*x^2)) + (3*b*d + a*e)*
(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(3/2)*(a + b*x^2)*Sqrt[(
a + b*x^2)^2])

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Maple [A]  time = 0.02, size = 186, normalized size = 1.2 \[{\frac{b{x}^{2}+a}{8\,{a}^{2}b} \left ( \arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){x}^{4}a{b}^{2}e+3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}{b}^{3}d+\sqrt{ab}{x}^{3}abe+3\,\sqrt{ab}{x}^{3}{b}^{2}d+2\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}be+6\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}a{b}^{2}d-\sqrt{ab}x{a}^{2}e+5\,\sqrt{ab}xabd+\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){a}^{3}e+3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){a}^{2}bd \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(arctan(x*b/(a*b)^(1/2))*x^4*a*b^2*e+3*arctan(x*b/(a*b)^(1/2))*x^4*b^3*d+(a*
b)^(1/2)*x^3*a*b*e+3*(a*b)^(1/2)*x^3*b^2*d+2*arctan(x*b/(a*b)^(1/2))*x^2*a^2*b*e
+6*arctan(x*b/(a*b)^(1/2))*x^2*a*b^2*d-(a*b)^(1/2)*x*a^2*e+5*(a*b)^(1/2)*x*a*b*d
+arctan(x*b/(a*b)^(1/2))*a^3*e+3*arctan(x*b/(a*b)^(1/2))*a^2*b*d)*(b*x^2+a)/(a*b
)^(1/2)/b/a^2/((b*x^2+a)^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272678, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, b^{3} d + a b^{2} e\right )} x^{4} + 3 \, a^{2} b d + a^{3} e + 2 \,{\left (3 \, a b^{2} d + a^{2} b e\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left ({\left (3 \, b^{2} d + a b e\right )} x^{3} +{\left (5 \, a b d - a^{2} e\right )} x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{-a b}}, \frac{{\left ({\left (3 \, b^{3} d + a b^{2} e\right )} x^{4} + 3 \, a^{2} b d + a^{3} e + 2 \,{\left (3 \, a b^{2} d + a^{2} b e\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left ({\left (3 \, b^{2} d + a b e\right )} x^{3} +{\left (5 \, a b d - a^{2} e\right )} x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(((3*b^3*d + a*b^2*e)*x^4 + 3*a^2*b*d + a^3*e + 2*(3*a*b^2*d + a^2*b*e)*x^
2)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*((3*b^2*d + a*b*e)*x^
3 + (5*a*b*d - a^2*e)*x)*sqrt(-a*b))/((a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*sqrt
(-a*b)), 1/8*(((3*b^3*d + a*b^2*e)*x^4 + 3*a^2*b*d + a^3*e + 2*(3*a*b^2*d + a^2*
b*e)*x^2)*arctan(sqrt(a*b)*x/a) + ((3*b^2*d + a*b*e)*x^3 + (5*a*b*d - a^2*e)*x)*
sqrt(a*b))/((a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x^{2}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x**2)/((a + b*x**2)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.627621, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x