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

Optimal. Leaf size=190 \[ -\frac{x (b d-a e)}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 \left (a+b x^2\right ) (5 b d-a e) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x (7 b d-3 a e)}{8 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

-((7*b*d - 3*a*e)*x)/(8*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - ((b*d - a*e)*x)/(
4*a^2*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(a + b*x^2))/(a^3*x*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*(5*b*d - a*e)*(a + b*x^2)*ArcTan[(Sqrt[b]*x)/S
qrt[a]])/(8*a^(7/2)*Sqrt[b]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.472031, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ -\frac{x (b d-a e)}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 \left (a+b x^2\right ) (5 b d-a e) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x (7 b d-3 a e)}{8 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-((7*b*d - 3*a*e)*x)/(8*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - ((b*d - a*e)*x)/(
4*a^2*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(a + b*x^2))/(a^3*x*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*(5*b*d - a*e)*(a + b*x^2)*ArcTan[(Sqrt[b]*x)/S
qrt[a]])/(8*a^(7/2)*Sqrt[b]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

_______________________________________________________________________________________

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)/x**2/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.13916, size = 124, normalized size = 0.65 \[ \frac{\sqrt{a} \sqrt{b} \left (a^2 \left (5 e x^2-8 d\right )+a b \left (3 e x^4-25 d x^2\right )-15 b^2 d x^4\right )+3 x \left (a+b x^2\right )^2 (a e-5 b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{b} x \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*Sqrt[b]*(-15*b^2*d*x^4 + a^2*(-8*d + 5*e*x^2) + a*b*(-25*d*x^2 + 3*e*x^
4)) + 3*(-5*b*d + a*e)*x*(a + b*x^2)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*S
qrt[b]*x*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.027, size = 206, normalized size = 1.1 \[{\frac{b{x}^{2}+a}{8\,x{a}^{3}} \left ( 3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{5}a{b}^{2}e-15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{5}{b}^{3}d+6\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}be-30\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}a{b}^{2}d+3\,\sqrt{ab}{x}^{4}abe-15\,\sqrt{ab}{x}^{4}{b}^{2}d+3\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) x{a}^{3}e-15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) x{a}^{2}bd+5\,\sqrt{ab}{x}^{2}{a}^{2}e-25\,\sqrt{ab}{x}^{2}abd-8\,\sqrt{ab}{a}^{2}d \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*arctan(x*b/(a*b)^(1/2))*x^5*a*b^2*e-15*arctan(x*b/(a*b)^(1/2))*x^5*b^3*d+
6*arctan(x*b/(a*b)^(1/2))*x^3*a^2*b*e-30*arctan(x*b/(a*b)^(1/2))*x^3*a*b^2*d+3*(
a*b)^(1/2)*x^4*a*b*e-15*(a*b)^(1/2)*x^4*b^2*d+3*arctan(x*b/(a*b)^(1/2))*x*a^3*e-
15*arctan(x*b/(a*b)^(1/2))*x*a^2*b*d+5*(a*b)^(1/2)*x^2*a^2*e-25*(a*b)^(1/2)*x^2*
a*b*d-8*(a*b)^(1/2)*a^2*d)*(b*x^2+a)/x/(a*b)^(1/2)/a^3/((b*x^2+a)^2)^(3/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.281941, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (5 \, b^{3} d - a b^{2} e\right )} x^{5} + 2 \,{\left (5 \, a b^{2} d - a^{2} b e\right )} x^{3} +{\left (5 \, a^{2} b d - a^{3} e\right )} x\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \,{\left (5 \, b^{2} d - a b e\right )} x^{4} + 8 \, a^{2} d + 5 \,{\left (5 \, a b d - a^{2} e\right )} x^{2}\right )} \sqrt{-a b}}{16 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt{-a b}}, -\frac{3 \,{\left ({\left (5 \, b^{3} d - a b^{2} e\right )} x^{5} + 2 \,{\left (5 \, a b^{2} d - a^{2} b e\right )} x^{3} +{\left (5 \, a^{2} b d - a^{3} e\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \,{\left (5 \, b^{2} d - a b e\right )} x^{4} + 8 \, a^{2} d + 5 \,{\left (5 \, a b d - a^{2} e\right )} x^{2}\right )} \sqrt{a b}}{8 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt{a b}}\right ] \]

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^2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.636847, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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^2),x, algorithm="giac")

[Out]

sage0*x