3.77 \(\int \frac{d+e x^2}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=97 \[ \frac{\left (a+b x^2\right ) (b d-a e) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e x \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0470934, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1148, 388, 205} \[ \frac{\left (a+b x^2\right ) (b d-a e) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e x \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1148

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^
4)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d + e*x^2)^q*(b/2 + c*x^2)^(2*p), x], x] /;
FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{d+e x^2}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (a b+b^2 x^2\right ) \int \frac{d+e x^2}{a b+b^2 x^2} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{e x \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (\left (-b^2 d+a b e\right ) \left (a b+b^2 x^2\right )\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{e x \left (a+b x^2\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{(b d-a e) \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0293192, size = 69, normalized size = 0.71 \[ -\frac{\left (a+b x^2\right ) \left ((a e-b d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )-\sqrt{a} \sqrt{b} e x\right )}{\sqrt{a} b^{3/2} \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 62, normalized size = 0.6 \begin{align*}{\frac{b{x}^{2}+a}{b} \left ( ex\sqrt{ab}-\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) ae+\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) bd \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)/((b*x^2+a)^2)^(1/2),x)

[Out]

(b*x^2+a)*(e*x*(a*b)^(1/2)-arctan(b*x/(a*b)^(1/2))*a*e+arctan(b*x/(a*b)^(1/2))*b*d)/((b*x^2+a)^2)^(1/2)/b/(a*b
)^(1/2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)/((b*x^2+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.51208, size = 223, normalized size = 2.3 \begin{align*} \left [\frac{2 \, a b e x + \sqrt{-a b}{\left (b d - a e\right )} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{2 \, a b^{2}}, \frac{a b e x + \sqrt{a b}{\left (b d - a e\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)/((b*x^2+a)^2)^(1/2),x, algorithm="fricas")

[Out]

[1/2*(2*a*b*e*x + sqrt(-a*b)*(b*d - a*e)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a*b^2), (a*b*e*x + sq
rt(a*b)*(b*d - a*e)*arctan(sqrt(a*b)*x/a))/(a*b^2)]

________________________________________________________________________________________

Sympy [A]  time = 0.4661, size = 82, normalized size = 0.85 \begin{align*} \frac{\sqrt{- \frac{1}{a b^{3}}} \left (a e - b d\right ) \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a b^{3}}} \left (a e - b d\right ) \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} + \frac{e x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)/((b*x**2+a)**2)**(1/2),x)

[Out]

sqrt(-1/(a*b**3))*(a*e - b*d)*log(-a*b*sqrt(-1/(a*b**3)) + x)/2 - sqrt(-1/(a*b**3))*(a*e - b*d)*log(a*b*sqrt(-
1/(a*b**3)) + x)/2 + e*x/b

________________________________________________________________________________________

Giac [A]  time = 1.13127, size = 80, normalized size = 0.82 \begin{align*} \frac{x e \mathrm{sgn}\left (b x^{2} + a\right )}{b} + \frac{{\left (b d \mathrm{sgn}\left (b x^{2} + a\right ) - a e \mathrm{sgn}\left (b x^{2} + a\right )\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)/((b*x^2+a)^2)^(1/2),x, algorithm="giac")

[Out]

x*e*sgn(b*x^2 + a)/b + (b*d*sgn(b*x^2 + a) - a*e*sgn(b*x^2 + a))*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b)