3.4 \(\int \frac{1}{(a^2+2 a b x^2+b^2 x^4)^{3/4}} \, dx\)

Optimal. Leaf size=34 \[ \frac{x \left (a+b x^2\right )}{a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0084038, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1089, 191} \[ \frac{x \left (a+b x^2\right )}{a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1089

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^2 + c*x^4)^FracPart[p]
)/(1 + (2*c*x^2)/b)^(2*FracPart[p]), Int[(1 + (2*c*x^2)/b)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2
- 4*a*c, 0] &&  !IntegerQ[2*p]

Rule 191

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

Rubi steps

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

Mathematica [A]  time = 0.0110457, size = 25, normalized size = 0.74 \[ \frac{x \left (a+b x^2\right )}{a \left (\left (a+b x^2\right )^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.054, size = 33, normalized size = 1. \begin{align*}{\frac{x \left ( b{x}^{2}+a \right ) }{a} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{-{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b^2*x^4+2*a*b*x^2+a^2)^(3/4),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{3}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(3/4),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.28183, size = 72, normalized size = 2.12 \begin{align*} \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} x}{a b x^{2} + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(3/4),x, algorithm="fricas")

[Out]

(b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4)*x/(a*b*x^2 + a^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(3/4),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16254, size = 32, normalized size = 0.94 \begin{align*} -\frac{x^{2}}{\sqrt{-b x^{2} - a} a{\left | x \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(3/4),x, algorithm="giac")

[Out]

-x^2/(sqrt(-b*x^2 - a)*a*abs(x))