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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, 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))

________________________________________________________________________________________

fricas [A]  time = 0.86, size = 34, normalized size = 1.00 \[ \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}} x}{a b x^{2} + a^{2}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.26, size = 19, normalized size = 0.56 \[ -\frac {1}{a \sqrt {-\frac {b x^{2} + a}{x^{2}}}} \]

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]

-1/(a*sqrt(-(b*x^2 + a)/x^2))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 33, normalized size = 0.97 \[ \frac {\left (b \,x^{2}+a \right ) x}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {3}{4}} a} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {3}{4}}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [B]  time = 4.14, size = 34, normalized size = 1.00 \[ \frac {x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{1/4}}{a\,\left (b\,x^2+a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {3}{4}}}\, dx \]

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)

________________________________________________________________________________________