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

Optimal. Leaf size=89 \[ \frac {x \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {a} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1112, 321, 205} \begin {gather*} \frac {x \left (a+b x^2\right )}{b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {a} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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]

Rule 321

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

Rule 1112

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 54, normalized size = 0.61 \begin {gather*} \frac {\left (a+b x^2\right ) \left (\sqrt {b} x-\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{b^{3/2} \sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 4.36, size = 52, normalized size = 0.58 \begin {gather*} \frac {\left (a+b x^2\right ) \left (\frac {x}{b}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}\right )}{\sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.79, size = 82, normalized size = 0.92 \begin {gather*} \left [\frac {\sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 2 \, x}{2 \, b}, -\frac {\sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - x}{b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.23, size = 42, normalized size = 0.47 \begin {gather*} -\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {a b} b} + \frac {x \mathrm {sgn}\left (b x^{2} + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 48, normalized size = 0.54 \begin {gather*} \frac {\left (b \,x^{2}+a \right ) \left (-a \arctan \left (\frac {b x}{\sqrt {a b}}\right )+\sqrt {a b}\, x \right )}{\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {a b}\, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.91, size = 26, normalized size = 0.29 \begin {gather*} -\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b) + x/b

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((a + b*x^2)^2)^(1/2),x)

[Out]

int(x^2/((a + b*x^2)^2)^(1/2), x)

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 56, normalized size = 0.63 \begin {gather*} \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (- b \sqrt {- \frac {a}{b^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (b \sqrt {- \frac {a}{b^{3}}} + x \right )}}{2} + \frac {x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________