3.206 \(\int \frac{(a-b x^2)^{3/2}}{\sqrt{a^2-b^2 x^4}} \, dx\)

Optimal. Leaf size=109 \[ \frac{3 a \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b} \sqrt{a^2-b^2 x^4}}-\frac{x \sqrt{a-b x^2} \left (a+b x^2\right )}{2 \sqrt{a^2-b^2 x^4}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0352851, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1152, 388, 217, 206} \[ \frac{3 a \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 \sqrt{b} \sqrt{a^2-b^2 x^4}}-\frac{x \sqrt{a-b x^2} \left (a+b x^2\right )}{2 \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1152

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + c*x^4)^FracPart[p]/((d + e*x
^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.115382, size = 110, normalized size = 1.01 \[ \frac{1}{2} \left (-\frac{x \sqrt{a^2-b^2 x^4}}{\sqrt{a-b x^2}}+\frac{3 a \log \left (\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right )}{\sqrt{b}}-\frac{3 a \log \left (b x^2-a\right )}{\sqrt{b}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 85, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,b{x}^{2}-2\,a}\sqrt{-b{x}^{2}+a}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}} \left ( -x\sqrt{b{x}^{2}+a}\sqrt{b}+3\,\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) a \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-b*x^2+a)^(1/2)*(-b^2*x^4+a^2)^(1/2)*(-x*(b*x^2+a)^(1/2)*b^(1/2)+3*ln(x*b^(1/2)+(b*x^2+a)^(1/2))*a)/(b*x
^2-a)/(b*x^2+a)^(1/2)/b^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.97421, size = 490, normalized size = 4.5 \begin{align*} \left [\frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} b x + 3 \,{\left (a b x^{2} - a^{2}\right )} \sqrt{b} \log \left (\frac{2 \, b^{2} x^{4} - a b x^{2} - 2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{b} x - a^{2}}{b x^{2} - a}\right )}{4 \,{\left (b^{2} x^{2} - a b\right )}}, \frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} b x + 3 \,{\left (a b x^{2} - a^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{b^{2} x^{3} - a b x}\right )}{2 \,{\left (b^{2} x^{2} - a b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*b*x + 3*(a*b*x^2 - a^2)*sqrt(b)*log((2*b^2*x^4 - a*b*x^2 - 2*sqr
t(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*sqrt(b)*x - a^2)/(b*x^2 - a)))/(b^2*x^2 - a*b), 1/2*(sqrt(-b^2*x^4 + a^2)*s
qrt(-b*x^2 + a)*b*x + 3*(a*b*x^2 - a^2)*sqrt(-b)*arctan(sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*sqrt(-b)/(b^2*x^
3 - a*b*x)))/(b^2*x^2 - a*b)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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