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

Optimal. Leaf size=60 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^2-b x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{a x^2-b x^4}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0816661, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2018, 640, 620, 203} \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^2-b x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{a x^2-b x^4}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a*x^2 - b*x^4],x]

[Out]

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

Rule 2018

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.0406761, size = 77, normalized size = 1.28 \[ \frac{x \left (\sqrt{b} x \left (b x^2-a\right )+a \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )\right )}{2 b^{3/2} \sqrt{x^2 \left (a-b x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a*x^2 - b*x^4],x]

[Out]

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

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Maple [A]  time = 0.049, size = 67, normalized size = 1.1 \begin{align*} -{\frac{x}{2}\sqrt{-b{x}^{2}+a} \left ( x\sqrt{-b{x}^{2}+a}{b}^{{\frac{3}{2}}}-a\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \right ) b \right ){\frac{1}{\sqrt{-b{x}^{4}+a{x}^{2}}}}{b}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x*(-b*x^2+a)^(1/2)*(x*(-b*x^2+a)^(1/2)*b^(3/2)-a*arctan(b^(1/2)*x/(-b*x^2+a)^(1/2))*b)/(-b*x^4+a*x^2)^(1/
2)/b^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26835, size = 271, normalized size = 4.52 \begin{align*} \left [-\frac{a \sqrt{-b} \log \left (2 \, b x^{2} - a - 2 \, \sqrt{-b x^{4} + a x^{2}} \sqrt{-b}\right ) + 2 \, \sqrt{-b x^{4} + a x^{2}} b}{4 \, b^{2}}, -\frac{a \sqrt{b} \arctan \left (\frac{\sqrt{-b x^{4} + a x^{2}} \sqrt{b}}{b x^{2} - a}\right ) + \sqrt{-b x^{4} + a x^{2}} b}{2 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(a*sqrt(-b)*log(2*b*x^2 - a - 2*sqrt(-b*x^4 + a*x^2)*sqrt(-b)) + 2*sqrt(-b*x^4 + a*x^2)*b)/b^2, -1/2*(a*
sqrt(b)*arctan(sqrt(-b*x^4 + a*x^2)*sqrt(b)/(b*x^2 - a)) + sqrt(-b*x^4 + a*x^2)*b)/b^2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19304, size = 92, normalized size = 1.53 \begin{align*} -\frac{a \log \left ({\left | 2 \,{\left (\sqrt{-b} x^{2} - \sqrt{-b x^{4} + a x^{2}}\right )} \sqrt{-b} + a \right |}\right )}{4 \, \sqrt{-b} b} - \frac{\sqrt{-b x^{4} + a x^{2}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*log(abs(2*(sqrt(-b)*x^2 - sqrt(-b*x^4 + a*x^2))*sqrt(-b) + a))/(sqrt(-b)*b) - 1/2*sqrt(-b*x^4 + a*x^2)/
b