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

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

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Rubi [A]  time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2018, 640, 620, 203} \begin {gather*} \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} \end {gather*}

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 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])

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 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 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]

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*}

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Mathematica [A]  time = 0.04, size = 77, normalized size = 1.28 \begin {gather*} \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 )}} \end {gather*}

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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IntegrateAlgebraic [B]  time = 0.30, size = 146, normalized size = 2.43 \begin {gather*} \frac {a \sqrt {-b} \log \left (a^2+4 a b x^2-8 \sqrt {-b} b x^2 \sqrt {a x^2-b x^4}-8 b^2 x^4\right )}{8 b^2}-\frac {a \tan ^{-1}\left (\frac {2 \sqrt {-b} \sqrt {b} x^2}{a}-\frac {2 \sqrt {b} \sqrt {a x^2-b x^4}}{a}\right )}{4 b^{3/2}}-\frac {\sqrt {a x^2-b x^4}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/Sqrt[a*x^2 - b*x^4],x]

[Out]

-1/2*Sqrt[a*x^2 - b*x^4]/b - (a*ArcTan[(2*Sqrt[-b]*Sqrt[b]*x^2)/a - (2*Sqrt[b]*Sqrt[a*x^2 - b*x^4])/a])/(4*b^(
3/2)) + (a*Sqrt[-b]*Log[a^2 + 4*a*b*x^2 - 8*b^2*x^4 - 8*Sqrt[-b]*b*x^2*Sqrt[a*x^2 - b*x^4]])/(8*b^2)

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fricas [A]  time = 0.92, size = 120, normalized size = 2.00 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.20, size = 68, normalized size = 1.13 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.01, size = 67, normalized size = 1.12 \begin {gather*} \frac {\sqrt {-b \,x^{2}+a}\, \left (a b \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right )-\sqrt {-b \,x^{2}+a}\, b^{\frac {3}{2}} x \right ) x}{2 \sqrt {-b \,x^{4}+a \,x^{2}}\, b^{\frac {5}{2}}} \end {gather*}

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 [A]  time = 3.03, size = 42, normalized size = 0.70 \begin {gather*} -\frac {a \arcsin \left (-\frac {2 \, b x^{2} - a}{a}\right )}{4 \, b^{\frac {3}{2}}} - \frac {\sqrt {-b x^{4} + a x^{2}}}{2 \, b} \end {gather*}

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]

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

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mupad [B]  time = 4.62, size = 60, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a\,x^2-b\,x^4}}{2\,b}-\frac {a\,\ln \left (\frac {\frac {a}{2}-b\,x^2}{\sqrt {-b}}+\sqrt {a\,x^2-b\,x^4}\right )}{4\,{\left (-b\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*x^2 - b*x^4)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\sqrt {- x^{2} \left (- a + b x^{2}\right )}}\, dx \end {gather*}

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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