3.150 \(\int \frac{x^3}{(1+a x) \sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{x^2 (1-a x)}{a^2 \sqrt{1-a^2 x^2}}+\frac{(4-3 a x) \sqrt{1-a^2 x^2}}{2 a^4}+\frac{3 \sin ^{-1}(a x)}{2 a^4} \]

[Out]

(x^2*(1 - a*x))/(a^2*Sqrt[1 - a^2*x^2]) + ((4 - 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^4) + (3*ArcSin[a*x])/(2*a^4)

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Rubi [A]  time = 0.0538498, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {850, 819, 780, 216} \[ \frac{x^2 (1-a x)}{a^2 \sqrt{1-a^2 x^2}}+\frac{(4-3 a x) \sqrt{1-a^2 x^2}}{2 a^4}+\frac{3 \sin ^{-1}(a x)}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

(x^2*(1 - a*x))/(a^2*Sqrt[1 - a^2*x^2]) + ((4 - 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^4) + (3*ArcSin[a*x])/(2*a^4)

Rule 850

Int[((x_)^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*x)/e)*(a + c*x
^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||
!IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 216

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

Rubi steps

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

Mathematica [A]  time = 0.0573529, size = 54, normalized size = 0.82 \[ \frac{\sqrt{1-a^2 x^2} \left (-a^2 x^2+a x+4\right )+3 (a x+1) \sin ^{-1}(a x)}{2 a^4 (a x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

(Sqrt[1 - a^2*x^2]*(4 + a*x - a^2*x^2) + 3*(1 + a*x)*ArcSin[a*x])/(2*a^4*(1 + a*x))

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Maple [A]  time = 0.056, size = 100, normalized size = 1.5 \begin{align*} -{\frac{x}{2\,{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3}{2\,{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{{a}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{1}{{a}^{5} \left ( x+{a}^{-1} \right ) }\sqrt{- \left ( x+{a}^{-1} \right ) ^{2}{a}^{2}+2\,a \left ( x+{a}^{-1} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46735, size = 92, normalized size = 1.39 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{5} x + a^{4}} - \frac{\sqrt{-a^{2} x^{2} + 1} x}{2 \, a^{3}} + \frac{3 \, \arcsin \left (a x\right )}{2 \, a^{4}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a^2*x^2 + 1)/(a^5*x + a^4) - 1/2*sqrt(-a^2*x^2 + 1)*x/a^3 + 3/2*arcsin(a*x)/a^4 + sqrt(-a^2*x^2 + 1)/a^4

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Fricas [A]  time = 1.67148, size = 169, normalized size = 2.56 \begin{align*} \frac{4 \, a x - 6 \,{\left (a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (a^{2} x^{2} - a x - 4\right )} \sqrt{-a^{2} x^{2} + 1} + 4}{2 \,{\left (a^{5} x + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*a*x - 6*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (a^2*x^2 - a*x - 4)*sqrt(-a^2*x^2 + 1) + 4)/
(a^5*x + a^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/(sqrt(-(a*x - 1)*(a*x + 1))*(a*x + 1)), x)

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Giac [A]  time = 1.2575, size = 105, normalized size = 1.59 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{x}{a^{3}} - \frac{2}{a^{4}}\right )} + \frac{3 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \, a^{3}{\left | a \right |}} - \frac{2}{a^{3}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*(x/a^3 - 2/a^4) + 3/2*arcsin(a*x)*sgn(a)/(a^3*abs(a)) - 2/(a^3*((sqrt(-a^2*x^2 + 1)*ab
s(a) + a)/(a^2*x) + 1)*abs(a))