3.3.10 \(\int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx\) [210]

Optimal. Leaf size=95 \[ \frac {2 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}-\frac {\sin ^{-1}(a x)}{a^3} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1651, 673, 665, 677, 222} \begin {gather*} -\frac {\text {ArcSin}(a x)}{a^3}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^3 (1-a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^4,x]

[Out]

(2*Sqrt[1 - a^2*x^2])/(a^3*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(5*a^3*(1 - a*x)^4) - (3*(1 - a^2*x^2)^(3/2))/(5*a
^3*(1 - a*x)^3) - ArcSin[a*x]/a^3

Rule 222

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]

Rule 665

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a + c*x^2)^(p + 1)/
(2*c*d*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*c*d*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rule 677

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

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^2)^p,
(d + e*x)^m*Pq, x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[m + Expon[Pq
, x] + 2*p + 1, 0] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx &=\int \left (\frac {\sqrt {1-a^2 x^2}}{a^2 (-1+a x)^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 (-1+a x)^3}+\frac {\sqrt {1-a^2 x^2}}{a^2 (-1+a x)^2}\right ) \, dx\\ &=\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^4} \, dx}{a^2}+\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^2} \, dx}{a^2}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{a^2}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3 (1-a x)^3}-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 a^2}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}-\frac {\sin ^{-1}(a x)}{a^3}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 82, normalized size = 0.86 \begin {gather*} \frac {\left (-8+19 a x-13 a^2 x^2\right ) \sqrt {1-a^2 x^2}}{5 a^3 (-1+a x)^3}-\frac {\log \left (-\sqrt {-a^2} x+\sqrt {1-a^2 x^2}\right )}{\left (-a^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^4,x]

[Out]

((-8 + 19*a*x - 13*a^2*x^2)*Sqrt[1 - a^2*x^2])/(5*a^3*(-1 + a*x)^3) - Log[-(Sqrt[-a^2]*x) + Sqrt[1 - a^2*x^2]]
/(-a^2)^(3/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs. \(2(85)=170\).
time = 0.07, size = 245, normalized size = 2.58

method	result	size
default	\(\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}+a \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{a^{4}}+\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x -\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x -\frac {1}{a}\right )^{3}}}{a^{6}}+\frac {2 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a^{6} \left (x -\frac {1}{a}\right )^{3}}\)	\(245\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-a^2*x^2+1)^(1/2)/(-a*x+1)^4,x,method=_RETURNVERBOSE)

[Out]

1/a^4*(1/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)+a*((-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-a/(a^2)^(1/2)*ar
ctan((a^2)^(1/2)*x/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))))+1/a^6*(1/5/a/(x-1/a)^4*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(
3/2)-1/15/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2))+2/3/a^6/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(3/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a^2*x^2 + 1)*x^2/(a*x - 1)^4, x)

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Fricas [A]
time = 1.60, size = 126, normalized size = 1.33 \begin {gather*} \frac {8 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 24 \, a x + 10 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (13 \, a^{2} x^{2} - 19 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} - 8}{5 \, {\left (a^{6} x^{3} - 3 \, a^{5} x^{2} + 3 \, a^{4} x - a^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(8*a^3*x^3 - 24*a^2*x^2 + 24*a*x + 10*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a
*x)) - (13*a^2*x^2 - 19*a*x + 8)*sqrt(-a^2*x^2 + 1) - 8)/(a^6*x^3 - 3*a^5*x^2 + 3*a^4*x - a^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (a x - 1\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))/(a*x - 1)**4, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [B]
time = 2.70, size = 220, normalized size = 2.32 \begin {gather*} \frac {4\,a^2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^7\,x^2-2\,a^6\,x+a^5\right )}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,\sqrt {-a^2}}-\frac {2\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (a\,\sqrt {-a^2}-3\,a^2\,x\,\sqrt {-a^2}+3\,a^3\,x^2\,\sqrt {-a^2}-a^4\,x^3\,\sqrt {-a^2}\right )}-\frac {13\,\sqrt {1-a^2\,x^2}}{5\,\left (a\,\sqrt {-a^2}-a^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {5\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,x^2-2\,a^4\,x+a^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a^2*(1 - a^2*x^2)^(1/2))/(15*(a^5 - 2*a^6*x + a^7*x^2)) - asinh(x*(-a^2)^(1/2))/(a^2*(-a^2)^(1/2)) - (2*(1
- a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(a*(-a^2)^(1/2) - 3*a^2*x*(-a^2)^(1/2) + 3*a^3*x^2*(-a^2)^(1/2) - a^4*x^3*(-
a^2)^(1/2))) - (13*(1 - a^2*x^2)^(1/2))/(5*(a*(-a^2)^(1/2) - a^2*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (5*(1 - a^2*x
^2)^(1/2))/(3*(a^3 - 2*a^4*x + a^5*x^2))

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