3.3.11 \(\int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx\)

Optimal. Leaf size=88 \[ \frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (1-a x)^3}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5} \]

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Rubi [A]  time = 0.12, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1639, 793, 659, 651} \begin {gather*} \frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (1-a x)^3}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^5,x]

[Out]

(1 - a^2*x^2)^(3/2)/(7*a^3*(1 - a*x)^5) - (12*(1 - a^2*x^2)^(3/2))/(35*a^3*(1 - a*x)^4) + (23*(1 - a^2*x^2)^(3
/2))/(105*a^3*(1 - a*x)^3)

Rule 651

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 659

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 793

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

Rule 1639

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx &=-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^4}+\frac {\int \frac {\left (4 a^2-3 a^3 x\right ) \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx}{a^4}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 (1-a x)^4}+\frac {23 \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^4} \, dx}{7 a^2}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 (1-a x)^4}+\frac {23 \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^3} \, dx}{35 a^2}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 (1-a x)^4}+\frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (1-a x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 50, normalized size = 0.57 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (23 a^3 x^3+13 a^2 x^2-8 a x+2\right )}{105 a^3 (a x-1)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^5,x]

[Out]

(Sqrt[1 - a^2*x^2]*(2 - 8*a*x + 13*a^2*x^2 + 23*a^3*x^3))/(105*a^3*(-1 + a*x)^4)

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IntegrateAlgebraic [A]  time = 0.72, size = 50, normalized size = 0.57 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (23 a^3 x^3+13 a^2 x^2-8 a x+2\right )}{105 a^3 (a x-1)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*Sqrt[1 - a^2*x^2])/(1 - a*x)^5,x]

[Out]

(Sqrt[1 - a^2*x^2]*(2 - 8*a*x + 13*a^2*x^2 + 23*a^3*x^3))/(105*a^3*(-1 + a*x)^4)

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fricas [A]  time = 0.40, size = 102, normalized size = 1.16 \begin {gather*} \frac {2 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 8 \, a x + {\left (23 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 8 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} + 2}{105 \, {\left (a^{7} x^{4} - 4 \, a^{6} x^{3} + 6 \, a^{5} x^{2} - 4 \, 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)^5,x, algorithm="fricas")

[Out]

1/105*(2*a^4*x^4 - 8*a^3*x^3 + 12*a^2*x^2 - 8*a*x + (23*a^3*x^3 + 13*a^2*x^2 - 8*a*x + 2)*sqrt(-a^2*x^2 + 1) +
2)/(a^7*x^4 - 4*a^6*x^3 + 6*a^5*x^2 - 4*a^4*x + a^3)

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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)^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.01, size = 44, normalized size = 0.50 \begin {gather*} \frac {\sqrt {-a^{2} x^{2}+1}\, \left (23 a^{2} x^{2}-10 a x +2\right ) \left (a x +1\right )}{105 \left (a x -1\right )^{4} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-a^2*x^2+1)^(1/2)/(-a*x+1)^5,x)

[Out]

1/105*(-a^2*x^2+1)^(1/2)*(23*a^2*x^2-10*a*x+2)*(a*x+1)/(a*x-1)^4/a^3

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maxima [B]  time = 0.44, size = 153, normalized size = 1.74 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{7} x^{4} - 4 \, a^{6} x^{3} + 6 \, a^{5} x^{2} - 4 \, a^{4} x + a^{3}\right )}} + \frac {29 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{6} x^{3} - 3 \, a^{5} x^{2} + 3 \, a^{4} x - a^{3}\right )}} + \frac {82 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{5} x^{2} - 2 \, a^{4} x + a^{3}\right )}} + \frac {23 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (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)^5,x, algorithm="maxima")

[Out]

2/7*sqrt(-a^2*x^2 + 1)/(a^7*x^4 - 4*a^6*x^3 + 6*a^5*x^2 - 4*a^4*x + a^3) + 29/35*sqrt(-a^2*x^2 + 1)/(a^6*x^3 -
3*a^5*x^2 + 3*a^4*x - a^3) + 82/105*sqrt(-a^2*x^2 + 1)/(a^5*x^2 - 2*a^4*x + a^3) + 23/105*sqrt(-a^2*x^2 + 1)/
(a^4*x - a^3)

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mupad [B]  time = 0.06, size = 287, normalized size = 3.26 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^7\,x^4-4\,a^6\,x^3+6\,a^5\,x^2-4\,a^4\,x+a^3\right )}+\frac {4\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,x^2-2\,a^4\,x+a^3\right )}+\frac {4\,a\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,x^2-2\,a^5\,x+a^4\right )}+\frac {29\,\sqrt {1-a^2\,x^2}}{35\,\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 {23\,\sqrt {1-a^2\,x^2}}{105\,\left (a\,\sqrt {-a^2}-a^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{3\,\left (a^7\,x^2-2\,a^6\,x+a^5\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)^5,x)

[Out]

(2*(1 - a^2*x^2)^(1/2))/(7*(a^3 - 4*a^4*x + 6*a^5*x^2 - 4*a^6*x^3 + a^7*x^4)) + (4*(1 - a^2*x^2)^(1/2))/(3*(a^
3 - 2*a^4*x + a^5*x^2)) + (4*a*(1 - a^2*x^2)^(1/2))/(35*(a^4 - 2*a^5*x + a^6*x^2)) + (29*(1 - a^2*x^2)^(1/2))/
(35*(-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))) + (2
3*(1 - a^2*x^2)^(1/2))/(105*(a*(-a^2)^(1/2) - a^2*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (2*a^2*(1 - a^2*x^2)^(1/2))/
(3*(a^5 - 2*a^6*x + a^7*x^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} - 5 a^{4} x^{4} + 10 a^{3} x^{3} - 10 a^{2} x^{2} + 5 a x - 1}\, 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)**5,x)

[Out]

-Integral(x**2*sqrt(-a**2*x**2 + 1)/(a**5*x**5 - 5*a**4*x**4 + 10*a**3*x**3 - 10*a**2*x**2 + 5*a*x - 1), x)

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