3.799 \(\int \frac{a+b x+c x^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{x \left (a d^2+c\right )+b}{d^2 \sqrt{1-d^2 x^2}}-\frac{c \sin ^{-1}(d x)}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.0511869, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {899, 1814, 12, 216} \[ \frac{x \left (a d^2+c\right )+b}{d^2 \sqrt{1-d^2 x^2}}-\frac{c \sin ^{-1}(d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 899

Int[((d_) + (e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :>
Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] &&
EqQ[e*f + d*g, 0] && (IntegerQ[m] || (GtQ[d, 0] && GtQ[f, 0]))

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A]  time = 0.0494655, size = 39, normalized size = 0.98 \[ \frac{\frac{d \left (x \left (a d^2+c\right )+b\right )}{\sqrt{1-d^2 x^2}}-c \sin ^{-1}(d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.23, size = 151, normalized size = 3.8 \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{ \left ( dx-1 \right ){d}^{3}} \left ( -\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ){d}^{3}xa-\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{- \left ( dx+1 \right ) \left ( dx-1 \right ) }}}} \right ){x}^{2}c{d}^{2}-{\it csgn} \left ( d \right ) d\sqrt{-{d}^{2}{x}^{2}+1}xc-{\it csgn} \left ( d \right ) d\sqrt{-{d}^{2}{x}^{2}+1}b+\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{- \left ( dx+1 \right ) \left ( dx-1 \right ) }}}} \right ) c \right ) \sqrt{-dx+1}{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}{\frac{1}{\sqrt{dx+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-d^2*x^2+1)^(1/2)*csgn(d)*d^3*x*a-arctan(csgn(d)*d*x/(-(d*x+1)*(d*x-1))^(1/2))*x^2*c*d^2-csgn(d)*d*(-d^2*x^
2+1)^(1/2)*x*c-csgn(d)*d*(-d^2*x^2+1)^(1/2)*b+arctan(csgn(d)*d*x/(-(d*x+1)*(d*x-1))^(1/2))*c)*(-d*x+1)^(1/2)*c
sgn(d)/(d*x-1)/(-d^2*x^2+1)^(1/2)/d^3/(d*x+1)^(1/2)

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Maxima [A]  time = 1.50029, size = 99, normalized size = 2.48 \begin{align*} \frac{a x}{\sqrt{-d^{2} x^{2} + 1}} + \frac{c x}{\sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{c \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}} d^{2}} + \frac{b}{\sqrt{-d^{2} x^{2} + 1} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/sqrt(-d^2*x^2 + 1) + c*x/(sqrt(-d^2*x^2 + 1)*d^2) - c*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^2) + b/(sqrt(-d
^2*x^2 + 1)*d^2)

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Fricas [B]  time = 1.60634, size = 215, normalized size = 5.38 \begin{align*} \frac{b d^{3} x^{2} -{\left (b d +{\left (a d^{3} + c d\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - b d + 2 \,{\left (c d^{2} x^{2} - c\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{d^{5} x^{2} - d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*d^3*x^2 - (b*d + (a*d^3 + c*d)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) - b*d + 2*(c*d^2*x^2 - c)*arctan((sqrt(d*x +
1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^5*x^2 - d^3)

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Sympy [C]  time = 159.678, size = 255, normalized size = 6.38 \begin{align*} a \left (- \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d}\right ) + b \left (- \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4}, 1 & 0, 1, \frac{3}{2} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{2}} - \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, 1 & \\- \frac{1}{4}, \frac{1}{4} & -1, - \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{2}}\right ) + c \left (\frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, \frac{1}{2}, 1, 1 \\- \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{3}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 1 & \\- \frac{3}{4}, - \frac{1}{4} & - \frac{3}{2}, -1, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-I*meijerg(((3/4, 5/4, 1), (1/2, 3/2, 2)), ((3/4, 1, 5/4, 3/2, 2), (0,)), 1/(d**2*x**2))/(2*pi**(3/2)*d) +
meijerg(((-1/2, 0, 1/4, 1/2, 3/4, 1), ()), ((1/4, 3/4), (-1/2, 0, 1, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(2*p
i**(3/2)*d)) + b*(-I*meijerg(((1/4, 3/4, 1), (0, 1, 3/2)), ((1/4, 1/2, 3/4, 1, 3/2), (0,)), 1/(d**2*x**2))/(2*
pi**(3/2)*d**2) - meijerg(((-1, -1/2, -1/4, 0, 1/4, 1), ()), ((-1/4, 1/4), (-1, -1/2, 1/2, 0)), exp_polar(-2*I
*pi)/(d**2*x**2))/(2*pi**(3/2)*d**2)) + c*(I*meijerg(((-1/4, 1/4), (-1/2, 1/2, 1, 1)), ((-1/4, 0, 1/4, 1/2, 1,
0), ()), 1/(d**2*x**2))/(2*pi**(3/2)*d**3) + meijerg(((-3/2, -1, -3/4, -1/2, -1/4, 1), ()), ((-3/4, -1/4), (-
3/2, -1, 0, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(2*pi**(3/2)*d**3))

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Giac [B]  time = 1.18268, size = 246, normalized size = 6.15 \begin{align*} -\frac{2 \, c \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{d^{3}} + \frac{\frac{a d^{2}{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}} - \frac{b d{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}} + \frac{c{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}}}{4 \, d^{3}} - \frac{{\left (a d^{2} - b d + c\right )} \sqrt{d x + 1}}{4 \, d^{3}{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}} - \frac{{\left (a d^{5} + b d^{4} + c d^{3}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}}{2 \,{\left (d x - 1\right )} d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^3 + 1/4*(a*d^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - b*d*(sqrt(2
) - sqrt(-d*x + 1))/sqrt(d*x + 1) + c*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1))/d^3 - 1/4*(a*d^2 - b*d + c)*sq
rt(d*x + 1)/(d^3*(sqrt(2) - sqrt(-d*x + 1))) - 1/2*(a*d^5 + b*d^4 + c*d^3)*sqrt(d*x + 1)*sqrt(-d*x + 1)/((d*x
- 1)*d^6)