3.15 \(\int \frac{x (a+b x+c x^2)}{\sqrt{1-d x} \sqrt{1+d x}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{\sqrt{1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac{b \sin ^{-1}(d x)}{2 d^3}-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2} \]

[Out]

-(c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - ((2*(2*c + 3*a*d^2) + 3*b*d^2*x)*Sqrt[1 - d^2*x^2])/(6*d^4) + (b*ArcSin[d
*x])/(2*d^3)

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Rubi [A]  time = 0.138713, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1609, 1809, 780, 216} \[ -\frac{\sqrt{1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac{b \sin ^{-1}(d x)}{2 d^3}-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - ((2*(2*c + 3*a*d^2) + 3*b*d^2*x)*Sqrt[1 - d^2*x^2])/(6*d^4) + (b*ArcSin[d
*x])/(2*d^3)

Rule 1609

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

Rule 1809

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

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 \left (a+b x+c x^2\right )}{\sqrt{1-d x} \sqrt{1+d x}} \, dx &=\int \frac{x \left (a+b x+c x^2\right )}{\sqrt{1-d^2 x^2}} \, dx\\ &=-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{\int \frac{x \left (-2 c-3 a d^2-3 b d^2 x\right )}{\sqrt{1-d^2 x^2}} \, dx}{3 d^2}\\ &=-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \sqrt{1-d^2 x^2}}{6 d^4}+\frac{b \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{2 d^2}\\ &=-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \sqrt{1-d^2 x^2}}{6 d^4}+\frac{b \sin ^{-1}(d x)}{2 d^3}\\ \end{align*}

Mathematica [A]  time = 0.0605943, size = 57, normalized size = 0.72 \[ \frac{3 b d \sin ^{-1}(d x)-\sqrt{1-d^2 x^2} \left (3 d^2 (2 a+b x)+2 c \left (d^2 x^2+2\right )\right )}{6 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[1 - d^2*x^2]*(3*d^2*(2*a + b*x) + 2*c*(2 + d^2*x^2))) + 3*b*d*ArcSin[d*x])/(6*d^4)

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Maple [C]  time = 0., size = 139, normalized size = 1.8 \begin{align*} -{\frac{{\it csgn} \left ( d \right ) }{6\,{d}^{4}}\sqrt{-dx+1}\sqrt{dx+1} \left ( 2\,{\it csgn} \left ( d \right ){x}^{2}c{d}^{2}\sqrt{-{d}^{2}{x}^{2}+1}+3\,\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ) xb{d}^{2}+6\,{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}a{d}^{2}+4\,{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}c-3\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) bd \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(2*csgn(d)*x^2*c*d^2*(-d^2*x^2+1)^(1/2)+3*(-d^2*x^2+1)^(1/2)*csgn(d)*x*b*d^2
+6*csgn(d)*(-d^2*x^2+1)^(1/2)*a*d^2+4*csgn(d)*(-d^2*x^2+1)^(1/2)*c-3*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*b*
d)*csgn(d)/d^4/(-d^2*x^2+1)^(1/2)

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Maxima [A]  time = 4.96562, size = 134, normalized size = 1.7 \begin{align*} -\frac{\sqrt{-d^{2} x^{2} + 1} c x^{2}}{3 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1} b x}{2 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1} a}{d^{2}} + \frac{b \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1} c}{3 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.14215, size = 189, normalized size = 2.39 \begin{align*} -\frac{6 \, b d \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right ) +{\left (2 \, c d^{2} x^{2} + 3 \, b d^{2} x + 6 \, a d^{2} + 4 \, c\right )} \sqrt{d x + 1} \sqrt{-d x + 1}}{6 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*b*d*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)) + (2*c*d^2*x^2 + 3*b*d^2*x + 6*a*d^2 + 4*c)*sqrt(
d*x + 1)*sqrt(-d*x + 1))/d^4

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Sympy [C]  time = 46.387, size = 313, normalized size = 3.96 \begin{align*} - \frac{i a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{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 )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{i b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} - \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*a*meijerg(((-1/4, 1/4), (0, 0, 1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0), ()), 1/(d**2*x**2))/(4*pi**(3/2)*d*
*2) - a*meijerg(((-1, -3/4, -1/2, -1/4, 0, 1), ()), ((-3/4, -1/4), (-1, -1/2, -1/2, 0)), exp_polar(-2*I*pi)/(d
**2*x**2))/(4*pi**(3/2)*d**2) - I*b*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0),
()), 1/(d**2*x**2))/(4*pi**(3/2)*d**3) + b*meijerg(((-3/2, -5/4, -1, -3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/
2, -1, -1, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d**3) - I*c*meijerg(((-5/4, -3/4), (-1, -1, -1/2,
1)), ((-3/2, -5/4, -1, -3/4, -1/2, 0), ()), 1/(d**2*x**2))/(4*pi**(3/2)*d**4) - c*meijerg(((-2, -7/4, -3/2, -
5/4, -1, 1), ()), ((-7/4, -5/4), (-2, -3/2, -3/2, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d**4)

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Giac [A]  time = 2.26382, size = 123, normalized size = 1.56 \begin{align*} \frac{6 \, b d^{10} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right ) -{\left (6 \, a d^{11} - 3 \, b d^{10} + 6 \, c d^{9} +{\left (2 \,{\left (d x + 1\right )} c d^{9} + 3 \, b d^{10} - 4 \, c d^{9}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}}{3840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(6*b*d^10*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)) - (6*a*d^11 - 3*b*d^10 + 6*c*d^9 + (2*(d*x + 1)*c*d^9 + 3*b
*d^10 - 4*c*d^9)*(d*x + 1))*sqrt(d*x + 1)*sqrt(-d*x + 1))/d