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

Optimal. Leaf size=166 \[ \frac{\sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )}{8 d^5}-\frac{x \sqrt{1-d^2 x^2} \left (c \left (8 a+\frac{3 c}{d^2}\right )+4 b^2\right )}{8 d^2}-\frac{2 b \sqrt{1-d^2 x^2} \left (3 a d^2+2 c\right )}{3 d^4}-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2} \]

[Out]

(-2*b*(2*c + 3*a*d^2)*Sqrt[1 - d^2*x^2])/(3*d^4) - ((4*b^2 + c*(8*a + (3*c)/d^2))*x*Sqrt[1 - d^2*x^2])/(8*d^2)
- (2*b*c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - (c^2*x^3*Sqrt[1 - d^2*x^2])/(4*d^2) + ((3*c^2 + 4*b^2*d^2 + 8*a*c*d
^2 + 8*a^2*d^4)*ArcSin[d*x])/(8*d^5)

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Rubi [A]  time = 0.31886, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {899, 1815, 641, 216} \[ \frac{\sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )}{8 d^5}-\frac{x \sqrt{1-d^2 x^2} \left (c \left (8 a+\frac{3 c}{d^2}\right )+4 b^2\right )}{8 d^2}-\frac{2 b \sqrt{1-d^2 x^2} \left (3 a d^2+2 c\right )}{3 d^4}-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(2*c + 3*a*d^2)*Sqrt[1 - d^2*x^2])/(3*d^4) - ((4*b^2 + c*(8*a + (3*c)/d^2))*x*Sqrt[1 - d^2*x^2])/(8*d^2)
- (2*b*c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - (c^2*x^3*Sqrt[1 - d^2*x^2])/(4*d^2) + ((3*c^2 + 4*b^2*d^2 + 8*a*c*d
^2 + 8*a^2*d^4)*ArcSin[d*x])/(8*d^5)

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 1815

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[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{\left (a+b x+c x^2\right )^2}{\sqrt{1-d x} \sqrt{1+d x}} \, dx &=\int \frac{\left (a+b x+c x^2\right )^2}{\sqrt{1-d^2 x^2}} \, dx\\ &=-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2}-\frac{\int \frac{-4 a^2 d^2-8 a b d^2 x-\left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x^2-8 b c d^2 x^3}{\sqrt{1-d^2 x^2}} \, dx}{4 d^2}\\ &=-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2}+\frac{\int \frac{12 a^2 d^4+8 b d^2 \left (2 c+3 a d^2\right ) x+3 d^2 \left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x^2}{\sqrt{1-d^2 x^2}} \, dx}{12 d^4}\\ &=-\frac{\left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^4}-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2}-\frac{\int \frac{-3 d^2 \left (3 c^2+4 b^2 d^2+8 a c d^2+8 a^2 d^4\right )-16 b d^4 \left (2 c+3 a d^2\right ) x}{\sqrt{1-d^2 x^2}} \, dx}{24 d^6}\\ &=-\frac{2 b \left (2 c+3 a d^2\right ) \sqrt{1-d^2 x^2}}{3 d^4}-\frac{\left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^4}-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2}+\frac{\left (3 c^2+4 b^2 d^2+8 a c d^2+8 a^2 d^4\right ) \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{8 d^4}\\ &=-\frac{2 b \left (2 c+3 a d^2\right ) \sqrt{1-d^2 x^2}}{3 d^4}-\frac{\left (3 c^2+4 b^2 d^2+8 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^4}-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2}+\frac{\left (3 c^2+4 b^2 d^2+8 a c d^2+8 a^2 d^4\right ) \sin ^{-1}(d x)}{8 d^5}\\ \end{align*}

Mathematica [A]  time = 0.118725, size = 114, normalized size = 0.69 \[ \frac{3 \sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )-d \sqrt{1-d^2 x^2} \left (16 b \left (3 a d^2+c d^2 x^2+2 c\right )+3 c x \left (8 a d^2+2 c d^2 x^2+3 c\right )+12 b^2 d^2 x\right )}{24 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(d*Sqrt[1 - d^2*x^2]*(12*b^2*d^2*x + 16*b*(2*c + 3*a*d^2 + c*d^2*x^2) + 3*c*x*(3*c + 8*a*d^2 + 2*c*d^2*x^2))
) + 3*(3*c^2 + 4*b^2*d^2 + 8*a*c*d^2 + 8*a^2*d^4)*ArcSin[d*x])/(24*d^5)

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Maple [C]  time = 0.171, size = 291, normalized size = 1.8 \begin{align*} -{\frac{{\it csgn} \left ( d \right ) }{24\,{d}^{5}}\sqrt{-dx+1}\sqrt{dx+1} \left ( 6\,{\it csgn} \left ( d \right ){x}^{3}{c}^{2}{d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}+16\,{\it csgn} \left ( d \right ){x}^{2}bc{d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}+24\,{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}xac+12\,{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}x{b}^{2}+48\,{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}ab-24\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){a}^{2}{d}^{4}+9\,{\it csgn} \left ( d \right ) d\sqrt{-{d}^{2}{x}^{2}+1}x{c}^{2}+32\,{\it csgn} \left ( d \right ) d\sqrt{-{d}^{2}{x}^{2}+1}bc-24\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) ac{d}^{2}-12\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){b}^{2}{d}^{2}-9\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){c}^{2} \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(6*csgn(d)*x^3*c^2*d^3*(-d^2*x^2+1)^(1/2)+16*csgn(d)*x^2*b*c*d^3*(-d^2*x^2+
1)^(1/2)+24*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x*a*c+12*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x*b^2+48*csgn(d)*d^3*(-d^2*
x^2+1)^(1/2)*a*b-24*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a^2*d^4+9*csgn(d)*d*(-d^2*x^2+1)^(1/2)*x*c^2+32*csg
n(d)*d*(-d^2*x^2+1)^(1/2)*b*c-24*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a*c*d^2-12*arctan(csgn(d)*d*x/(-d^2*x^
2+1)^(1/2))*b^2*d^2-9*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*c^2)*csgn(d)/d^5/(-d^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.99035, size = 275, normalized size = 1.66 \begin{align*} -\frac{\sqrt{-d^{2} x^{2} + 1} c^{2} x^{3}}{4 \, d^{2}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1} b c x^{2}}{3 \, d^{2}} + \frac{a^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1} a b}{d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1}{\left (b^{2} + 2 \, a c\right )} x}{2 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1} c^{2} x}{8 \, d^{4}} + \frac{{\left (b^{2} + 2 \, a c\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} - \frac{4 \, \sqrt{-d^{2} x^{2} + 1} b c}{3 \, d^{4}} + \frac{3 \, c^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-d^2*x^2 + 1)*c^2*x^3/d^2 - 2/3*sqrt(-d^2*x^2 + 1)*b*c*x^2/d^2 + a^2*arcsin(d^2*x/sqrt(d^2))/sqrt(d^
2) - 2*sqrt(-d^2*x^2 + 1)*a*b/d^2 - 1/2*sqrt(-d^2*x^2 + 1)*(b^2 + 2*a*c)*x/d^2 - 3/8*sqrt(-d^2*x^2 + 1)*c^2*x/
d^4 + 1/2*(b^2 + 2*a*c)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^2) - 4/3*sqrt(-d^2*x^2 + 1)*b*c/d^4 + 3/8*c^2*arc
sin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^4)

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Fricas [A]  time = 1.39013, size = 313, normalized size = 1.89 \begin{align*} -\frac{{\left (6 \, c^{2} d^{3} x^{3} + 16 \, b c d^{3} x^{2} + 48 \, a b d^{3} + 32 \, b c d + 3 \,{\left (4 \,{\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 6 \,{\left (8 \, a^{2} d^{4} + 4 \,{\left (b^{2} + 2 \, a c\right )} d^{2} + 3 \, c^{2}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{24 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((6*c^2*d^3*x^3 + 16*b*c*d^3*x^2 + 48*a*b*d^3 + 32*b*c*d + 3*(4*(b^2 + 2*a*c)*d^3 + 3*c^2*d)*x)*sqrt(d*x
+ 1)*sqrt(-d*x + 1) + 6*(8*a^2*d^4 + 4*(b^2 + 2*a*c)*d^2 + 3*c^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(
d*x)))/d^5

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Sympy [C]  time = 141.49, size = 631, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*a**2*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), 1/(d**2*x**2))/(4*pi**(3/2)*d)
+ a**2*meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), exp_polar(-2*I*pi)/(d**2*x
**2))/(4*pi**(3/2)*d) - I*a*b*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))/(2*pi**(3/2)*d**2) - a*b*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))/(2*pi**(3/2)*d**2) - I*a*c*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))/(2*pi**(3/2)*d**3) + a*c*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))/(2*pi**(3/2)*d**3) - I*b**2*mei
jerg(((-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**2*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*b*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))/(2*pi**(3/2)*d**4) - b*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))/(2*pi**(3/2)*d**4) - I*c**2*meijerg(((-7/4, -5/4), (-3/
2, -3/2, -1, 1)), ((-2, -7/4, -3/2, -5/4, -1, 0), ()), 1/(d**2*x**2))/(4*pi**(3/2)*d**5) + c**2*meijerg(((-5/2
, -9/4, -2, -7/4, -3/2, 1), ()), ((-9/4, -7/4), (-5/2, -2, -2, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/
2)*d**5)

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Giac [A]  time = 1.7079, size = 243, normalized size = 1.46 \begin{align*} -\frac{{\left (48 \, a b d^{19} - 12 \, b^{2} d^{18} - 24 \, a c d^{18} + 48 \, b c d^{17} - 15 \, c^{2} d^{16} +{\left (12 \, b^{2} d^{18} + 24 \, a c d^{18} - 32 \, b c d^{17} + 27 \, c^{2} d^{16} + 2 \,{\left (3 \,{\left (d x + 1\right )} c^{2} d^{16} + 8 \, b c d^{17} - 9 \, c^{2} d^{16}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 6 \,{\left (8 \, a^{2} d^{20} + 4 \, b^{2} d^{18} + 8 \, a c d^{18} + 3 \, c^{2} d^{16}\right )} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{344064 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/344064*((48*a*b*d^19 - 12*b^2*d^18 - 24*a*c*d^18 + 48*b*c*d^17 - 15*c^2*d^16 + (12*b^2*d^18 + 24*a*c*d^18 -
32*b*c*d^17 + 27*c^2*d^16 + 2*(3*(d*x + 1)*c^2*d^16 + 8*b*c*d^17 - 9*c^2*d^16)*(d*x + 1))*(d*x + 1))*sqrt(d*x
+ 1)*sqrt(-d*x + 1) - 6*(8*a^2*d^20 + 4*b^2*d^18 + 8*a*c*d^18 + 3*c^2*d^16)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))
)/d