3.3 \(\int (A+B x+C x^2) \sqrt{d^2-e^2 x^2} \, dx\)

Optimal. Leaf size=125 \[ \frac{1}{8} x \sqrt{d^2-e^2 x^2} \left (4 A+\frac{C d^2}{e^2}\right )+\frac{d^2 \left (4 A e^2+C d^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2} \]

[Out]

((4*A + (C*d^2)/e^2)*x*Sqrt[d^2 - e^2*x^2])/8 - (B*(d^2 - e^2*x^2)^(3/2))/(3*e^2) - (C*x*(d^2 - e^2*x^2)^(3/2)
)/(4*e^2) + (d^2*(C*d^2 + 4*A*e^2)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^3)

________________________________________________________________________________________

Rubi [A]  time = 0.068973, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1815, 641, 195, 217, 203} \[ \frac{1}{8} x \sqrt{d^2-e^2 x^2} \left (4 A+\frac{C d^2}{e^2}\right )+\frac{d^2 \left (4 A e^2+C d^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2],x]

[Out]

((4*A + (C*d^2)/e^2)*x*Sqrt[d^2 - e^2*x^2])/8 - (B*(d^2 - e^2*x^2)^(3/2))/(3*e^2) - (C*x*(d^2 - e^2*x^2)^(3/2)
)/(4*e^2) + (d^2*(C*d^2 + 4*A*e^2)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^3)

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \left (A+B x+C x^2\right ) \sqrt{d^2-e^2 x^2} \, dx &=-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{\int \left (-C d^2-4 A e^2-4 B e^2 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{4 e^2}\\ &=-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{\left (-C d^2-4 A e^2\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{4 e^2}\\ &=\frac{\left (C d^2+4 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{\left (d^2 \left (-C d^2-4 A e^2\right )\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=\frac{\left (C d^2+4 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{\left (d^2 \left (-C d^2-4 A e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^2}\\ &=\frac{\left (C d^2+4 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}+\frac{d^2 \left (C d^2+4 A e^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}\\ \end{align*}

Mathematica [A]  time = 0.143962, size = 121, normalized size = 0.97 \[ \frac{\sqrt{d^2-e^2 x^2} \left (e \sqrt{1-\frac{e^2 x^2}{d^2}} \left (12 A e^2 x-8 B d^2+8 B e^2 x^2-3 C d^2 x+6 C e^2 x^3\right )+3 \left (4 A d e^2+C d^3\right ) \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{24 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)*Sqrt[d^2 - e^2*x^2],x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(e*Sqrt[1 - (e^2*x^2)/d^2]*(-8*B*d^2 - 3*C*d^2*x + 12*A*e^2*x + 8*B*e^2*x^2 + 6*C*e^2*x^3
) + 3*(C*d^3 + 4*A*d*e^2)*ArcSin[(e*x)/d]))/(24*e^3*Sqrt[1 - (e^2*x^2)/d^2])

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 154, normalized size = 1.2 \begin{align*} -{\frac{Cx}{4\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{C{d}^{2}x}{8\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{C{d}^{4}}{8\,{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{B}{3\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{Ax}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{A{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/4*C*x*(-e^2*x^2+d^2)^(3/2)/e^2+1/8*C*d^2/e^2*x*(-e^2*x^2+d^2)^(1/2)+1/8*C*d^4/e^2/(e^2)^(1/2)*arctan((e^2)^
(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/3*B*(-e^2*x^2+d^2)^(3/2)/e^2+1/2*A*x*(-e^2*x^2+d^2)^(1/2)+1/2*A*d^2/(e^2)^(1/2
)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.52992, size = 188, normalized size = 1.5 \begin{align*} \frac{A d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} + \frac{C d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{2}} + \frac{1}{2} \, \sqrt{-e^{2} x^{2} + d^{2}} A x + \frac{\sqrt{-e^{2} x^{2} + d^{2}} C d^{2} x}{8 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} C x}{4 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} B}{3 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*A*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) + 1/8*C*d^4*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^2) + 1/2*
sqrt(-e^2*x^2 + d^2)*A*x + 1/8*sqrt(-e^2*x^2 + d^2)*C*d^2*x/e^2 - 1/4*(-e^2*x^2 + d^2)^(3/2)*C*x/e^2 - 1/3*(-e
^2*x^2 + d^2)^(3/2)*B/e^2

________________________________________________________________________________________

Fricas [A]  time = 2.0362, size = 227, normalized size = 1.82 \begin{align*} -\frac{6 \,{\left (C d^{4} + 4 \, A d^{2} e^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (6 \, C e^{3} x^{3} + 8 \, B e^{3} x^{2} - 8 \, B d^{2} e - 3 \,{\left (C d^{2} e - 4 \, A e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/24*(6*(C*d^4 + 4*A*d^2*e^2)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (6*C*e^3*x^3 + 8*B*e^3*x^2 - 8*B*d^
2*e - 3*(C*d^2*e - 4*A*e^3)*x)*sqrt(-e^2*x^2 + d^2))/e^3

________________________________________________________________________________________

Sympy [C]  time = 6.47671, size = 347, normalized size = 2.78 \begin{align*} A \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + B \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) + C \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)*(-e**2*x**2+d**2)**(1/2),x)

[Out]

A*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**3/(2*d*sqrt(-1 + e**
2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True))
+ B*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) + C*Piecewise((-
I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 3*I*d*x**3/(8*sqrt(-1 + e**2*x**2
/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**4*asin(e*x/d)/(8*e**
3) - d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) - e**2*x**5/(4*d*sqrt(1
- e**2*x**2/d**2)), True))

________________________________________________________________________________________

Giac [A]  time = 1.14138, size = 115, normalized size = 0.92 \begin{align*} \frac{1}{8} \,{\left (C d^{4} + 4 \, A d^{2} e^{2}\right )} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{24} \,{\left (8 \, B d^{2} e^{\left (-2\right )} -{\left (2 \,{\left (3 \, C x + 4 \, B\right )} x - 3 \,{\left (C d^{2} e^{2} - 4 \, A e^{4}\right )} e^{\left (-4\right )}\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

1/8*(C*d^4 + 4*A*d^2*e^2)*arcsin(x*e/d)*e^(-3)*sgn(d) - 1/24*(8*B*d^2*e^(-2) - (2*(3*C*x + 4*B)*x - 3*(C*d^2*e
^2 - 4*A*e^4)*e^(-4))*x)*sqrt(-x^2*e^2 + d^2)