3.30 \(\int \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx\)

Optimal. Leaf size=177 \[ \frac{\left (a^2 C+2 A b^2\right ) \sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{B \left (a^2-b^2 x^2\right )}{b^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C x \left (a^2-b^2 x^2\right )}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}} \]

[Out]

-((B*(a^2 - b^2*x^2))/(b^2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])) - (C*x*(a^2 - b^2*x^2))/(2*b^2*Sqrt[a + b*x]*Sqrt
[a*c - b*c*x]) + ((2*A*b^2 + a^2*C)*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(2*
b^3*Sqrt[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.124376, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {901, 1815, 641, 217, 203} \[ \frac{\left (a^2 C+2 A b^2\right ) \sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{B \left (a^2-b^2 x^2\right )}{b^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C x \left (a^2-b^2 x^2\right )}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]

[Out]

-((B*(a^2 - b^2*x^2))/(b^2*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])) - (C*x*(a^2 - b^2*x^2))/(2*b^2*Sqrt[a + b*x]*Sqrt
[a*c - b*c*x]) + ((2*A*b^2 + a^2*C)*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(2*
b^3*Sqrt[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

Rule 901

Int[((d_) + (e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :>
Dist[((d + e*x)^FracPart[m]*(f + g*x)^FracPart[m])/(d*f + e*g*x^2)^FracPart[m], Int[(d*f + e*g*x^2)^m*(a + b*x
+ c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m - n, 0] && EqQ[e*f + d*g, 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 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 \frac{A+B x+C x^2}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx &=\frac{\sqrt{a^2 c-b^2 c x^2} \int \frac{A+B x+C x^2}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{\sqrt{a+b x} \sqrt{a c-b c x}}\\ &=-\frac{C x \left (a^2-b^2 x^2\right )}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{\sqrt{a^2 c-b^2 c x^2} \int \frac{-c \left (2 A b^2+a^2 C\right )-2 b^2 B c x}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{2 b^2 c \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=-\frac{B \left (a^2-b^2 x^2\right )}{b^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C x \left (a^2-b^2 x^2\right )}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (\left (2 A b^2+a^2 C\right ) \sqrt{a^2 c-b^2 c x^2}\right ) \int \frac{1}{\sqrt{a^2 c-b^2 c x^2}} \, dx}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=-\frac{B \left (a^2-b^2 x^2\right )}{b^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C x \left (a^2-b^2 x^2\right )}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (\left (2 A b^2+a^2 C\right ) \sqrt{a^2 c-b^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1+b^2 c x^2} \, dx,x,\frac{x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}\\ &=-\frac{B \left (a^2-b^2 x^2\right )}{b^2 \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C x \left (a^2-b^2 x^2\right )}{2 b^2 \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (2 A b^2+a^2 C\right ) \sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right )}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}\\ \end{align*}

Mathematica [A]  time = 0.414243, size = 169, normalized size = 0.95 \[ -\frac{\sqrt{a-b x} \left (\sqrt{\frac{b x}{a}+1} \left (4 \tan ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{a+b x}}\right ) \left (a (a C-b B)+A b^2\right )+b \sqrt{a-b x} \sqrt{a+b x} (2 B+C x)\right )-2 \sqrt{a} \sqrt{a+b x} (a C-2 b B) \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )\right )}{2 b^3 \sqrt{\frac{b x}{a}+1} \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]),x]

[Out]

-(Sqrt[a - b*x]*(-2*Sqrt[a]*(-2*b*B + a*C)*Sqrt[a + b*x]*ArcSin[Sqrt[a - b*x]/(Sqrt[2]*Sqrt[a])] + Sqrt[1 + (b
*x)/a]*(b*Sqrt[a - b*x]*Sqrt[a + b*x]*(2*B + C*x) + 4*(A*b^2 + a*(-(b*B) + a*C))*ArcTan[Sqrt[a - b*x]/Sqrt[a +
b*x]])))/(2*b^3*Sqrt[c*(a - b*x)]*Sqrt[1 + (b*x)/a])

________________________________________________________________________________________

Maple [A]  time = 0.018, size = 180, normalized size = 1. \begin{align*}{\frac{1}{2\,{b}^{2}c}\sqrt{bx+a}\sqrt{-c \left ( bx-a \right ) } \left ( 2\,A\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ){b}^{2}c+C\arctan \left ({x\sqrt{{b}^{2}c}{\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}} \right ){a}^{2}c-C\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }x-2\,B\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{{b}^{2}c} \right ){\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}{\frac{1}{\sqrt{{b}^{2}c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)

[Out]

1/2*(b*x+a)^(1/2)*(-c*(b*x-a))^(1/2)/b^2*(2*A*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*b^2*c+C*arctan(
(b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*a^2*c-C*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)*x-2*B*(-c*(b^2*x^2-a^
2))^(1/2)*(b^2*c)^(1/2))/(-c*(b^2*x^2-a^2))^(1/2)/c/(b^2*c)^(1/2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.6435, size = 460, normalized size = 2.6 \begin{align*} \left [-\frac{{\left (C a^{2} + 2 \, A b^{2}\right )} \sqrt{-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) + 2 \,{\left (C b x + 2 \, B b\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{4 \, b^{3} c}, -\frac{{\left (C a^{2} + 2 \, A b^{2}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{c} x}{b^{2} c x^{2} - a^{2} c}\right ) +{\left (C b x + 2 \, B b\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{2 \, b^{3} c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*((C*a^2 + 2*A*b^2)*sqrt(-c)*log(2*b^2*c*x^2 - 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c) +
2*(C*b*x + 2*B*b)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(b^3*c), -1/2*((C*a^2 + 2*A*b^2)*sqrt(c)*arctan(sqrt(-b*c
*x + a*c)*sqrt(b*x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c)) + (C*b*x + 2*B*b)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(
b^3*c)]

________________________________________________________________________________________

Sympy [C]  time = 25.8753, size = 338, normalized size = 1.91 \begin{align*} - \frac{i A{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{c}} + \frac{A{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{c}} - \frac{i B 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{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b^{2} \sqrt{c}} - \frac{B 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{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b^{2} \sqrt{c}} - \frac{i C a^{2}{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{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b^{3} \sqrt{c}} + \frac{C a^{2}{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{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b^{3} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out