3.1184 \(\int \frac{\sqrt{a+b x+c x^2}}{b d+2 c d x} \, dx\)

Optimal. Leaf size=83 \[ \frac{\sqrt{a+b x+c x^2}}{2 c d}-\frac{\sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{4 c^{3/2} d} \]

[Out]

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

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Rubi [A]  time = 0.166222, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{a+b x+c x^2}}{2 c d}-\frac{\sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{4 c^{3/2} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.8945, size = 71, normalized size = 0.86 \[ \frac{\sqrt{a + b x + c x^{2}}}{2 c d} - \frac{\sqrt{- 4 a c + b^{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{4 c^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d),x)

[Out]

sqrt(a + b*x + c*x**2)/(2*c*d) - sqrt(-4*a*c + b**2)*atan(2*sqrt(c)*sqrt(a + b*x
+ c*x**2)/sqrt(-4*a*c + b**2))/(4*c**(3/2)*d)

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Mathematica [A]  time = 0.225447, size = 137, normalized size = 1.65 \[ \frac{\left (b^2-4 a c\right ) \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )+2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \log (b+2 c x)}{4 c^{3/2} d \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[-b^2 + 4*a*c]*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*Log[b + 2*c*
x] + (b^2 - 4*a*c)*Log[-(b^2*Sqrt[c]) + 4*a*c^(3/2) + 2*c*Sqrt[-b^2 + 4*a*c]*Sqr
t[a + x*(b + c*x)]])/(4*c^(3/2)*Sqrt[-b^2 + 4*a*c]*d)

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Maple [B]  time = 0.018, size = 244, normalized size = 2.9 \[{\frac{1}{4\,cd}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{a}{cd}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{{b}^{2}}{4\,{c}^{2}d}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d),x)

[Out]

1/4/d/c*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)-1/d/c/((4*a*c-b^2)/c)^(1/2)*ln((
1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1
/2))/(x+1/2*b/c))*a+1/4/d/c^2/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((
4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24789, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, \sqrt{c x^{2} + b x + a}}{8 \, c d}, -\frac{\sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (-\frac{b^{2} - 4 \, a c}{2 \, \sqrt{c x^{2} + b x + a} c \sqrt{\frac{b^{2} - 4 \, a c}{c}}}\right ) - 2 \, \sqrt{c x^{2} + b x + a}}{4 \, c d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="fricas")

[Out]

[1/8*(sqrt(-(b^2 - 4*a*c)/c)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(c*
x^2 + b*x + a)*c*sqrt(-(b^2 - 4*a*c)/c))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*sqrt(c
*x^2 + b*x + a))/(c*d), -1/4*(sqrt((b^2 - 4*a*c)/c)*arctan(-1/2*(b^2 - 4*a*c)/(s
qrt(c*x^2 + b*x + a)*c*sqrt((b^2 - 4*a*c)/c))) - 2*sqrt(c*x^2 + b*x + a))/(c*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d),x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(b + 2*c*x), x)/d

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GIAC/XCAS [A]  time = 0.223106, size = 131, normalized size = 1.58 \[ -\frac{{\left (b^{2} - 4 \, a c\right )} \arctan \left (-\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right )}{2 \, \sqrt{b^{2} c - 4 \, a c^{2}} c d} + \frac{\sqrt{c x^{2} + b x + a}}{2 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="giac")

[Out]

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