3.1227 \(\int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{\sqrt{c} d \sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.108271, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{\sqrt{c} d \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.8178, size = 51, normalized size = 0.93 \[ \frac{\operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{c} d \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.120188, size = 83, normalized size = 1.51 \[ \frac{\log (b+2 c x)-\log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )}{\sqrt{c} d \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 101, normalized size = 1.8 \[ -{\frac{1}{cd}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258336, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c\right )} \sqrt{-b^{2} c + 4 \, a c^{2}} + 4 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right )}{2 \, \sqrt{-b^{2} c + 4 \, a c^{2}} d}, -\frac{\arctan \left (\frac{\sqrt{b^{2} c - 4 \, a c^{2}}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{\sqrt{b^{2} c - 4 \, a c^{2}} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{b \sqrt{a + b x + c x^{2}} + 2 c x \sqrt{a + b x + c x^{2}}}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(b*sqrt(a + b*x + c*x**2) + 2*c*x*sqrt(a + b*x + c*x**2)), x)/d

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GIAC/XCAS [A]  time = 0.230201, size = 88, normalized size = 1.6 \[ \frac{2 \, \arctan \left (-\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right )}{\sqrt{b^{2} c - 4 \, a c^{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)),x, algorithm="giac")

[Out]

2*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)*d)