3.1185 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^2} \, dx\)

Optimal. Leaf size=75 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 c d^2 (b+2 c x)} \]

[Out]

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

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Rubi [A]  time = 0.0870053, antiderivative size = 75, 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 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2} d^2}-\frac{\sqrt{a+b x+c x^2}}{2 c d^2 (b+2 c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.8113, size = 65, normalized size = 0.87 \[ - \frac{\sqrt{a + b x + c x^{2}}}{2 c d^{2} \left (b + 2 c x\right )} + \frac{\operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{4 c^{\frac{3}{2}} d^{2}} \]

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)**2,x)

[Out]

-sqrt(a + b*x + c*x**2)/(2*c*d**2*(b + 2*c*x)) + atanh((b + 2*c*x)/(2*sqrt(c)*sq
rt(a + b*x + c*x**2)))/(4*c**(3/2)*d**2)

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Mathematica [A]  time = 0.128162, size = 70, normalized size = 0.93 \[ \frac{\frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{4 c^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{2 b c+4 c^2 x}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 291, normalized size = 3.9 \[ -{\frac{1}{c{d}^{2} \left ( 4\,ac-{b}^{2} \right ) } \left ( \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}}+{\frac{x}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}+{\frac{b}{2\,c{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}}}+{\frac{a}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ( \sqrt{c} \left ( x+{\frac{b}{2\,c}} \right ) +\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{b}^{2}}{4\,{d}^{2} \left ( 4\,ac-{b}^{2} \right ) }\ln \left ( \sqrt{c} \left ( x+{\frac{b}{2\,c}} \right ) +\sqrt{ \left ( x+{\frac{b}{2\,c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{4\,c}}} \right ){c}^{-{\frac{3}{2}}}} \]

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)^2,x)

[Out]

-1/c/d^2/(4*a*c-b^2)/(x+1/2*b/c)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)+1/d^2
/(4*a*c-b^2)*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*x+1/2/c/d^2/(4*a*c-b^2)*(
(x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)*b+1/c^(1/2)/d^2/(4*a*c-b^2)*ln(c^(1/2)*
(x+1/2*b/c)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*a-1/4/c^(3/2)/d^2/(4*a*c-
b^2)*ln(c^(1/2)*(x+1/2*b/c)+((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2))*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254599, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, c x + b\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) - 4 \, \sqrt{c x^{2} + b x + a} \sqrt{c}}{8 \,{\left (2 \, c^{2} d^{2} x + b c d^{2}\right )} \sqrt{c}}, \frac{{\left (2 \, c x + b\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) - 2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c}}{4 \,{\left (2 \, c^{2} d^{2} x + b c d^{2}\right )} \sqrt{-c}}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx}{d^{2}} \]

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)**2,x)

[Out]

Integral(sqrt(a + b*x + c*x**2)/(b**2 + 4*b*c*x + 4*c**2*x**2), x)/d**2

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GIAC/XCAS [A]  time = 0.228096, size = 261, normalized size = 3.48 \[ -\frac{1}{4} \, d^{2}{\left (\frac{{\left (\frac{c \arctan \left (\frac{\sqrt{-\frac{b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \sqrt{-\frac{b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}\right )}{\rm sign}\left (\frac{1}{2 \, c d x + b d}\right ){\rm sign}\left (c\right ){\rm sign}\left (d\right )}{c^{2} d^{4}{\left | c \right |}} - \frac{{\left (c \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) + \sqrt{-c} \sqrt{c}\right )}{\rm sign}\left (\frac{1}{2 \, c d x + b d}\right ){\rm sign}\left (c\right ){\rm sign}\left (d\right )}{\sqrt{-c} c^{2} d^{4}{\left | c \right |}}\right )}{\left | c \right |} \]

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)^2,x, algorithm="giac")

[Out]

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