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

Optimal. Leaf size=116 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a f+b e)+3 b^2 f+8 c^2 d\right )}{8 c^{5/2}}+\frac{\sqrt{a+b x+c x^2} (4 c e-3 b f)}{4 c^2}+\frac{f x \sqrt{a+b x+c x^2}}{2 c} \]

[Out]

((4*c*e - 3*b*f)*Sqrt[a + b*x + c*x^2])/(4*c^2) + (f*x*Sqrt[a + b*x + c*x^2])/(2
*c) + ((8*c^2*d + 3*b^2*f - 4*c*(b*e + a*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])])/(8*c^(5/2))

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Rubi [A]  time = 0.214815, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a f+b e)+3 b^2 f+8 c^2 d\right )}{8 c^{5/2}}+\frac{\sqrt{a+b x+c x^2} (4 c e-3 b f)}{4 c^2}+\frac{f x \sqrt{a+b x+c x^2}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

((4*c*e - 3*b*f)*Sqrt[a + b*x + c*x^2])/(4*c^2) + (f*x*Sqrt[a + b*x + c*x^2])/(2
*c) + ((8*c^2*d + 3*b^2*f - 4*c*(b*e + a*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt
[a + b*x + c*x^2])])/(8*c^(5/2))

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Rubi in Sympy [A]  time = 15.7553, size = 97, normalized size = 0.84 \[ - \frac{\sqrt{a + b x + c x^{2}} \left (\frac{3 b f}{2} - 2 c e - c f x\right )}{2 c^{2}} + \frac{\left (- 4 a c f + 3 b^{2} f - 4 b c e + 8 c^{2} d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x + c*x**2)*(3*b*f/2 - 2*c*e - c*f*x)/(2*c**2) + (-4*a*c*f + 3*b**2*
f - 4*b*c*e + 8*c**2*d)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(8
*c**(5/2))

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Mathematica [A]  time = 0.147425, size = 95, normalized size = 0.82 \[ \frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (-4 c (a f+b e)+3 b^2 f+8 c^2 d\right )+2 \sqrt{c} \sqrt{a+x (b+c x)} (-3 b f+4 c e+2 c f x)}{8 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*(4*c*e - 3*b*f + 2*c*f*x)*Sqrt[a + x*(b + c*x)] + (8*c^2*d + 3*b^2*f
- 4*c*(b*e + a*f))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(8*c^(5/2))

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Maple [A]  time = 0.009, size = 185, normalized size = 1.6 \[{d\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{e}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{be}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{fx}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,bf}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}f}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{fa}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+e/c*(c*x^2+b*x+a)^(1/2)-1/
2*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/2*f*x*(c*x^2+b*x+a)^
(1/2)/c-3/4*f*b/c^2*(c*x^2+b*x+a)^(1/2)+3/8*f*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))-1/2*f*a/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.571157, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, c f x + 4 \, c e - 3 \, b f\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} -{\left (8 \, c^{2} d - 4 \, b c e +{\left (3 \, b^{2} - 4 \, a c\right )} f\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{16 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (2 \, c f x + 4 \, c e - 3 \, b f\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} +{\left (8 \, c^{2} d - 4 \, b c e +{\left (3 \, b^{2} - 4 \, a c\right )} f\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{8 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*c*f*x + 4*c*e - 3*b*f)*sqrt(c*x^2 + b*x + a)*sqrt(c) - (8*c^2*d - 4*
b*c*e + (3*b^2 - 4*a*c)*f)*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)))/c^(5/2), 1/8*(2*(2*c*f*x + 4*c*e - 3*b*f)
*sqrt(c*x^2 + b*x + a)*sqrt(-c) + (8*c^2*d - 4*b*c*e + (3*b^2 - 4*a*c)*f)*arctan
(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282181, size = 132, normalized size = 1.14 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (\frac{2 \, f x}{c} - \frac{3 \, b f - 4 \, c e}{c^{2}}\right )} - \frac{{\left (8 \, c^{2} d + 3 \, b^{2} f - 4 \, a c f - 4 \, b c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*x^2 + b*x + a)*(2*f*x/c - (3*b*f - 4*c*e)/c^2) - 1/8*(8*c^2*d + 3*b^2
*f - 4*a*c*f - 4*b*c*e)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) -
b))/c^(5/2)