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

Optimal. Leaf size=102 \[ \frac{\left (8 a f-b \left (\frac{4 d f}{e}+e\right )\right ) \tanh ^{-1}\left (\frac{e+2 f x}{2 \sqrt{f} \sqrt{d+e x+f x^2}}\right )}{8 f^{3/2}}+\frac{b x \sqrt{d+e x+f x^2}}{2 e}+\frac{b \sqrt{d+e x+f x^2}}{4 f} \]

[Out]

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

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Rubi [A]  time = 0.222452, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{\left (8 a f-b \left (\frac{4 d f}{e}+e\right )\right ) \tanh ^{-1}\left (\frac{e+2 f x}{2 \sqrt{f} \sqrt{d+e x+f x^2}}\right )}{8 f^{3/2}}+\frac{b x \sqrt{d+e x+f x^2}}{2 e}+\frac{b \sqrt{d+e x+f x^2}}{4 f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.5069, size = 82, normalized size = 0.8 \[ \frac{b \left (\frac{e}{2} + f x\right ) \sqrt{d + e x + f x^{2}}}{2 e f} - \frac{\left (- 8 a e f + 4 b d f + b e^{2}\right ) \operatorname{atanh}{\left (\frac{e + 2 f x}{2 \sqrt{f} \sqrt{d + e x + f x^{2}}} \right )}}{8 e f^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*(e/2 + f*x)*sqrt(d + e*x + f*x**2)/(2*e*f) - (-8*a*e*f + 4*b*d*f + b*e**2)*ata
nh((e + 2*f*x)/(2*sqrt(f)*sqrt(d + e*x + f*x**2)))/(8*e*f**(3/2))

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Mathematica [A]  time = 0.242723, size = 85, normalized size = 0.83 \[ \frac{\left (8 a e f-b \left (4 d f+e^2\right )\right ) \log \left (2 \sqrt{f} \sqrt{d+x (e+f x)}+e+2 f x\right )+2 b \sqrt{f} (e+2 f x) \sqrt{d+x (e+f x)}}{8 e f^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 136, normalized size = 1.3 \[{a\ln \left ({1 \left ({\frac{e}{2}}+fx \right ){\frac{1}{\sqrt{f}}}}+\sqrt{f{x}^{2}+ex+d} \right ){\frac{1}{\sqrt{f}}}}+{\frac{b}{4\,f}\sqrt{f{x}^{2}+ex+d}}-{\frac{be}{8}\ln \left ({1 \left ({\frac{e}{2}}+fx \right ){\frac{1}{\sqrt{f}}}}+\sqrt{f{x}^{2}+ex+d} \right ){f}^{-{\frac{3}{2}}}}+{\frac{bx}{2\,e}\sqrt{f{x}^{2}+ex+d}}-{\frac{bd}{2\,e}\ln \left ({1 \left ({\frac{e}{2}}+fx \right ){\frac{1}{\sqrt{f}}}}+\sqrt{f{x}^{2}+ex+d} \right ){\frac{1}{\sqrt{f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*ln((1/2*e+f*x)/f^(1/2)+(f*x^2+e*x+d)^(1/2))/f^(1/2)+1/4*b*(f*x^2+e*x+d)^(1/2)/
f-1/8*e*b/f^(3/2)*ln((1/2*e+f*x)/f^(1/2)+(f*x^2+e*x+d)^(1/2))+1/2*b*x*(f*x^2+e*x
+d)^(1/2)/e-1/2/e*b/f^(1/2)*d*ln((1/2*e+f*x)/f^(1/2)+(f*x^2+e*x+d)^(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((b*f*x^2/e + b*x + a)/sqrt(f*x^2 + e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.369598, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b f x + b e\right )} \sqrt{f x^{2} + e x + d} \sqrt{f} -{\left (b e^{2} + 4 \,{\left (b d - 2 \, a e\right )} f\right )} \log \left (-4 \,{\left (2 \, f^{2} x + e f\right )} \sqrt{f x^{2} + e x + d} -{\left (8 \, f^{2} x^{2} + 8 \, e f x + e^{2} + 4 \, d f\right )} \sqrt{f}\right )}{16 \, e f^{\frac{3}{2}}}, \frac{2 \,{\left (2 \, b f x + b e\right )} \sqrt{f x^{2} + e x + d} \sqrt{-f} -{\left (b e^{2} + 4 \,{\left (b d - 2 \, a e\right )} f\right )} \arctan \left (\frac{{\left (2 \, f x + e\right )} \sqrt{-f}}{2 \, \sqrt{f x^{2} + e x + d} f}\right )}{8 \, e \sqrt{-f} f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.284918, size = 113, normalized size = 1.11 \[ \frac{1}{4} \, \sqrt{f x^{2} + x e + d}{\left (2 \, b x e^{\left (-1\right )} + \frac{b}{f}\right )} + \frac{{\left (4 \, b d f - 8 \, a f e + b e^{2}\right )} e^{\left (-1\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{f} x - \sqrt{f x^{2} + x e + d}\right )} \sqrt{f} - e \right |}\right )}{8 \, f^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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