3.2365 \(\int \frac{d+e x}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+b x+c x^2}}{c} \]

[Out]

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

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Rubi [A]  time = 0.0744271, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+b x+c x^2}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.67094, size = 60, normalized size = 0.88 \[ \frac{e \sqrt{a + b x + c x^{2}}}{c} - \frac{\left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.141485, size = 64, normalized size = 0.94 \[ \frac{(2 c d-b e) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{2 c^{3/2}}+\frac{e \sqrt{a+x (b+c x)}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 81, normalized size = 1.2 \[{d\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{e}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{be}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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*x^2+b*x+a)^(1/2)/c-1/
2*e*b/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((e*x + d)/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281656, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c x^{2} + b x + a} \sqrt{c} e -{\left (2 \, c d - b e\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{4 \, c^{\frac{3}{2}}}, \frac{2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c} e +{\left (2 \, c d - b e\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{2 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224335, size = 88, normalized size = 1.29 \[ \frac{\sqrt{c x^{2} + b x + a} e}{c} - \frac{{\left (2 \, c d - b e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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