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

Optimal. Leaf size=55 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}+\frac{e \sqrt{b x+c x^2}}{c} \]

[Out]

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

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Rubi [A]  time = 0.0701547, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}+\frac{e \sqrt{b x+c x^2}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.09382, size = 48, normalized size = 0.87 \[ \frac{e \sqrt{b x + c x^{2}}}{c} - \frac{\left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0682738, size = 79, normalized size = 1.44 \[ \frac{\sqrt{x} \sqrt{b+c x} (2 c d-b e) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )+\sqrt{c} e x (b+c x)}{c^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 78, normalized size = 1.4 \[{d\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}+{\frac{e}{c}\sqrt{c{x}^{2}+bx}}-{\frac{be}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229963, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{c x^{2} + b x} \sqrt{c} e -{\left (2 \, c d - b e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{2 \, c^{\frac{3}{2}}}, \frac{\sqrt{c x^{2} + b x} \sqrt{-c} e +{\left (2 \, c d - b e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{\sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(c*x^2 + b*x)*sqrt(c)*e - (2*c*d - b*e)*log((2*c*x + b)*sqrt(c) - 2*
sqrt(c*x^2 + b*x)*c))/c^(3/2), (sqrt(c*x^2 + b*x)*sqrt(-c)*e + (2*c*d - b*e)*arc
tan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23081, size = 85, normalized size = 1.55 \[ \frac{\sqrt{c x^{2} + b x} e}{c} - \frac{{\left (2 \, c d - b e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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