3.77 \(\int \frac{\sqrt{b x+c x^2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 \sqrt{b x+c x^2}}{\sqrt{x}}-2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right ) \]

[Out]

(2*Sqrt[b*x + c*x^2])/Sqrt[x] - 2*Sqrt[b]*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqr
t[x])]

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Rubi [A]  time = 0.0678044, antiderivative size = 53, 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 \[ \frac{2 \sqrt{b x+c x^2}}{\sqrt{x}}-2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^(3/2),x]

[Out]

(2*Sqrt[b*x + c*x^2])/Sqrt[x] - 2*Sqrt[b]*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqr
t[x])]

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Rubi in Sympy [A]  time = 8.00949, size = 48, normalized size = 0.91 \[ - 2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 \sqrt{b x + c x^{2}}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/x**(3/2),x)

[Out]

-2*sqrt(b)*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x))) + 2*sqrt(b*x + c*x**2)/sq
rt(x)

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Mathematica [A]  time = 0.0475165, size = 60, normalized size = 1.13 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (\sqrt{b+c x}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{\sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/x^(3/2),x]

[Out]

(2*Sqrt[x]*Sqrt[b + c*x]*(Sqrt[b + c*x] - Sqrt[b]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]
))/Sqrt[x*(b + c*x)]

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Maple [A]  time = 0.017, size = 48, normalized size = 0.9 \[ -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{\sqrt{x}\sqrt{cx+b}} \left ( \sqrt{b}{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) -\sqrt{cx+b} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^(3/2),x)

[Out]

-2*(x*(c*x+b))^(1/2)/x^(1/2)*(b^(1/2)*arctanh((c*x+b)^(1/2)/b^(1/2))-(c*x+b)^(1/
2))/(c*x+b)^(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(sqrt(c*x^2 + b*x)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247749, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, c x^{2} + \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \, b x}{\sqrt{c x^{2} + b x} \sqrt{x}}, \frac{2 \,{\left (c x^{2} - \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} \arctan \left (\frac{b \sqrt{x}}{\sqrt{c x^{2} + b x} \sqrt{-b}}\right ) + b x\right )}}{\sqrt{c x^{2} + b x} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(3/2),x, algorithm="fricas")

[Out]

[(2*c*x^2 + sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x)*log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2
+ b*x)*sqrt(b)*sqrt(x))/x^2) + 2*b*x)/(sqrt(c*x^2 + b*x)*sqrt(x)), 2*(c*x^2 - s
qrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x)*arctan(b*sqrt(x)/(sqrt(c*x^2 + b*x)*sqrt(-b)))
+ b*x)/(sqrt(c*x^2 + b*x)*sqrt(x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/x**(3/2),x)

[Out]

Integral(sqrt(x*(b + c*x))/x**(3/2), x)

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GIAC/XCAS [A]  time = 0.211401, size = 82, normalized size = 1.55 \[ \frac{2 \, b \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{c x + b} - \frac{2 \,{\left (b \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b} \sqrt{b}\right )}}{\sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(3/2),x, algorithm="giac")

[Out]

2*b*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) + 2*sqrt(c*x + b) - 2*(b*arctan(sqrt
(b)/sqrt(-b)) + sqrt(-b)*sqrt(b))/sqrt(-b)