3.78 \(\int \frac{\sqrt{b x+c x^2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}-\frac{\sqrt{b x+c x^2}}{x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0692107, antiderivative size = 54, 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 \[ -\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}-\frac{\sqrt{b x+c x^2}}{x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.97257, size = 48, normalized size = 0.89 \[ - \frac{\sqrt{b x + c x^{2}}}{x^{\frac{3}{2}}} - \frac{c \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/x**(5/2),x)

[Out]

-sqrt(b*x + c*x**2)/x**(3/2) - c*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/sqr
t(b)

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Mathematica [A]  time = 0.0340337, size = 68, normalized size = 1.26 \[ -\frac{\sqrt{x (b+c x)}}{x^{3/2}}-\frac{c \sqrt{x (b+c x)} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{x} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/x^(5/2),x]

[Out]

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

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Maple [A]  time = 0.013, size = 53, normalized size = 1. \[{1 \left ( -{\it Artanh} \left ({1\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ) cx-\sqrt{cx+b}\sqrt{b} \right ) \sqrt{x \left ( cx+b \right ) }{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^(5/2),x)

[Out]

(-arctanh((c*x+b)^(1/2)/b^(1/2))*c*x-(c*x+b)^(1/2)*b^(1/2))*(x*(c*x+b))^(1/2)/x^
(3/2)/(c*x+b)^(1/2)/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^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238076, size = 1, normalized size = 0.02 \[ \left [\frac{c x^{2} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{2 \, \sqrt{b} x^{2}}, -\frac{c x^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{\sqrt{-b} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(5/2),x, algorithm="fricas")

[Out]

[1/2*(c*x^2*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^2) -
2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/(sqrt(b)*x^2), -(c*x^2*arctan(sqrt(-b)*sqr
t(x)/sqrt(c*x^2 + b*x)) + sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x))/(sqrt(-b)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/x**(5/2),x)

[Out]

Integral(sqrt(x*(b + c*x))/x**(5/2), x)

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GIAC/XCAS [A]  time = 0.228687, size = 51, normalized size = 0.94 \[ c{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{c x + b}}{c x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(5/2),x, algorithm="giac")

[Out]

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