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

Optimal. Leaf size=56 \[ \frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0700286, antiderivative size = 56, 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 \[ \frac{2 \sqrt{x}}{b \sqrt{b x+c x^2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.46243, size = 49, normalized size = 0.88 \[ \frac{2 \sqrt{x}}{b \sqrt{b x + c x^{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.039948, size = 56, normalized size = 1. \[ \frac{2 \sqrt{x} \left (\sqrt{b}-\sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{b^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 51, normalized size = 0.9 \[ -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{{b}^{3/2}\sqrt{x} \left ( cx+b \right ) } \left ({\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}-\sqrt{b} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(x*(c*x+b))^(1/2)/b^(3/2)*(arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)-b^(1/
2))/x^(1/2)/(c*x+b)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235416, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (c x^{2} + b x\right )} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{{\left (b c x^{2} + b^{2} x\right )} \sqrt{b}}, -\frac{2 \,{\left ({\left (c x^{2} + b x\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) - \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}\right )}}{{\left (b c x^{2} + b^{2} x\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213929, size = 90, normalized size = 1.61 \[ \frac{2 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{2 \,{\left (\sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b}\right )}}{\sqrt{-b} b^{\frac{3}{2}}} + \frac{2}{\sqrt{c x + b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b) - 2*(sqrt(b)*arctan(sqrt(b)/sqrt(-
b)) + sqrt(-b))/(sqrt(-b)*b^(3/2)) + 2/(sqrt(c*x + b)*b)