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

Optimal. Leaf size=76 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}-\frac{3 b \sqrt{b x+c x^2}}{4 c^2}+\frac{x \sqrt{b x+c x^2}}{2 c} \]

[Out]

(-3*b*Sqrt[b*x + c*x^2])/(4*c^2) + (x*Sqrt[b*x + c*x^2])/(2*c) + (3*b^2*ArcTanh[
(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(5/2))

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Rubi [A]  time = 0.0877493, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}-\frac{3 b \sqrt{b x+c x^2}}{4 c^2}+\frac{x \sqrt{b x+c x^2}}{2 c} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[b*x + c*x^2],x]

[Out]

(-3*b*Sqrt[b*x + c*x^2])/(4*c^2) + (x*Sqrt[b*x + c*x^2])/(2*c) + (3*b^2*ArcTanh[
(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(5/2))

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Rubi in Sympy [A]  time = 9.83411, size = 68, normalized size = 0.89 \[ \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{5}{2}}} - \frac{3 b \sqrt{b x + c x^{2}}}{4 c^{2}} + \frac{x \sqrt{b x + c x^{2}}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b**2*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(4*c**(5/2)) - 3*b*sqrt(b*x + c*x**2)
/(4*c**2) + x*sqrt(b*x + c*x**2)/(2*c)

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Mathematica [A]  time = 0.0552627, size = 90, normalized size = 1.18 \[ \frac{\sqrt{c} x \left (-3 b^2-b c x+2 c^2 x^2\right )+3 b^2 \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{4 c^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*x*(-3*b^2 - b*c*x + 2*c^2*x^2) + 3*b^2*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt
[x] + Sqrt[c]*Sqrt[b + c*x]])/(4*c^(5/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.007, size = 68, normalized size = 0.9 \[{\frac{x}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,b}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^2+b*x)^(1/2),x)

[Out]

1/2*x*(c*x^2+b*x)^(1/2)/c-3/4*b*(c*x^2+b*x)^(1/2)/c^2+3/8*b^2/c^(5/2)*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(x^2/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226823, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x - 3 \, b\right )} \sqrt{c}}{8 \, c^{\frac{5}{2}}}, \frac{3 \, b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x}{\left (2 \, c x - 3 \, b\right )} \sqrt{-c}}{4 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*b^2*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) + 2*sqrt(c*x^2 + b*
x)*(2*c*x - 3*b)*sqrt(c))/c^(5/2), 1/4*(3*b^2*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/
(c*x)) + sqrt(c*x^2 + b*x)*(2*c*x - 3*b)*sqrt(-c))/(sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.227192, size = 88, normalized size = 1.16 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (\frac{2 \, x}{c} - \frac{3 \, b}{c^{2}}\right )} - \frac{3 \, b^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*x^2 + b*x)*(2*x/c - 3*b/c^2) - 3/8*b^2*ln(abs(-2*(sqrt(c)*x - sqrt(c*
x^2 + b*x))*sqrt(c) - b))/c^(5/2)