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

Optimal. Leaf size=48 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 x}{c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0581041, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 x}{c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.16403, size = 42, normalized size = 0.88 \[ - \frac{2 x}{c \sqrt{b x + c x^{2}}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0387115, size = 66, normalized size = 1.38 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )-2 \sqrt{c} x}{c^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 47, normalized size = 1. \[ -2\,{\frac{x}{c\sqrt{c{x}^{2}+bx}}}+{1\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^2+b*x)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232481, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{c} x - \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{\sqrt{c x^{2} + b x} c^{\frac{3}{2}}}, -\frac{2 \,{\left (\sqrt{-c} x - \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )\right )}}{\sqrt{c x^{2} + b x} \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError