3.312 \(\int \frac{x^2}{\sqrt{a x^3+b x^4}} \, dx\)

Optimal. Leaf size=56 \[ \frac{\sqrt{a x^3+b x^4}}{b x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.138274, 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 \[ \frac{\sqrt{a x^3+b x^4}}{b x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a*x^3 + b*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 12.1087, size = 46, normalized size = 0.82 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a x^{3} + b x^{4}}} \right )}}{b^{\frac{3}{2}}} + \frac{\sqrt{a x^{3} + b x^{4}}}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**4+a*x**3)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0415812, size = 75, normalized size = 1.34 \[ \frac{\sqrt{b} x^2 (a+b x)-a x^{3/2} \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{3/2} \sqrt{x^3 (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[a*x^3 + b*x^4],x]

[Out]

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

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Maple [A]  time = 0.008, size = 78, normalized size = 1.4 \[{\frac{x}{2}\sqrt{x \left ( bx+a \right ) } \left ( 2\,\sqrt{b{x}^{2}+ax}{b}^{3/2}-a\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{b{x}^{2}+ax}\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) b \right ){\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^4+a*x^3)^(1/2),x)

[Out]

1/2*x*(x*(b*x+a))^(1/2)*(2*(b*x^2+a*x)^(1/2)*b^(3/2)-a*ln(1/2*(2*(b*x^2+a*x)^(1/
2)*b^(1/2)+2*b*x+a)/b^(1/2))*b)/(b*x^4+a*x^3)^(1/2)/b^(5/2)

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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(b*x^4 + a*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237094, size = 1, normalized size = 0.02 \[ \left [\frac{a \sqrt{b} x \log \left (\frac{{\left (2 \, b x^{2} + a x\right )} \sqrt{b} - 2 \, \sqrt{b x^{4} + a x^{3}} b}{x}\right ) + 2 \, \sqrt{b x^{4} + a x^{3}} b}{2 \, b^{2} x}, \frac{a \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{4} + a x^{3}} \sqrt{-b}}{b x^{2}}\right ) + \sqrt{b x^{4} + a x^{3}} b}{b^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(b*x^4 + a*x^3),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**4+a*x**3)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.236795, size = 55, normalized size = 0.98 \[ \frac{\sqrt{b + \frac{a}{x}} x}{b} + \frac{a \arctan \left (\frac{\sqrt{b + \frac{a}{x}}}{\sqrt{-b}}\right )}{\sqrt{-b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(b*x^4 + a*x^3),x, algorithm="giac")

[Out]

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