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

Optimal. Leaf size=86 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{a x^3+b x^4}}{4 b^2 x}+\frac{\sqrt{a x^3+b x^4}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.219217, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{a x^3+b x^4}}{4 b^2 x}+\frac{\sqrt{a x^3+b x^4}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a*x^3 + b*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 18.6513, size = 75, normalized size = 0.87 \[ \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a x^{3} + b x^{4}}} \right )}}{4 b^{\frac{5}{2}}} - \frac{3 a \sqrt{a x^{3} + b x^{4}}}{4 b^{2} x} + \frac{\sqrt{a x^{3} + b x^{4}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0499583, size = 94, normalized size = 1.09 \[ \frac{\sqrt{b} x^2 \left (-3 a^2-a b x+2 b^2 x^2\right )+3 a^2 x^{3/2} \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{5/2} \sqrt{x^3 (a+b x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 98, normalized size = 1.1 \[{\frac{x}{8}\sqrt{x \left ( bx+a \right ) } \left ( 4\,x\sqrt{b{x}^{2}+ax}{b}^{5/2}-6\,\sqrt{b{x}^{2}+ax}{b}^{3/2}a+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{b{x}^{2}+ax}\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}b \right ){\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^4+a*x^3)^(1/2),x)

[Out]

1/8*x*(x*(b*x+a))^(1/2)*(4*x*(b*x^2+a*x)^(1/2)*b^(5/2)-6*(b*x^2+a*x)^(1/2)*b^(3/
2)*a+3*ln(1/2*(2*(b*x^2+a*x)^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^2*b)/(b*x^4+a*x^3
)^(1/2)/b^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235787, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \sqrt{b} x \log \left (\frac{{\left (2 \, b x^{2} + a x\right )} \sqrt{b} + 2 \, \sqrt{b x^{4} + a x^{3}} b}{x}\right ) + 2 \, \sqrt{b x^{4} + a x^{3}}{\left (2 \, b^{2} x - 3 \, a b\right )}}{8 \, b^{3} x}, -\frac{3 \, a^{2} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{4} + a x^{3}} \sqrt{-b}}{b x^{2}}\right ) - \sqrt{b x^{4} + a x^{3}}{\left (2 \, b^{2} x - 3 \, a b\right )}}{4 \, b^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*a^2*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)*(2*b^2*x - 3*a*b))/(b^3*x), -1/4*(3*a^2*sqrt(-b)*x*arct
an(sqrt(b*x^4 + a*x^3)*sqrt(-b)/(b*x^2)) - sqrt(b*x^4 + a*x^3)*(2*b^2*x - 3*a*b)
)/(b^3*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{b x^{4} + a x^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/sqrt(b*x^4 + a*x^3), x)