3.270 \(\int \frac{x^{5/2}}{\sqrt{a x^2+b x^3}} \, dx\)

Optimal. Leaf size=95 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x^2+b x^3}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{a x^2+b x^3}}{4 b^2 \sqrt{x}}+\frac{\sqrt{x} \sqrt{a x^2+b x^3}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.219881, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x^2+b x^3}}\right )}{4 b^{5/2}}-\frac{3 a \sqrt{a x^2+b x^3}}{4 b^2 \sqrt{x}}+\frac{\sqrt{x} \sqrt{a x^2+b x^3}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/Sqrt[a*x^2 + b*x^3],x]

[Out]

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

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Rubi in Sympy [A]  time = 20.7359, size = 85, normalized size = 0.89 \[ \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a x^{2} + b x^{3}}} \right )}}{4 b^{\frac{5}{2}}} - \frac{3 a \sqrt{a x^{2} + b x^{3}}}{4 b^{2} \sqrt{x}} + \frac{\sqrt{x} \sqrt{a x^{2} + b x^{3}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(b*x**3+a*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0510325, size = 92, normalized size = 0.97 \[ \frac{\sqrt{b} x^{3/2} \left (-3 a^2-a b x+2 b^2 x^2\right )+3 a^2 x \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{5/2} \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/Sqrt[a*x^2 + b*x^3],x]

[Out]

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

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Maple [A]  time = 0.009, size = 92, normalized size = 1. \[{\frac{1}{8}\sqrt{x} \left ( 4\,{b}^{7/2}{x}^{3}-2\,{b}^{5/2}{x}^{2}a-6\,{b}^{3/2}x{a}^{2}+3\,\sqrt{x \left ( bx+a \right ) }\ln \left ( 1/2\,{\frac{2\,\sqrt{b{x}^{2}+ax}\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}b \right ){\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x^3+a*x^2)^(1/2),x)

[Out]

1/8*x^(1/2)*(4*b^(7/2)*x^3-2*b^(5/2)*x^2*a-6*b^(3/2)*x*a^2+3*(x*(b*x+a))^(1/2)*l
n(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^3+a*x^2)^(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^(5/2)/sqrt(b*x^3 + a*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22852, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \sqrt{b} x \log \left (\frac{2 \, \sqrt{b x^{3} + a x^{2}} b \sqrt{x} +{\left (2 \, b x^{2} + a x\right )} \sqrt{b}}{x}\right ) + 2 \, \sqrt{b x^{3} + a x^{2}}{\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt{x}}{8 \, b^{3} x}, -\frac{3 \, a^{2} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-b}}{b x^{\frac{3}{2}}}\right ) - \sqrt{b x^{3} + a x^{2}}{\left (2 \, b^{2} x - 3 \, a b\right )} \sqrt{x}}{4 \, b^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/sqrt(b*x^3 + a*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}}}{\sqrt{x^{2} \left (a + b x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(b*x**3+a*x**2)**(1/2),x)

[Out]

Integral(x**(5/2)/sqrt(x**2*(a + b*x)), x)

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GIAC/XCAS [A]  time = 0.226209, size = 70, normalized size = 0.74 \[ \frac{1}{4} \, \sqrt{b x + a} \sqrt{x}{\left (\frac{2 \, x}{b} - \frac{3 \, a}{b^{2}}\right )} - \frac{3 \, a^{2}{\rm ln}\left ({\left | -\sqrt{b} \sqrt{x} + \sqrt{b x + a} \right |}\right )}{4 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/sqrt(b*x^3 + a*x^2),x, algorithm="giac")

[Out]

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