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

Optimal. Leaf size=53 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b} \]

[Out]

(-2*a*Sqrt[x])/b^2 + (2*x^(3/2))/(3*b) + (2*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqr
t[a]])/b^(5/2)

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Rubi [A]  time = 0.0452411, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(a + b*x),x]

[Out]

(-2*a*Sqrt[x])/b^2 + (2*x^(3/2))/(3*b) + (2*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqr
t[a]])/b^(5/2)

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Rubi in Sympy [A]  time = 8.7056, size = 49, normalized size = 0.92 \[ \frac{2 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{2 a \sqrt{x}}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0306921, size = 49, normalized size = 0.92 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{2 \sqrt{x} (b x-3 a)}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.009, size = 43, normalized size = 0.8 \[{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-2\,{\frac{a\sqrt{x}}{{b}^{2}}}+2\,{\frac{{a}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221336, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (b x - 3 \, a\right )} \sqrt{x}}{3 \, b^{2}}, \frac{2 \,{\left (3 \, a \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (b x - 3 \, a\right )} \sqrt{x}\right )}}{3 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.45422, size = 49, normalized size = 0.92 \[ \frac{2 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{2 a \sqrt{x}}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.205621, size = 61, normalized size = 1.15 \[ \frac{2 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \,{\left (b^{2} x^{\frac{3}{2}} - 3 \, a b \sqrt{x}\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a^2*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^2) + 2/3*(b^2*x^(3/2) - 3*a*b*sqr
t(x))/b^3