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

Optimal. Leaf size=21 \[ \frac{2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

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Rubi [A]  time = 0.0125158, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + b*x)^(5/2),x]

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

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Rubi in Sympy [A]  time = 2.3188, size = 17, normalized size = 0.81 \[ \frac{2 x^{\frac{3}{2}}}{3 a \left (a + b x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/(3*a*(a + b*x)**(3/2))

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Mathematica [A]  time = 0.0188992, size = 21, normalized size = 1. \[ \frac{2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

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Maple [A]  time = 0.007, size = 16, normalized size = 0.8 \[{\frac{2}{3\,a}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(b*x+a)^(5/2),x)

[Out]

2/3*x^(3/2)/a/(b*x+a)^(3/2)

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Maxima [A]  time = 1.33624, size = 20, normalized size = 0.95 \[ \frac{2 \, x^{\frac{3}{2}}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x^(3/2)/((b*x + a)^(3/2)*a)

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Fricas [A]  time = 0.21385, size = 30, normalized size = 1.43 \[ \frac{2 \, x^{\frac{3}{2}}}{3 \,{\left (a b x + a^{2}\right )} \sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x^(3/2)/((a*b*x + a^2)*sqrt(b*x + a))

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Sympy [A]  time = 13.1119, size = 42, normalized size = 2. \[ \frac{2 x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/(3*a**(5/2)*sqrt(1 + b*x/a) + 3*a**(3/2)*b*x*sqrt(1 + b*x/a))

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GIAC/XCAS [A]  time = 0.217909, size = 116, normalized size = 5.52 \[ \frac{4 \,{\left (3 \,{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b} + a^{2} b^{\frac{5}{2}}\right )}{\left | b \right |}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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