3.587 \(\int \frac{1}{\sqrt{x} (a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{4 \sqrt{x}}{3 a^2 \sqrt{a+b x}}+\frac{2 \sqrt{x}}{3 a (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0287422, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{4 \sqrt{x}}{3 a^2 \sqrt{a+b x}}+\frac{2 \sqrt{x}}{3 a (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.05024, size = 37, normalized size = 0.86 \[ \frac{2 \sqrt{x}}{3 a \left (a + b x\right )^{\frac{3}{2}}} + \frac{4 \sqrt{x}}{3 a^{2} \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0188422, size = 29, normalized size = 0.67 \[ \frac{2 \sqrt{x} (3 a+2 b x)}{3 a^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 24, normalized size = 0.6 \[{\frac{4\,bx+6\,a}{3\,{a}^{2}}\sqrt{x} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(5/2)/x^(1/2),x)

[Out]

2/3*x^(1/2)*(2*b*x+3*a)/(b*x+a)^(3/2)/a^2

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Maxima [A]  time = 1.32884, size = 36, normalized size = 0.84 \[ -\frac{2 \,{\left (b - \frac{3 \,{\left (b x + a\right )}}{x}\right )} x^{\frac{3}{2}}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*sqrt(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.21676, size = 47, normalized size = 1.09 \[ \frac{2 \,{\left (2 \, b x^{2} + 3 \, a x\right )}}{3 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{b x + a} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*sqrt(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 23.5804, size = 92, normalized size = 2.14 \[ \frac{6 a}{3 a^{3} \sqrt{b} \sqrt{\frac{a}{b x} + 1} + 3 a^{2} b^{\frac{3}{2}} x \sqrt{\frac{a}{b x} + 1}} + \frac{4 b x}{3 a^{3} \sqrt{b} \sqrt{\frac{a}{b x} + 1} + 3 a^{2} b^{\frac{3}{2}} x \sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*a/(3*a**3*sqrt(b)*sqrt(a/(b*x) + 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) + 1)) + 4
*b*x/(3*a**3*sqrt(b)*sqrt(a/(b*x) + 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) + 1))

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GIAC/XCAS [A]  time = 0.210933, size = 109, normalized size = 2.53 \[ \frac{8 \,{\left (3 \,{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{\frac{5}{2}}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*sqrt(x)),x, algorithm="giac")

[Out]

8/3*(3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*b^(5/2)/(((sqr
t(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*abs(b))