3.581 \(\int \frac{1}{x^{3/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{a \sqrt{x} \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{a^2 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0261449, antiderivative size = 39, 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 \[ \frac{2}{a \sqrt{x} \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{a^2 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.39674, size = 34, normalized size = 0.87 \[ \frac{2}{a \sqrt{x} \sqrt{a + b x}} - \frac{4 \sqrt{a + b x}}{a^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0209557, size = 25, normalized size = 0.64 \[ -\frac{2 (a+2 b x)}{a^2 \sqrt{x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 22, normalized size = 0.6 \[ -2\,{\frac{2\,bx+a}{{a}^{2}\sqrt{x}\sqrt{bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(b*x+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.32152, size = 43, normalized size = 1.1 \[ -\frac{2 \, b \sqrt{x}}{\sqrt{b x + a} a^{2}} - \frac{2 \, \sqrt{b x + a}}{a^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(3/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212735, size = 28, normalized size = 0.72 \[ -\frac{2 \,{\left (2 \, b x + a\right )}}{\sqrt{b x + a} a^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 15.9795, size = 41, normalized size = 1.05 \[ - \frac{2}{a \sqrt{b} x \sqrt{\frac{a}{b x} + 1}} - \frac{4 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208678, size = 111, normalized size = 2.85 \[ -\frac{4 \, b^{\frac{5}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a{\left | b \right |}} - \frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{{\left (b x + a\right )} b - a b} a^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(3/2)),x, algorithm="giac")

[Out]

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