3.127 \(\int \frac{1}{\sqrt{x} \left (b \sqrt{x}+a x\right )^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{4 \left (2 a \sqrt{x}+b\right )}{b^2 \sqrt{a x+b \sqrt{x}}} \]

[Out]

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

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Rubi [A]  time = 0.0808706, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{4 \left (2 a \sqrt{x}+b\right )}{b^2 \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.46269, size = 29, normalized size = 0.97 \[ - \frac{2 \left (4 a \sqrt{x} + 2 b\right )}{b^{2} \sqrt{a x + b \sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0398779, size = 45, normalized size = 1.5 \[ -\frac{4 \left (2 a \sqrt{x}+b\right ) \sqrt{a x+b \sqrt{x}}}{a b^2 x+b^3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 112, normalized size = 3.7 \[ 4\,{\frac{\sqrt{b\sqrt{x}+ax} \left ( -x \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{2}+{a}^{2} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}x-2\,\sqrt{x} \left ( b\sqrt{x}+ax \right ) ^{3/2}ab- \left ( b\sqrt{x}+ax \right ) ^{3/2}{b}^{2} \right ) }{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }x{b}^{3} \left ( b+\sqrt{x}a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.49694, size = 34, normalized size = 1.13 \[ -\frac{4 \,{\left (2 \, a \sqrt{x} + b\right )}}{\sqrt{a \sqrt{x} + b} b^{2} x^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.269941, size = 73, normalized size = 2.43 \[ \frac{4 \,{\left (a b x -{\left (2 \, a^{2} x - b^{2}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{a^{2} b^{2} x^{2} - b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x} \left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220243, size = 35, normalized size = 1.17 \[ -\frac{4 \,{\left (\frac{2 \, a \sqrt{x}}{b^{2}} + \frac{1}{b}\right )}}{\sqrt{a x + b \sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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