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

Optimal. Leaf size=25 \[ \frac{4 \sqrt{x}}{b \sqrt{a x+b \sqrt{x}}} \]

[Out]

(4*Sqrt[x])/(b*Sqrt[b*Sqrt[x] + a*x])

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Rubi [A]  time = 0.0051318, antiderivative size = 25, 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, Rules used = {2000} \[ \frac{4 \sqrt{x}}{b \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Sqrt[x] + a*x)^(-3/2),x]

[Out]

(4*Sqrt[x])/(b*Sqrt[b*Sqrt[x] + a*x])

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=\frac{4 \sqrt{x}}{b \sqrt{b \sqrt{x}+a x}}\\ \end{align*}

Mathematica [A]  time = 0.0164931, size = 25, normalized size = 1. \[ \frac{4 \sqrt{x}}{b \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Sqrt[x] + a*x)^(-3/2),x]

[Out]

(4*Sqrt[x])/(b*Sqrt[b*Sqrt[x] + a*x])

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Maple [C]  time = 0.008, size = 404, normalized size = 16.2 \begin{align*}{\frac{1}{{b}^{2}}\sqrt{b\sqrt{x}+ax} \left ( 2\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}x+\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ) x{a}^{2}b+2\,{a}^{5/2}x\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b \right ){\frac{1}{\sqrt{a}}}} \right ) x{a}^{2}b+4\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}\sqrt{x}b+2\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ) \sqrt{x}a{b}^{2}+4\,{a}^{3/2}\sqrt{x}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }b-4\,{a}^{3/2} \left ( \sqrt{x} \left ( b+a\sqrt{x} \right ) \right ) ^{3/2}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ) \sqrt{x}a{b}^{2}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}{b}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{3}+2\,\sqrt{a}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b \right ){\frac{1}{\sqrt{a}}}} \right ){b}^{3} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}} \left ( b+a\sqrt{x} \right ) ^{-2}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^(1/2)+a*x)^(1/2)*(2*(b*x^(1/2)+a*x)^(1/2)*a^(5/2)*x+ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b
)/a^(1/2))*x*a^2*b+2*a^(5/2)*x*(x^(1/2)*(b+a*x^(1/2)))^(1/2)-ln(1/2*(2*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*a^(1/2)+2
*a*x^(1/2)+b)/a^(1/2))*x*a^2*b+4*(b*x^(1/2)+a*x)^(1/2)*a^(3/2)*x^(1/2)*b+2*ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*
x)^(1/2)*a^(1/2)+b)/a^(1/2))*x^(1/2)*a*b^2+4*a^(3/2)*x^(1/2)*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*b-4*a^(3/2)*(x^(1/2
)*(b+a*x^(1/2)))^(3/2)-2*ln(1/2*(2*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2)+b)/a^(1/2))*x^(1/2)*a*b^2
+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)*b^2+ln(1/2*(2*a*x^(1/2)+2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+b)/a^(1/2))*b^3+2*a^(
1/2)*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*b^2-ln(1/2*(2*(x^(1/2)*(b+a*x^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2)+b)/a^(1/2))
*b^3)/(x^(1/2)*(b+a*x^(1/2)))^(1/2)/b^2/(b+a*x^(1/2))^2/a^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.28203, size = 77, normalized size = 3.08 \begin{align*} \frac{4 \, \sqrt{a x + b \sqrt{x}}{\left (a \sqrt{x} - b\right )}}{a^{2} b x - b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14704, size = 46, normalized size = 1.84 \begin{align*} \frac{4}{{\left (\sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + b\right )} \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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