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

Optimal. Leaf size=19 \[ \frac {2 \sqrt {x}}{a \sqrt {a+b x}} \]

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {37} \begin {gather*} \frac {2 \sqrt {x}}{a \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*x)^(3/2)),x]

[Out]

(2*Sqrt[x])/(a*Sqrt[a + b*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x}}{a \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x])/(a*Sqrt[a + b*x])

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IntegrateAlgebraic [A]  time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x}}{a \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x])/(a*Sqrt[a + b*x])

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fricas [A]  time = 1.01, size = 22, normalized size = 1.16 \begin {gather*} \frac {2 \, \sqrt {b x + a} \sqrt {x}}{a b x + a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)*sqrt(x)/(a*b*x + a^2)

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giac [B]  time = 1.11, size = 45, normalized size = 2.37 \begin {gather*} \frac {4 \, b^{\frac {3}{2}}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} \frac {2 \sqrt {x}}{\sqrt {b x +a}\, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 15, normalized size = 0.79 \begin {gather*} \frac {2 \, \sqrt {x}}{\sqrt {b x + a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)/(sqrt(b*x + a)*a)

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mupad [B]  time = 0.33, size = 22, normalized size = 1.16 \begin {gather*} \frac {2\,\sqrt {x}\,\sqrt {a+b\,x}}{a^2+b\,x\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a + b*x)^(3/2)),x)

[Out]

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

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sympy [A]  time = 0.88, size = 17, normalized size = 0.89 \begin {gather*} \frac {2}{a \sqrt {b} \sqrt {\frac {a}{b x} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a*sqrt(b)*sqrt(a/(b*x) + 1))

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