3.7.43 \(\int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx\)

Optimal. Leaf size=19 \[ \frac {x^{3/2}}{3 (2-b x)^{3/2}} \]

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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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37} \begin {gather*} \frac {x^{3/2}}{3 (2-b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(2 - b*x)^(5/2),x]

[Out]

x^(3/2)/(3*(2 - b*x)^(3/2))

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 {\sqrt {x}}{(2-b x)^{5/2}} \, dx &=\frac {x^{3/2}}{3 (2-b x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {x^{3/2}}{3 (2-b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(2 - b*x)^(5/2),x]

[Out]

x^(3/2)/(3*(2 - b*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.11, size = 26, normalized size = 1.37 \begin {gather*} \frac {x^{3/2} \sqrt {2-b x}}{3 (b x-2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x]/(2 - b*x)^(5/2),x]

[Out]

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

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fricas [B]  time = 1.03, size = 28, normalized size = 1.47 \begin {gather*} \frac {\sqrt {-b x + 2} x^{\frac {3}{2}}}{3 \, {\left (b^{2} x^{2} - 4 \, b x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(-b*x+2)^(5/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.25, size = 95, normalized size = 5.00 \begin {gather*} \frac {4 \, {\left (3 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} \sqrt {-b} + 4 \, \sqrt {-b} b^{2}\right )} {\left | b \right |}}{3 \, {\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(-b*x+2)^(5/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.74 \begin {gather*} \frac {x^{\frac {3}{2}}}{3 \left (-b x +2\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(-b*x+2)^(5/2),x)

[Out]

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

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maxima [A]  time = 1.38, size = 13, normalized size = 0.68 \begin {gather*} \frac {x^{\frac {3}{2}}}{3 \, {\left (-b x + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(-b*x+2)^(5/2),x, algorithm="maxima")

[Out]

1/3*x^(3/2)/(-b*x + 2)^(3/2)

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mupad [B]  time = 0.23, size = 13, normalized size = 0.68 \begin {gather*} \frac {x^{3/2}}{3\,{\left (2-b\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(2 - b*x)^(5/2),x)

[Out]

x^(3/2)/(3*(2 - b*x)^(3/2))

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sympy [B]  time = 1.46, size = 65, normalized size = 3.42 \begin {gather*} \begin {cases} \frac {i x^{\frac {3}{2}}}{3 b x \sqrt {b x - 2} - 6 \sqrt {b x - 2}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\- \frac {x^{\frac {3}{2}}}{3 b x \sqrt {- b x + 2} - 6 \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(-b*x+2)**(5/2),x)

[Out]

Piecewise((I*x**(3/2)/(3*b*x*sqrt(b*x - 2) - 6*sqrt(b*x - 2)), Abs(b*x)/2 > 1), (-x**(3/2)/(3*b*x*sqrt(-b*x +
2) - 6*sqrt(-b*x + 2)), True))

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