3.7.34 \(\int \frac {1}{x^{5/2} \sqrt {2-b x}} \, dx\)

Optimal. Leaf size=40 \[ -\frac {\sqrt {2-b x}}{3 x^{3/2}}-\frac {b \sqrt {2-b x}}{3 \sqrt {x}} \]

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Rubi [A]  time = 0.00, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {45, 37} \begin {gather*} -\frac {\sqrt {2-b x}}{3 x^{3/2}}-\frac {b \sqrt {2-b x}}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[2 - b*x]),x]

[Out]

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

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{x^{5/2} \sqrt {2-b x}} \, dx &=-\frac {\sqrt {2-b x}}{3 x^{3/2}}+\frac {1}{3} b \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx\\ &=-\frac {\sqrt {2-b x}}{3 x^{3/2}}-\frac {b \sqrt {2-b x}}{3 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.60 \begin {gather*} -\frac {\sqrt {2-b x} (b x+1)}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*Sqrt[2 - b*x]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.09, size = 25, normalized size = 0.62 \begin {gather*} \frac {(-b x-1) \sqrt {2-b x}}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^(5/2)*Sqrt[2 - b*x]),x]

[Out]

((-1 - b*x)*Sqrt[2 - b*x])/(3*x^(3/2))

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fricas [A]  time = 0.87, size = 18, normalized size = 0.45 \begin {gather*} -\frac {{\left (b x + 1\right )} \sqrt {-b x + 2}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b*x + 1)*sqrt(-b*x + 2)/x^(3/2)

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giac [A]  time = 1.12, size = 43, normalized size = 1.08 \begin {gather*} -\frac {{\left ({\left (b x - 2\right )} b^{3} + 3 \, b^{3}\right )} \sqrt {-b x + 2} b}{3 \, {\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac {3}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 19, normalized size = 0.48 \begin {gather*} -\frac {\left (b x +1\right ) \sqrt {-b x +2}}{3 x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 28, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {-b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (-b x + 2\right )}^{\frac {3}{2}}}{6 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-b*x + 2)*b/sqrt(x) - 1/6*(-b*x + 2)^(3/2)/x^(3/2)

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mupad [B]  time = 0.29, size = 19, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {2-b\,x}\,\left (\frac {b\,x}{3}+\frac {1}{3}\right )}{x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(2 - b*x)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.96, size = 139, normalized size = 3.48 \begin {gather*} \begin {cases} - \frac {b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}}{3} - \frac {\sqrt {b} \sqrt {-1 + \frac {2}{b x}}}{3 x} & \text {for}\: \frac {2}{\left |{b x}\right |} > 1 \\\frac {i b^{\frac {7}{2}} x^{2} \sqrt {1 - \frac {2}{b x}}}{- 3 b^{2} x^{2} + 6 b x} - \frac {i b^{\frac {5}{2}} x \sqrt {1 - \frac {2}{b x}}}{- 3 b^{2} x^{2} + 6 b x} - \frac {2 i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}}{- 3 b^{2} x^{2} + 6 b x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-b**(3/2)*sqrt(-1 + 2/(b*x))/3 - sqrt(b)*sqrt(-1 + 2/(b*x))/(3*x), 2/Abs(b*x) > 1), (I*b**(7/2)*x**
2*sqrt(1 - 2/(b*x))/(-3*b**2*x**2 + 6*b*x) - I*b**(5/2)*x*sqrt(1 - 2/(b*x))/(-3*b**2*x**2 + 6*b*x) - 2*I*b**(3
/2)*sqrt(1 - 2/(b*x))/(-3*b**2*x**2 + 6*b*x), True))

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