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

Optimal. Leaf size=46 \[ -\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 46, 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 {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[a - b*x]),x]

[Out]

(-2*Sqrt[a - b*x])/(3*a*x^(3/2)) - (4*b*Sqrt[a - b*x])/(3*a^2*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 {a-b x}} \, dx &=-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}+\frac {(2 b) \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}-\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 28, normalized size = 0.61 \begin {gather*} -\frac {2 \sqrt {a-b x} (a+2 b x)}{3 a^2 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.11, size = 28, normalized size = 0.61 \begin {gather*} -\frac {2 \sqrt {a-b x} (a+2 b x)}{3 a^2 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.77, size = 22, normalized size = 0.48 \begin {gather*} -\frac {2 \, {\left (2 \, b x + a\right )} \sqrt {-b x + a}}{3 \, a^{2} x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.45, size = 54, normalized size = 1.17 \begin {gather*} -\frac {2 \, {\left (\frac {2 \, {\left (b x - a\right )} b^{3}}{a^{2}} + \frac {3 \, b^{3}}{a}\right )} \sqrt {-b x + a} b}{3 \, {\left ({\left (b x - a\right )} b + a 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+a)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 23, normalized size = 0.50 \begin {gather*} -\frac {2 \sqrt {-b x +a}\, \left (2 b x +a \right )}{3 a^{2} x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 32, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (\frac {3 \, \sqrt {-b x + a} b}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.35, size = 26, normalized size = 0.57 \begin {gather*} -\frac {\left (\frac {2}{3\,a}+\frac {4\,b\,x}{3\,a^2}\right )\,\sqrt {a-b\,x}}{x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a - b*x)^(1/2)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 2.06, size = 177, normalized size = 3.85 \begin {gather*} \begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 a x} - \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a^{2}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {2 i a^{2} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} + \frac {2 i a b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} - \frac {4 i b^{\frac {7}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(b)*sqrt(a/(b*x) - 1)/(3*a*x) - 4*b**(3/2)*sqrt(a/(b*x) - 1)/(3*a**2), Abs(a/(b*x)) > 1), (2
*I*a**2*b**(3/2)*sqrt(-a/(b*x) + 1)/(-3*a**3*b*x + 3*a**2*b**2*x**2) + 2*I*a*b**(5/2)*x*sqrt(-a/(b*x) + 1)/(-3
*a**3*b*x + 3*a**2*b**2*x**2) - 4*I*b**(7/2)*x**2*sqrt(-a/(b*x) + 1)/(-3*a**3*b*x + 3*a**2*b**2*x**2), True))

________________________________________________________________________________________