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

Optimal. Leaf size=45 \[ \frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}+\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 45, 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} \[ \frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}+\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a - b*x)^(5/2)),x]

[Out]

(2*Sqrt[x])/(3*a*(a - b*x)^(3/2)) + (4*Sqrt[x])/(3*a^2*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]

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}{\sqrt {x} (a-b x)^{5/2}} \, dx &=\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}}+\frac {2 \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx}{3 a}\\ &=\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}}+\frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.67 \[ \frac {2 \sqrt {x} (3 a-2 b x)}{3 a^2 (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 44, normalized size = 0.98 \[ -\frac {2 \, {\left (2 \, b x - 3 \, a\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.43, size = 96, normalized size = 2.13 \[ \frac {8 \, {\left (3 \, {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} \sqrt {-b} b^{2}}{3 \, {\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*(3*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)*sqrt(-b)*b^2/(((sqrt(-b*x + a)*sqrt(-b) -
sqrt((b*x - a)*b + a*b))^2 - a*b)^3*abs(b))

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maple [A]  time = 0.00, size = 25, normalized size = 0.56 \[ \frac {2 \left (-2 b x +3 a \right ) \sqrt {x}}{3 \left (-b x +a \right )^{\frac {3}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 30, normalized size = 0.67 \[ \frac {2 \, {\left (b - \frac {3 \, {\left (b x - a\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.41, size = 56, normalized size = 1.24 \[ \frac {6\,a\,\sqrt {x}\,\sqrt {a-b\,x}-4\,b\,x^{3/2}\,\sqrt {a-b\,x}}{3\,a^4-6\,a^3\,b\,x+3\,a^2\,b^2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a - b*x)^(5/2)),x)

[Out]

(6*a*x^(1/2)*(a - b*x)^(1/2) - 4*b*x^(3/2)*(a - b*x)^(1/2))/(3*a^4 + 3*a^2*b^2*x^2 - 6*a^3*b*x)

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sympy [C]  time = 2.00, size = 211, normalized size = 4.69 \[ \begin {cases} \frac {6 i a}{3 i a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} - 3 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} - \frac {4 i b x}{3 i a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} - 3 i a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {6 a b}{3 i a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} - 3 i a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} - \frac {4 b^{2} x}{3 i a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} - 3 i a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((6*I*a/(3*I*a**3*sqrt(b)*sqrt(a/(b*x) - 1) - 3*I*a**2*b**(3/2)*x*sqrt(a/(b*x) - 1)) - 4*I*b*x/(3*I*a
**3*sqrt(b)*sqrt(a/(b*x) - 1) - 3*I*a**2*b**(3/2)*x*sqrt(a/(b*x) - 1)), Abs(a/(b*x)) > 1), (6*a*b/(3*I*a**3*b*
*(3/2)*sqrt(-a/(b*x) + 1) - 3*I*a**2*b**(5/2)*x*sqrt(-a/(b*x) + 1)) - 4*b**2*x/(3*I*a**3*b**(3/2)*sqrt(-a/(b*x
) + 1) - 3*I*a**2*b**(5/2)*x*sqrt(-a/(b*x) + 1)), True))

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