∫ 1_____√ x (a−bx)5∕2 dx [605]

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

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, 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 = {47, 37} \begin {gather*} \frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}+\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}} \end {gather*}

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

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

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]))) && (

\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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 30, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {x} (3 a-2 b x)}{3 a^2 (a-b x)^{3/2}} \end {gather*}

________________________________________________________________________________________


method	result	size

gosper	\(\frac {2 \sqrt {x}\, \left (-2 b x +3 a \right )}{3 \left (-b x +a \right )^{\frac {3}{2}} a^{2}}\)	\(25\)
default	\(\frac {2 \sqrt {x}}{3 a \left (-b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {-b x +a}}\)	\(34\)

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

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 30, normalized size = 0.67 \begin {gather*} \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}} \end {gather*}

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

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

________________________________________________________________________________________

Fricas [A]
time = 0.54, size = 44, normalized size = 0.98 \begin {gather*} -\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 )}} \end {gather*}

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

-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)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 1.06, size = 197, normalized size = 4.38 \begin {gather*} \begin {cases} - \frac {6 a}{- 3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} + \frac {4 b x}{- 3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {6 i a b}{- 3 a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} + 3 a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} - \frac {4 i b^{2} x}{- 3 a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} + 3 a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \end {gather*}

Piecewise((-6*a/(-3*a**3*sqrt(b)*sqrt(a/(b*x) - 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) - 1)) + 4*b*x/(-3*a**3*sqr

t(b)*sqrt(a/(b*x) - 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) - 1)), Abs(a/(b*x)) > 1), (6*I*a*b/(-3*a**3*b**(3/2)*s

qrt(-a/(b*x) + 1) + 3*a**2*b**(5/2)*x*sqrt(-a/(b*x) + 1)) - 4*I*b**2*x/(-3*a**3*b**(3/2)*sqrt(-a/(b*x) + 1) +

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (33) = 66\).
time = 2.38, size = 96, normalized size = 2.13 \begin {gather*} \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 |}} \end {gather*}

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

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) -

________________________________________________________________________________________

Mupad [B]
time = 0.41, size = 56, normalized size = 1.24 \begin {gather*} \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} \end {gather*}

(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)

________________________________________________________________________________________

3.7.5 \(\int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx\) [605]