\(\int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx\) [600]

Optimal result

Integrand size = 16, antiderivative size = 41 \[ \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx=\frac {2}{a \sqrt {x} \sqrt {a-b x}}-\frac {4 \sqrt {a-b x}}{a^2 \sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \[ \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx=\frac {2}{a \sqrt {x} \sqrt {a-b x}}-\frac {4 \sqrt {a-b x}}{a^2 \sqrt {x}} \]

[In]

Int[1/(x^(3/2)*(a - b*x)^(3/2)),x]

[Out]

2/(a*Sqrt[x]*Sqrt[a - b*x]) - (4*Sqrt[a - b*x])/(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 47

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx=-\frac {2 (a-2 b x)}{a^2 \sqrt {x} \sqrt {a-b x}} \]

[In]

Integrate[1/(x^(3/2)*(a - b*x)^(3/2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.56

method	result	size
gosper	\(-\frac {2 \left (-2 b x +a \right )}{\sqrt {x}\, \sqrt {-b x +a}\, a^{2}}\)	\(23\)
default	\(-\frac {2}{a \sqrt {x}\, \sqrt {-b x +a}}+\frac {4 b \sqrt {x}}{a^{2} \sqrt {-b x +a}}\)	\(35\)
risch	\(-\frac {2 \sqrt {-b x +a}}{a^{2} \sqrt {x}}+\frac {2 b \sqrt {x}}{a^{2} \sqrt {-b x +a}}\)	\(35\)

[In]

int(1/x^(3/2)/(-b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx=-\frac {2 \, {\left (2 \, b x - a\right )} \sqrt {-b x + a} \sqrt {x}}{a^{2} b x^{2} - a^{3} x} \]

[In]

integrate(1/x^(3/2)/(-b*x+a)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.86 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx=\begin {cases} - \frac {2}{a \sqrt {b} x \sqrt {\frac {a}{b x} - 1}} + \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {2 i a b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{- a^{3} b + a^{2} b^{2} x} - \frac {4 i b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- a^{3} b + a^{2} b^{2} x} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx=\frac {2 \, b \sqrt {x}}{\sqrt {-b x + a} a^{2}} - \frac {2 \, \sqrt {-b x + a}}{a^{2} \sqrt {x}} \]

[In]

integrate(1/x^(3/2)/(-b*x+a)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (33) = 66\).

Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.29 \[ \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx=-\frac {4 \, \sqrt {-b} b^{2}}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} a {\left | b \right |}} - \frac {2 \, \sqrt {-b x + a} b^{2}}{\sqrt {{\left (b x - a\right )} b + a b} a^{2} {\left | b \right |}} \]

[In]

integrate(1/x^(3/2)/(-b*x+a)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx=-\frac {2\,a\,\sqrt {a-b\,x}-4\,b\,x\,\sqrt {a-b\,x}}{\sqrt {x}\,\left (a^3-a^2\,b\,x\right )} \]

[In]

int(1/(x^(3/2)*(a - b*x)^(3/2)),x)

[Out]

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