\(\int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx\) [819]

Optimal result

Integrand size = 25, antiderivative size = 28 \[ \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx=-\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {23, 30} \[ \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx=-\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}} \]

[In]

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

[Out]

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx=-\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82

method	result	size
gosper	\(-\frac {\sqrt {b x +a}}{2 x^{2} \sqrt {-b x -a}}\)	\(23\)
default	\(\frac {\sqrt {-b x -a}}{2 \sqrt {b x +a}\, x^{2}}\)	\(23\)
risch	\(\frac {i \sqrt {\frac {-b x -a}{b x +a}}\, \sqrt {b x +a}}{2 \sqrt {-b x -a}\, x^{2}}\)	\(42\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx=\frac {\sqrt {-b^{2}}}{2 \, b x^{2}} \]

[In]

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

[Out]

1/2*sqrt(-b^2)/(b*x^2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.99 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx=\frac {2 i a b^{3} \left (\frac {a}{b} + x\right )}{2 a^{4} - 4 a^{3} b \left (\frac {a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac {a}{b} + x\right )^{2}} - \frac {i b^{4} \left (\frac {a}{b} + x\right )^{2}}{2 a^{4} - 4 a^{3} b \left (\frac {a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac {a}{b} + x\right )^{2}} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (22) = 44\).

Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx=-\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}} b}{2 \, a^{2} x} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}}}{2 \, a x^{2}} \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx=\frac {i}{2 \, x^{2}} \]

[In]

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

[Out]

1/2*I/x^2

Mupad [B] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx=\frac {\sqrt {-a-b\,x}}{2\,x^2\,\sqrt {a+b\,x}} \]

[In]

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

[Out]

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