\(\int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx\) [820]

Optimal result

Integrand size = 27, antiderivative size = 36 \[ \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx=\frac {x^{1-m} \sqrt {a+b x}}{(1-m) \sqrt {-a-b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {23, 30} \[ \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx=\frac {x^{1-m} \sqrt {a+b x}}{(1-m) \sqrt {-a-b x}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx=\frac {x^{1-m} \sqrt {a+b x}}{(1-m) \sqrt {-a-b x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86

method	result	size
gosper	\(-\frac {x \sqrt {b x +a}\, x^{-m}}{\left (-1+m \right ) \sqrt {-b x -a}}\)	\(31\)
risch	\(\frac {i \sqrt {\frac {-b x -a}{b x +a}}\, \sqrt {b x +a}\, x \,x^{-m}}{\sqrt {-b x -a}\, \left (-1+m \right )}\)	\(50\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx=\frac {\sqrt {b x + a} \sqrt {-b x - a} x}{{\left (a m + {\left (b m - b\right )} x - a\right )} x^{m}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.48 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.69 \[ \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx=\begin {cases} \frac {i a^{1 - m} b^{m - 1} \left (-1 + \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{1 - m} e^{i \pi m}}{m e^{i \pi m} - e^{i \pi m}} & \text {for}\: \left |{\frac {b \left (\frac {a}{b} + x\right )}{a}}\right | > 1 \\- \frac {i a^{1 - m} b^{m - 1} \left (1 - \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{1 - m}}{m e^{i \pi m} - e^{i \pi m}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((I*a**(1 - m)*b**(m - 1)*(-1 + b*(a/b + x)/a)**(1 - m)*exp(I*pi*m)/(m*exp(I*pi*m) - exp(I*pi*m)), Ab
s(b*(a/b + x)/a) > 1), (-I*a**(1 - m)*b**(m - 1)*(1 - b*(a/b + x)/a)**(1 - m)/(m*exp(I*pi*m) - exp(I*pi*m)), T
rue))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.42 \[ \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx=-\frac {x}{{\left (i \, m - i\right )} x^{m}} \]

[In]

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

[Out]

-x/((I*m - I)*x^m)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.39 \[ \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx=\frac {i \, x^{-m + 1}}{m - 1} \]

[In]

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

[Out]

I*x^(-m + 1)/(m - 1)

Mupad [B] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {x^{-m} \sqrt {a+b x}}{\sqrt {-a-b x}} \, dx=-\frac {x^{1-m}\,\sqrt {a+b\,x}}{\left (m-1\right )\,\sqrt {-a-b\,x}} \]

[In]

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

[Out]

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