\(\int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx\) [581]

Optimal result

Integrand size = 26, antiderivative size = 36 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2 \sqrt {b-\frac {a}{x}} x^{1+m}}{(1+2 m) \sqrt {a-b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {529, 23, 30} \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2 x^{m+1} \sqrt {b-\frac {a}{x}}}{(2 m+1) \sqrt {a-b x}} \]

[In]

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

[Out]

(2*Sqrt[b - a/x]*x^(1 + m))/((1 + 2*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]

Rule 529

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Dist[x^(n*FracPar
t[q])*((c + d/x^n)^FracPart[q]/(d + c*x^n)^FracPart[q]), Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x], x] /
; FreeQ[{a, b, c, d, m, n, p, q}, x] && EqQ[mn, -n] &&  !IntegerQ[q] &&  !IntegerQ[p]

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

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {\sqrt {b-\frac {a}{x}} x^{1+m}}{\left (\frac {1}{2}+m\right ) \sqrt {a-b x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {2 x^{1+m} \sqrt {-\frac {-b x +a}{x}}}{\left (1+2 m \right ) \sqrt {-b x +a}}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2 \, \sqrt {-b x + a} x x^{m} \sqrt {\frac {b x - a}{x}}}{2 \, a m - {\left (2 \, b m + b\right )} x + a} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**m*sqrt(-a/x + b)/sqrt(a - b*x), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2 \, \sqrt {x} x^{m}}{2 i \, m + i} \]

[In]

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

[Out]

2*sqrt(x)*x^m/(2*I*m + I)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 17.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx=\frac {2\,x^{m+1}\,\sqrt {b-\frac {a}{x}}}{\left (2\,m+1\right )\,\sqrt {a-b\,x}} \]

[In]

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

[Out]

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