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

Optimal. Leaf size=36 \[ \frac {2 x^{m+1} \sqrt {b-\frac {a}{x}}}{(2 m+1) \sqrt {a-b x}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {515, 23, 30} \[ \frac {2 x^{m+1} \sqrt {b-\frac {a}{x}}}{(2 m+1) \sqrt {a-b x}} \]

Antiderivative was successfully verified.

[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 515

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Dist[(x^(n*FracPa
rt[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*} \int \frac {\sqrt {b-\frac {a}{x}} x^m}{\sqrt {a-b x}} \, dx &=\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*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 0.97 \[ \frac {x^{m+1} \sqrt {b-\frac {a}{x}}}{\left (m+\frac {1}{2}\right ) \sqrt {a-b x}} \]

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 0.43, size = 44, normalized size = 1.22 \[ \frac {2 \, \sqrt {-b x + a} x x^{m} \sqrt {\frac {b x - a}{x}}}{2 \, a m - {\left (2 \, b m + b\right )} x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):
Check [abs(t_nostep)]Undef/Unsigned Inf encountered in limitLimit: Max order reached or unable to make series
expansion Error: Bad Argument Value

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maple [A]  time = 0.01, size = 36, normalized size = 1.00 \[ \frac {2 \sqrt {-\frac {-b x +a}{x}}\, x^{m +1}}{\left (2 m +1\right ) \sqrt {-b x +a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.71, size = 15, normalized size = 0.42 \[ \frac {2 \, \sqrt {x} x^{m}}{2 i \, m + i} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 3.26, size = 32, normalized size = 0.89 \[ \frac {2\,x^{m+1}\,\sqrt {b-\frac {a}{x}}}{\left (2\,m+1\right )\,\sqrt {a-b\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {- \frac {a}{x} + b}}{\sqrt {a - b x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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