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

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {23, 30} \begin {gather*} \frac {x^{1-m} \sqrt {a+b x}}{(1-m) \sqrt {-a-b x}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.00 \begin {gather*} \frac {x^{1-m} \sqrt {a+b x}}{(1-m) \sqrt {-a-b x}} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [A]
time = 0.07, size = 31, normalized size = 0.86

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [C] Result contains complex when optimal does not.
time = 0.32, size = 15, normalized size = 0.42 \begin {gather*} -\frac {x}{{\left (i \, m - i\right )} x^{m}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/((I*m - I)*x^m)

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Fricas [A]
time = 1.10, size = 42, normalized size = 1.17 \begin {gather*} \frac {\sqrt {b x + a} \sqrt {-b x - a} x}{{\left (a m + {\left (b m - b\right )} x - a\right )} x^{m}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [C] Result contains complex when optimal does not.
time = 1.58, size = 180, normalized size = 5.00 \begin {gather*} \begin {cases} - \frac {i a a^{- m} b^{m} \left (-1 + \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{- m} e^{i \pi m}}{b \left (m e^{i \pi m} - e^{i \pi m}\right )} + \frac {i a^{- m} b^{m} \left (-1 + \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{- m} \left (\frac {a}{b} + x\right ) 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 a^{- m} b^{m} \left (1 - \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{- m}}{b \left (m e^{i \pi m} - e^{i \pi m}\right )} + \frac {i a^{- m} b^{m} \left (1 - \frac {b \left (\frac {a}{b} + x\right )}{a}\right )^{- m} \left (\frac {a}{b} + x\right )}{m e^{i \pi m} - e^{i \pi m}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*a*b**m*exp(I*pi*m)/(a**m*b*(-1 + b*(a/b + x)/a)**m*(m*exp(I*pi*m) - exp(I*pi*m))) + I*b**m*(a/b
+ x)*exp(I*pi*m)/(a**m*(-1 + b*(a/b + x)/a)**m*(m*exp(I*pi*m) - exp(I*pi*m))), Abs(b*(a/b + x)/a) > 1), (-I*a*
b**m/(a**m*b*(1 - b*(a/b + x)/a)**m*(m*exp(I*pi*m) - exp(I*pi*m))) + I*b**m*(a/b + x)/(a**m*(1 - b*(a/b + x)/a
)**m*(m*exp(I*pi*m) - exp(I*pi*m))), True))

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Giac [C] Result contains complex when optimal does not.
time = 3.10, size = 14, normalized size = 0.39 \begin {gather*} \frac {i \, x^{-m + 1}}{m - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x^(-m + 1)/(m - 1)

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Mupad [B]
time = 1.20, size = 31, normalized size = 0.86 \begin {gather*} -\frac {x^{1-m}\,\sqrt {a+b\,x}}{\left (m-1\right )\,\sqrt {-a-b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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