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

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

[Out]

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

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Rubi [A]  time = 0.006489, antiderivative size = 36, normalized size of antiderivative = 1., 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} \[ \frac{x^{1-m} \sqrt{a+b x}}{(1-m) \sqrt{-a-b x}} \]

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*}

Mathematica [A]  time = 0.0113113, size = 36, normalized size = 1. \[ \frac{x^{1-m} \sqrt{a+b x}}{(1-m) \sqrt{-a-b x}} \]

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 = 0.001, size = 31, normalized size = 0.9 \begin{align*} -{\frac{x}{ \left ( -1+m \right ){x}^{m}}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \end{align*}

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)

[Out]

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

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Maxima [C]  time = 1.31847, size = 20, normalized size = 0.56 \begin{align*} -\frac{x}{{\left (i \, m - i\right )} x^{m}} \end{align*}

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 = 1.7015, size = 84, normalized size = 2.33 \begin{align*} \frac{\sqrt{b x + a} \sqrt{-b x - a} x}{{\left (a m +{\left (b m - b\right )} x - a\right )} x^{m}} \end{align*}

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]  time = 6.19846, size = 144, normalized size = 4. \begin{align*} \begin{cases} - \frac{i a a^{- m} b^{m} \left (-1 + \frac{b \left (\frac{a}{b} + x\right )}{a}\right )^{- m}}{b \left (m - 1\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 - 1} & \text{for}\: \frac{\left |{b \left (\frac{a}{b} + x\right )}\right |}{\left |{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{align*}

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x + a}}{\sqrt{-b x - a} x^{m}}\,{d x} \end{align*}

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]

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