3.817 \(\int \frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \, dx\)

Optimal. Leaf size=24 \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]

[Out]

(Sqrt[a + b*x]*Log[x])/Sqrt[-a - b*x]

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Rubi [A]  time = 0.0063245, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {23, 29} \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*Log[x])/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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \, dx &=\frac{\sqrt{a+b x} \int \frac{1}{x} \, dx}{\sqrt{-a-b x}}\\ &=\frac{\sqrt{a+b x} \log (x)}{\sqrt{-a-b x}}\\ \end{align*}

Mathematica [A]  time = 0.0043989, size = 24, normalized size = 1. \[ \frac{\log (x) \sqrt{a+b x}}{\sqrt{-a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*Log[x])/Sqrt[-a - b*x]

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Maple [A]  time = 0.006, size = 22, normalized size = 0.9 \begin{align*} -{\ln \left ( x \right ) \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/(-b*x-a)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78665, size = 4, normalized size = 0.17 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [C]  time = 2.20083, size = 39, normalized size = 1.62 \begin{align*} \begin{cases} - i \log{\left (-1 + \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{for}\: \frac{\left |{b \left (\frac{a}{b} + x\right )}\right |}{\left |{a}\right |} > 1 \\- i \log{\left (1 - \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.34537, size = 18, normalized size = 0.75 \begin{align*} -i \, \log \left ({\left | b x \right |}\right ) + i \, \log \left ({\left | a \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(abs(b*x)) + I*log(abs(a))