Optimal. Leaf size=26 \[ -\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {23, 30}
\begin {gather*} -\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 23
Rule 30
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{x^2 \sqrt {-a-b x}} \, dx &=\frac {\sqrt {a+b x} \int \frac {1}{x^2} \, dx}{\sqrt {-a-b x}}\\ &=-\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+b x}}{x \sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 22, normalized size = 0.85
method | result | size |
default | \(\frac {\sqrt {-b x -a}}{\sqrt {b x +a}\, x}\) | \(22\) |
gosper | \(-\frac {\sqrt {b x +a}}{x \sqrt {-b x -a}}\) | \(23\) |
risch | \(\frac {i \sqrt {\frac {-b x -a}{b x +a}}\, \sqrt {b x +a}}{\sqrt {-b x -a}\, x}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 28, normalized size = 1.08 \begin {gather*} \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.85, size = 20, normalized size = 0.77 \begin {gather*} \frac {i b^{2} \left (\frac {a}{b} + x\right )}{- a^{2} + a b \left (\frac {a}{b} + x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 2.08, size = 5, normalized size = 0.19 \begin {gather*} \frac {i}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.14, size = 21, normalized size = 0.81 \begin {gather*} \frac {\sqrt {-a-b\,x}}{x\,\sqrt {a+b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________