Optimal. Leaf size=41 \[ -\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6269, 6263,
862, 37} \begin {gather*} -\frac {2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 862
Rule 6263
Rule 6269
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {1-a x}}{x^{5/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1-a^2 x^2}}{x^{5/2} \sqrt {1-a x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{5/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 41, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 45, normalized size = 1.10
method | result | size |
gosper | \(-\frac {2 \left (a x +1\right )^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \sqrt {-a^{2} x^{2}+1}}\) | \(40\) |
default | \(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (a x +1\right )}{3 x \left (a x -1\right )}\) | \(45\) |
risch | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a^{2} x^{2}+2 a x +1\right )}{3 \sqrt {-a^{2} x^{2}+1}\, x \sqrt {-a c x \left (a x +1\right )}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 47, normalized size = 1.15 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1} {\left (a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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]
________________________________________________________________________________________
Mupad [B]
time = 1.12, size = 44, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {2\,a^2\,x^2}{3}+\frac {4\,a\,x}{3}+\frac {2}{3}\right )}{x\,\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________