Optimal. Leaf size=123 \[ \frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {46 a^2 \sqrt {c-\frac {c}{a x}} x}{3 \sqrt {1-a x} \sqrt {1+a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6269, 6264, 91,
79, 37} \begin {gather*} \frac {46 a^2 x \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {a x+1}}+\frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {a x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 79
Rule 91
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int \frac {e^{-3 \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^{-3 \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 {(1-a x)^2}{x^{5/2} (1+a x)^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {-5 a+\frac {3 a^2 x}{2}}{x^{3/2} (1+a x)^{3/2}} \, dx}{3 \sqrt {1-a x}}\\ &=\frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (23 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} (1+a x)^{3/2}} \, dx}{3 \sqrt {1-a x}}\\ &=\frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {46 a^2 \sqrt {c-\frac {c}{a x}} x}{3 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 50, normalized size = 0.41 \begin {gather*} \frac {2 \sqrt {c-\frac {c}{a x}} \left (-1+10 a x+23 a^2 x^2\right )}{3 x \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.88, size = 61, normalized size = 0.50
method | result | size |
gosper | \(\frac {2 \left (23 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x \left (a x +1\right )^{2} \left (a x -1\right )^{2}}\) | \(61\) |
default | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (23 a^{2} x^{2}+10 a x -1\right )}{3 x \left (a x +1\right ) \left (a x -1\right )}\) | \(61\) |
risch | \(\frac {2 \left (11 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{3 x \sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}+\frac {8 a^{2} x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}\) | \(151\) |
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 = 0.35, size = 58, normalized size = 0.47 \begin {gather*} -\frac {2 \, {\left (23 \, a^{2} x^{2} + 10 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a^{2} x^{3} - 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 (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a x + 1\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.12, size = 80, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {46\,x^2\,\sqrt {1-a^2\,x^2}}{3}-\frac {2\,\sqrt {1-a^2\,x^2}}{3\,a^2}+\frac {20\,x\,\sqrt {1-a^2\,x^2}}{3\,a}\right )}{\frac {x}{a^2}-x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________