Optimal. Leaf size=51 \[ \frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {98, 21, 105, 41, 216, 92, 208} \[ \frac {4 \sqrt {a x+1}}{\sqrt {1-a x}}-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 41
Rule 92
Rule 98
Rule 105
Rule 208
Rule 216
Rubi steps
\begin {align*} \int \frac {(1+a x)^{3/2}}{x (1-a x)^{3/2}} \, dx &=\frac {4 \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 \int \frac {-\frac {a}{2}+\frac {a^2 x}{2}}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{a}\\ &=\frac {4 \sqrt {1+a x}}{\sqrt {1-a x}}+\int \frac {\sqrt {1-a x}}{x \sqrt {1+a x}} \, dx\\ &=\frac {4 \sqrt {1+a x}}{\sqrt {1-a x}}-a \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx+\int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {4 \sqrt {1+a x}}{\sqrt {1-a x}}-a \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-a \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )\\ &=\frac {4 \sqrt {1+a x}}{\sqrt {1-a x}}-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 72, normalized size = 1.41 \[ \frac {2 \left (\sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+2 a x+2\right )}{\sqrt {1-a^2 x^2}}-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.42, size = 93, normalized size = 1.82 \[ \frac {4 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {a x + 1} \sqrt {-a x + 1} - 1}{a x}\right ) + {\left (a x - 1\right )} \log \left (\frac {\sqrt {a x + 1} \sqrt {-a x + 1} - 1}{x}\right ) - 4 \, \sqrt {a x + 1} \sqrt {-a x + 1} - 4}{a x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.04, size = 134, normalized size = 2.63 \[ \frac {\left (-a x \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\relax (a )-a x \arctan \left (\frac {a x \,\mathrm {csgn}\relax (a )}{\sqrt {-\left (a x +1\right ) \left (a x -1\right )}}\right )+\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\relax (a )+\arctan \left (\frac {a x \,\mathrm {csgn}\relax (a )}{\sqrt {-\left (a x +1\right ) \left (a x -1\right )}}\right )-4 \sqrt {-a^{2} x^{2}+1}\, \mathrm {csgn}\relax (a )\right ) \sqrt {-a x +1}\, \sqrt {a x +1}\, \mathrm {csgn}\relax (a )}{\left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.50, size = 65, normalized size = 1.27 \[ \frac {4 \, a x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4}{\sqrt {-a^{2} x^{2} + 1}} - \arcsin \left (a x\right ) - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a\,x+1\right )}^{3/2}}{x\,{\left (1-a\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x + 1\right )^{\frac {3}{2}}}{x \left (- a x + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________