Optimal. Leaf size=45 \[ \frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\sin ^{-1}(a x) \]
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Rubi [A] time = 0.09, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1805, 844, 216, 266, 63, 208} \begin {gather*} \frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\sin ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1805
Rubi steps
\begin {align*} \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\int \frac {-1+a x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-a \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\sin ^{-1}(a x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\sin ^{-1}(a x)-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^2}\\ &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 59, normalized size = 1.31 \begin {gather*} \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1-a^2 x^2\right )-\sqrt {1-a^2 x^2} \sin ^{-1}(a x)+4 a x+3}{\sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.41, size = 97, normalized size = 2.16 \begin {gather*} -\frac {4 \sqrt {1-a^2 x^2}}{a x-1}-\frac {\sqrt {-a^2} \log \left (\sqrt {1-a^2 x^2}-\sqrt {-a^2} x\right )}{a}+2 \tanh ^{-1}\left (\sqrt {-a^2} x-\sqrt {1-a^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 82, normalized size = 1.82 \begin {gather*} \frac {4 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} - 4}{a x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.54, size = 87, normalized size = 1.93 \begin {gather*} -\frac {a \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {8 \, a}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 1.67 \begin {gather*} \frac {4 a x}{\sqrt {-a^{2} x^{2}+1}}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {4}{\sqrt {-a^{2} x^{2}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.08, size = 65, normalized size = 1.44 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 82, normalized size = 1.82 \begin {gather*} \frac {4\,a\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x + 1\right )^{3}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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