Optimal. Leaf size=64 \[ -\frac{\sqrt{1-a x} (a x+1)^{3/2}}{2 a}-\frac{3 \sqrt{1-a x} \sqrt{a x+1}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.0117152, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {50, 41, 216} \[ -\frac{\sqrt{1-a x} (a x+1)^{3/2}}{2 a}-\frac{3 \sqrt{1-a x} \sqrt{a x+1}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{(1+a x)^{3/2}}{\sqrt{1-a x}} \, dx &=-\frac{\sqrt{1-a x} (1+a x)^{3/2}}{2 a}+\frac{3}{2} \int \frac{\sqrt{1+a x}}{\sqrt{1-a x}} \, dx\\ &=-\frac{3 \sqrt{1-a x} \sqrt{1+a x}}{2 a}-\frac{\sqrt{1-a x} (1+a x)^{3/2}}{2 a}+\frac{3}{2} \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 \sqrt{1-a x} \sqrt{1+a x}}{2 a}-\frac{\sqrt{1-a x} (1+a x)^{3/2}}{2 a}+\frac{3}{2} \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1-a x} \sqrt{1+a x}}{2 a}-\frac{\sqrt{1-a x} (1+a x)^{3/2}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0401253, size = 47, normalized size = 0.73 \[ -\frac{\sqrt{1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 98, normalized size = 1.5 \begin{align*} -{\frac{1}{2\,a} \left ( ax+1 \right ) ^{{\frac{3}{2}}}\sqrt{-ax+1}}-{\frac{3}{2\,a}\sqrt{-ax+1}\sqrt{ax+1}}+{\frac{3}{2}\sqrt{ \left ( ax+1 \right ) \left ( -ax+1 \right ) }\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47734, size = 69, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} x + \frac{3 \, \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64875, size = 138, normalized size = 2.16 \begin{align*} -\frac{{\left (a x + 4\right )} \sqrt{a x + 1} \sqrt{-a x + 1} + 6 \, \arctan \left (\frac{\sqrt{a x + 1} \sqrt{-a x + 1} - 1}{a x}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.9349, size = 75, normalized size = 1.17 \begin{align*} \begin{cases} \frac{2 \left (\begin{cases} - \frac{a x \sqrt{- a x + 1} \sqrt{a x + 1}}{4} - \sqrt{- a x + 1} \sqrt{a x + 1} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{a x + 1}}{2} \right )}}{2} & \text{for}\: a x - 1 \geq -2 \wedge a x - 1 < 0 \end{cases}\right )}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0726, size = 57, normalized size = 0.89 \begin{align*} -\frac{{\left (a x + 4\right )} \sqrt{a x + 1} \sqrt{-a x + 1} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{a x + 1}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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