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 = 0.01, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 41
Rule 50
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.45, size = 55, normalized size = 0.86 \[ -\frac {{\left (a x + 4\right )} \sqrt {a x + 1} \sqrt {-a x + 1} + 6 \, \arctan \left (\frac {\sqrt {a x + 1} \sqrt {-a x + 1} - 1}{a x}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.70, size = 42, normalized size = 0.66 \[ -\frac {{\left (a x + 4\right )} \sqrt {a x + 1} \sqrt {-a x + 1} - 6 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {a x + 1}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 98, normalized size = 1.53 \[ \frac {3 \sqrt {\left (a x +1\right ) \left (-a x +1\right )}\, \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a x +1}\, \sqrt {-a x +1}\, \sqrt {a^{2}}}-\frac {\left (a x +1\right )^{\frac {3}{2}} \sqrt {-a x +1}}{2 a}-\frac {3 \sqrt {-a x +1}\, \sqrt {a x +1}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 42, normalized size = 0.66 \[ -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{2 \, a} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a\,x+1\right )}^{3/2}}{\sqrt {1-a\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 33.75, size = 75, normalized size = 1.17 \[ \begin {cases} \frac {2 \left (\begin {cases} - \frac {a x \sqrt {- a x + 1} \sqrt {a x + 1}}{4} - \sqrt {- a x + 1} \sqrt {a x + 1} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {a x + 1}}{2} \right )}}{2} & \text {for}\: a x - 1 \geq -2 \wedge a x - 1 < 0 \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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