Optimal. Leaf size=64 \[ -\frac {3 \sqrt {1-a x} \sqrt {1+a x}}{2 a}-\frac {\sqrt {1-a x} (1+a x)^{3/2}}{2 a}+\frac {3 \sin ^{-1}(a x)}{2 a} \]
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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 = {52, 41, 222}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 52
Rule 222
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 = 0.09, size = 64, normalized size = 1.00 \begin {gather*} -\frac {\frac {\sqrt {1-a x} \left (4+5 a x+a^2 x^2\right )}{\sqrt {1+a x}}+6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: } \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.15, size = 98, normalized size = 1.53
method | result | size |
default | \(-\frac {\left (a x +1\right )^{\frac {3}{2}} \sqrt {-a x +1}}{2 a}-\frac {3 \sqrt {-a x +1}\, \sqrt {a x +1}}{2 a}+\frac {3 \sqrt {\left (a x +1\right ) \left (-a x +1\right )}\, \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a x +1}\, \sqrt {-a x +1}\, \sqrt {a^{2}}}\) | \(98\) |
risch | \(\frac {\left (a x +4\right ) \sqrt {a x +1}\, \left (a x -1\right ) \sqrt {\left (a x +1\right ) \left (-a x +1\right )}}{2 a \sqrt {-\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {-a x +1}}+\frac {3 \sqrt {\left (a x +1\right ) \left (-a x +1\right )}\, \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a x +1}\, \sqrt {-a x +1}\, \sqrt {a^{2}}}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 42, normalized size = 0.66 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 55, normalized size = 0.86 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.28, size = 88, normalized size = 1.38 \begin {gather*} \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}\: \sqrt {a x + 1} > - \sqrt {2} \wedge \sqrt {a x + 1} < \sqrt {2} \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 71, normalized size = 1.11 \begin {gather*} \frac {2 \left (-\frac {3}{4}-\frac {1}{4} \sqrt {a x+1} \sqrt {a x+1}\right ) \sqrt {a x+1} \sqrt {-a x+1}+3 \arcsin \left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )}{a} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a\,x+1\right )}^{3/2}}{\sqrt {1-a\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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