Optimal. Leaf size=45 \[ -\frac {10 a \sqrt {1-a x}}{3 \sqrt {a x}}-\frac {2 a \sqrt {1-a x}}{3 (a x)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {16, 78, 37} \[ -\frac {10 a \sqrt {1-a x}}{3 \sqrt {a x}}-\frac {2 a \sqrt {1-a x}}{3 (a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {1+a x}{x^2 \sqrt {a x} \sqrt {1-a x}} \, dx &=a^2 \int \frac {1+a x}{(a x)^{5/2} \sqrt {1-a x}} \, dx\\ &=-\frac {2 a \sqrt {1-a x}}{3 (a x)^{3/2}}+\frac {1}{3} \left (5 a^2\right ) \int \frac {1}{(a x)^{3/2} \sqrt {1-a x}} \, dx\\ &=-\frac {2 a \sqrt {1-a x}}{3 (a x)^{3/2}}-\frac {10 a \sqrt {1-a x}}{3 \sqrt {a x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.64 \[ -\frac {2 \sqrt {-a x (a x-1)} (5 a x+1)}{3 a x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 27, normalized size = 0.60 \[ -\frac {2 \, {\left (5 \, a x + 1\right )} \sqrt {a x} \sqrt {-a x + 1}}{3 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.27, size = 88, normalized size = 1.96 \[ -\frac {\frac {a^{2} {\left (\sqrt {-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac {3}{2}}} + \frac {21 \, a^{2} {\left (\sqrt {-a x + 1} - 1\right )}}{\sqrt {a x}} - \frac {{\left (a^{2} + \frac {21 \, a {\left (\sqrt {-a x + 1} - 1\right )}^{2}}{x}\right )} \left (a x\right )^{\frac {3}{2}}}{{\left (\sqrt {-a x + 1} - 1\right )}^{3}}}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.56 \[ -\frac {2 \left (5 a x +1\right ) \sqrt {-a x +1}}{3 \sqrt {a x}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 42, normalized size = 0.93 \[ -\frac {10 \, \sqrt {-a^{2} x^{2} + a x}}{3 \, x} - \frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{3 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.75, size = 24, normalized size = 0.53 \[ -\frac {\sqrt {1-a\,x}\,\left (\frac {10\,a\,x}{3}+\frac {2}{3}\right )}{x\,\sqrt {a\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 15.24, size = 107, normalized size = 2.38 \[ a \left (\begin {cases} - 2 \sqrt {-1 + \frac {1}{a x}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- 2 i \sqrt {1 - \frac {1}{a x}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {4 a \sqrt {-1 + \frac {1}{a x}}}{3} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{3 x} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {4 i a \sqrt {1 - \frac {1}{a x}}}{3} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{3 x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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