Optimal. Leaf size=39 \[ \frac {2}{a \sqrt {x} \sqrt {a+b x}}-\frac {4 \sqrt {a+b x}}{a^2 \sqrt {x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37}
\begin {gather*} \frac {2}{a \sqrt {x} \sqrt {a+b x}}-\frac {4 \sqrt {a+b x}}{a^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx &=\frac {2}{a \sqrt {x} \sqrt {a+b x}}+\frac {2 \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{a}\\ &=\frac {2}{a \sqrt {x} \sqrt {a+b x}}-\frac {4 \sqrt {a+b x}}{a^2 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 25, normalized size = 0.64 \begin {gather*} -\frac {2 (a+2 b x)}{a^2 \sqrt {x} \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.62, size = 37, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {b} \left (-a-2 b x\right ) \sqrt {\frac {a+b x}{b x}}}{a^2 \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 33, normalized size = 0.85
method | result | size |
gosper | \(-\frac {2 \left (2 b x +a \right )}{\sqrt {x}\, \sqrt {b x +a}\, a^{2}}\) | \(22\) |
default | \(-\frac {2}{a \sqrt {x}\, \sqrt {b x +a}}-\frac {4 b \sqrt {x}}{a^{2} \sqrt {b x +a}}\) | \(33\) |
risch | \(-\frac {2 \sqrt {b x +a}}{a^{2} \sqrt {x}}-\frac {2 b \sqrt {x}}{a^{2} \sqrt {b x +a}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 32, normalized size = 0.82 \begin {gather*} -\frac {2 \, b \sqrt {x}}{\sqrt {b x + a} a^{2}} - \frac {2 \, \sqrt {b x + a}}{a^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 34, normalized size = 0.87 \begin {gather*} -\frac {2 \, {\left (2 \, b x + a\right )} \sqrt {b x + a} \sqrt {x}}{a^{2} b x^{2} + a^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.88, size = 41, normalized size = 1.05 \begin {gather*} - \frac {2}{a \sqrt {b} x \sqrt {\frac {a}{b x} + 1}} - \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 73, normalized size = 1.87 \begin {gather*} 2 \left (-\frac {\frac {1}{2}\cdot 2 b \sqrt {x} \sqrt {a+b x}}{a^{2} \left (a+b x\right )}+\frac {4 \sqrt {b}}{2 a \left (\left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )^{2}-a\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 39, normalized size = 1.00 \begin {gather*} -\frac {2\,a\,\sqrt {a+b\,x}+4\,b\,x\,\sqrt {a+b\,x}}{\sqrt {x}\,\left (a^3+b\,x\,a^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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