Optimal. Leaf size=55 \[ -\frac {2 \sqrt {b x+2}}{3 \sqrt {x}}+\frac {2}{3 \sqrt {x} \sqrt {b x+2}}+\frac {1}{3 \sqrt {x} (b x+2)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac {2 \sqrt {b x+2}}{3 \sqrt {x}}+\frac {2}{3 \sqrt {x} \sqrt {b x+2}}+\frac {1}{3 \sqrt {x} (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (2+b x)^{5/2}} \, dx &=\frac {1}{3 \sqrt {x} (2+b x)^{3/2}}+\frac {2}{3} \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx\\ &=\frac {1}{3 \sqrt {x} (2+b x)^{3/2}}+\frac {2}{3 \sqrt {x} \sqrt {2+b x}}+\frac {2}{3} \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{3 \sqrt {x} (2+b x)^{3/2}}+\frac {2}{3 \sqrt {x} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.58 \[ \frac {-2 b^2 x^2-6 b x-3}{3 \sqrt {x} (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 45, normalized size = 0.82 \[ -\frac {{\left (2 \, b^{2} x^{2} + 6 \, b x + 3\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{3} + 4 \, b x^{2} + 4 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.36, size = 145, normalized size = 2.64 \[ -\frac {\sqrt {b x + 2} b^{2}}{4 \, \sqrt {{\left (b x + 2\right )} b - 2 \, b} {\left | b \right |}} - \frac {3 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} b^{\frac {5}{2}} + 24 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac {7}{2}} + 20 \, b^{\frac {9}{2}}}{3 \, {\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.49 \[ -\frac {2 b^{2} x^{2}+6 b x +3}{3 \left (b x +2\right )^{\frac {3}{2}} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 40, normalized size = 0.73 \[ \frac {{\left (b^{2} - \frac {6 \, {\left (b x + 2\right )} b}{x}\right )} x^{\frac {3}{2}}}{12 \, {\left (b x + 2\right )}^{\frac {3}{2}}} - \frac {\sqrt {b x + 2}}{4 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 57, normalized size = 1.04 \[ -\frac {3\,\sqrt {b\,x+2}+6\,b\,x\,\sqrt {b\,x+2}+2\,b^2\,x^2\,\sqrt {b\,x+2}}{\sqrt {x}\,\left (x\,\left (3\,x\,b^2+12\,b\right )+12\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.89, size = 117, normalized size = 2.13 \[ - \frac {2 b^{\frac {13}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} - \frac {6 b^{\frac {11}{2}} x \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} - \frac {3 b^{\frac {9}{2}} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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