Optimal. Leaf size=59 \[ -\frac {2 b^2 \sqrt {b x+2}}{15 \sqrt {x}}+\frac {2 b \sqrt {b x+2}}{15 x^{3/2}}-\frac {\sqrt {b x+2}}{5 x^{5/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 59, 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} \[ -\frac {2 b^2 \sqrt {b x+2}}{15 \sqrt {x}}+\frac {2 b \sqrt {b x+2}}{15 x^{3/2}}-\frac {\sqrt {b x+2}}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \sqrt {2+b x}} \, dx &=-\frac {\sqrt {2+b x}}{5 x^{5/2}}-\frac {1}{5} (2 b) \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx\\ &=-\frac {\sqrt {2+b x}}{5 x^{5/2}}+\frac {2 b \sqrt {2+b x}}{15 x^{3/2}}+\frac {1}{15} \left (2 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\\ &=-\frac {\sqrt {2+b x}}{5 x^{5/2}}+\frac {2 b \sqrt {2+b x}}{15 x^{3/2}}-\frac {2 b^2 \sqrt {2+b x}}{15 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.54 \[ -\frac {\sqrt {b x+2} \left (2 b^2 x^2-2 b x+3\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 26, normalized size = 0.44 \[ -\frac {{\left (2 \, b^{2} x^{2} - 2 \, b x + 3\right )} \sqrt {b x + 2}}{15 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 55, normalized size = 0.93 \[ -\frac {{\left (15 \, b^{5} + 2 \, {\left ({\left (b x + 2\right )} b^{5} - 5 \, b^{5}\right )} {\left (b x + 2\right )}\right )} \sqrt {b x + 2} b}{15 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {5}{2}} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.46 \[ -\frac {\sqrt {b x +2}\, \left (2 b^{2} x^{2}-2 b x +3\right )}{15 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 41, normalized size = 0.69 \[ -\frac {\sqrt {b x + 2} b^{2}}{4 \, \sqrt {x}} + \frac {{\left (b x + 2\right )}^{\frac {3}{2}} b}{6 \, x^{\frac {3}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {5}{2}}}{20 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 26, normalized size = 0.44 \[ -\frac {\sqrt {b\,x+2}\,\left (\frac {2\,b^2\,x^2}{15}-\frac {2\,b\,x}{15}+\frac {1}{5}\right )}{x^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.10, size = 224, normalized size = 3.80 \[ - \frac {2 b^{\frac {17}{2}} x^{4} \sqrt {1 + \frac {2}{b x}}}{15 b^{6} x^{4} + 60 b^{5} x^{3} + 60 b^{4} x^{2}} - \frac {6 b^{\frac {15}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{15 b^{6} x^{4} + 60 b^{5} x^{3} + 60 b^{4} x^{2}} - \frac {3 b^{\frac {13}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{15 b^{6} x^{4} + 60 b^{5} x^{3} + 60 b^{4} x^{2}} - \frac {4 b^{\frac {11}{2}} x \sqrt {1 + \frac {2}{b x}}}{15 b^{6} x^{4} + 60 b^{5} x^{3} + 60 b^{4} x^{2}} - \frac {12 b^{\frac {9}{2}} \sqrt {1 + \frac {2}{b x}}}{15 b^{6} x^{4} + 60 b^{5} x^{3} + 60 b^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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