Optimal. Leaf size=59 \[ -\frac {2 b^2 (b x+2)^{3/2}}{105 x^{3/2}}+\frac {2 b (b x+2)^{3/2}}{35 x^{5/2}}-\frac {(b x+2)^{3/2}}{7 x^{7/2}} \]
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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} \[ -\frac {2 b^2 (b x+2)^{3/2}}{105 x^{3/2}}+\frac {2 b (b x+2)^{3/2}}{35 x^{5/2}}-\frac {(b x+2)^{3/2}}{7 x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {\sqrt {2+b x}}{x^{9/2}} \, dx &=-\frac {(2+b x)^{3/2}}{7 x^{7/2}}-\frac {1}{7} (2 b) \int \frac {\sqrt {2+b x}}{x^{7/2}} \, dx\\ &=-\frac {(2+b x)^{3/2}}{7 x^{7/2}}+\frac {2 b (2+b x)^{3/2}}{35 x^{5/2}}+\frac {1}{35} \left (2 b^2\right ) \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx\\ &=-\frac {(2+b x)^{3/2}}{7 x^{7/2}}+\frac {2 b (2+b x)^{3/2}}{35 x^{5/2}}-\frac {2 b^2 (2+b x)^{3/2}}{105 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.54 \[ -\frac {(b x+2)^{3/2} \left (2 b^2 x^2-6 b x+15\right )}{105 x^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 34, normalized size = 0.58 \[ -\frac {{\left (2 \, b^{3} x^{3} - 2 \, b^{2} x^{2} + 3 \, b x + 30\right )} \sqrt {b x + 2}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 55, normalized size = 0.93 \[ -\frac {{\left (35 \, b^{7} + 2 \, {\left ({\left (b x + 2\right )} b^{7} - 7 \, b^{7}\right )} {\left (b x + 2\right )}\right )} {\left (b x + 2\right )}^{\frac {3}{2}} b}{105 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {7}{2}} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.46 \[ -\frac {\left (b x +2\right )^{\frac {3}{2}} \left (2 b^{2} x^{2}-6 b x +15\right )}{105 x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 41, normalized size = 0.69 \[ -\frac {{\left (b x + 2\right )}^{\frac {3}{2}} b^{2}}{12 \, x^{\frac {3}{2}}} + \frac {{\left (b x + 2\right )}^{\frac {5}{2}} b}{10 \, x^{\frac {5}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {7}{2}}}{28 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 34, normalized size = 0.58 \[ -\frac {\sqrt {b\,x+2}\,\left (\frac {2\,b^3\,x^3}{105}-\frac {2\,b^2\,x^2}{105}+\frac {b\,x}{35}+\frac {2}{7}\right )}{x^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.80, size = 270, normalized size = 4.58 \[ - \frac {2 b^{\frac {19}{2}} x^{5} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {6 b^{\frac {17}{2}} x^{4} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {3 b^{\frac {15}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {34 b^{\frac {13}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {132 b^{\frac {11}{2}} x \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} - \frac {120 b^{\frac {9}{2}} \sqrt {1 + \frac {2}{b x}}}{105 b^{6} x^{5} + 420 b^{5} x^{4} + 420 b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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