Optimal. Leaf size=74 \[ -\frac {2 b^2 \sqrt {b x+2}}{5 \sqrt {x}}+\frac {2 b \sqrt {b x+2}}{5 x^{3/2}}-\frac {3 \sqrt {b x+2}}{5 x^{5/2}}+\frac {1}{x^{5/2} \sqrt {b x+2}} \]
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Rubi [A] time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac {2 b^2 \sqrt {b x+2}}{5 \sqrt {x}}+\frac {2 b \sqrt {b x+2}}{5 x^{3/2}}-\frac {3 \sqrt {b x+2}}{5 x^{5/2}}+\frac {1}{x^{5/2} \sqrt {b x+2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} (2+b x)^{3/2}} \, dx &=\frac {1}{x^{5/2} \sqrt {2+b x}}+3 \int \frac {1}{x^{7/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{x^{5/2} \sqrt {2+b x}}-\frac {3 \sqrt {2+b x}}{5 x^{5/2}}-\frac {1}{5} (6 b) \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{x^{5/2} \sqrt {2+b x}}-\frac {3 \sqrt {2+b x}}{5 x^{5/2}}+\frac {2 b \sqrt {2+b x}}{5 x^{3/2}}+\frac {1}{5} \left (2 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{x^{5/2} \sqrt {2+b x}}-\frac {3 \sqrt {2+b x}}{5 x^{5/2}}+\frac {2 b \sqrt {2+b x}}{5 x^{3/2}}-\frac {2 b^2 \sqrt {2+b x}}{5 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.53 \[ \frac {-2 b^3 x^3-2 b^2 x^2+b x-1}{5 x^{5/2} \sqrt {b x+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 47, normalized size = 0.64 \[ -\frac {{\left (2 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - b x + 1\right )} \sqrt {b x + 2} \sqrt {x}}{5 \, {\left (b x^{4} + 2 \, x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.11, size = 107, normalized size = 1.45 \[ -\frac {b^{\frac {9}{2}}}{2 \, {\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} {\left | b \right |}} - \frac {{\left (\frac {60 \, b^{6}}{{\left | b \right |}} + {\left (\frac {11 \, {\left (b x + 2\right )} b^{6}}{{\left | b \right |}} - \frac {50 \, b^{6}}{{\left | b \right |}}\right )} {\left (b x + 2\right )}\right )} \sqrt {b x + 2}}{40 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.47 \[ -\frac {2 b^{3} x^{3}+2 b^{2} x^{2}-b x +1}{5 \sqrt {b x +2}\, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 56, normalized size = 0.76 \[ -\frac {b^{3} \sqrt {x}}{8 \, \sqrt {b x + 2}} - \frac {3 \, \sqrt {b x + 2} b^{2}}{8 \, \sqrt {x}} + \frac {{\left (b x + 2\right )}^{\frac {3}{2}} b}{8 \, x^{\frac {3}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {5}{2}}}{40 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 46, normalized size = 0.62 \[ -\frac {\sqrt {b\,x+2}\,\left (\frac {2\,b\,x^2}{5}-\frac {x}{5}+\frac {1}{5\,b}+\frac {2\,b^2\,x^3}{5}\right )}{x^{7/2}+\frac {2\,x^{5/2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 10.97, size = 269, normalized size = 3.64 \[ - \frac {2 b^{\frac {29}{2}} x^{5} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} - \frac {10 b^{\frac {27}{2}} x^{4} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} - \frac {15 b^{\frac {25}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} - \frac {5 b^{\frac {23}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} - \frac {4 b^{\frac {19}{2}} \sqrt {1 + \frac {2}{b x}}}{5 b^{12} x^{5} + 30 b^{11} x^{4} + 60 b^{10} x^{3} + 40 b^{9} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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