Optimal. Leaf size=71 \[ -\frac {2 \sqrt {b x+2}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {b x+2}}+\frac {1}{3 x^{3/2} (b x+2)^{3/2}}+\frac {2 b \sqrt {b x+2}}{3 \sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 71, 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} \begin {gather*} -\frac {2 \sqrt {b x+2}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {b x+2}}+\frac {1}{3 x^{3/2} (b x+2)^{3/2}}+\frac {2 b \sqrt {b x+2}}{3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (2+b x)^{5/2}} \, dx &=\frac {1}{3 x^{3/2} (2+b x)^{3/2}}+\int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx\\ &=\frac {1}{3 x^{3/2} (2+b x)^{3/2}}+\frac {1}{x^{3/2} \sqrt {2+b x}}+2 \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{3 x^{3/2} (2+b x)^{3/2}}+\frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}-\frac {1}{3} (2 b) \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx\\ &=\frac {1}{3 x^{3/2} (2+b x)^{3/2}}+\frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}+\frac {2 b \sqrt {2+b x}}{3 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.56 \begin {gather*} \frac {2 b^3 x^3+6 b^2 x^2+3 b x-1}{3 x^{3/2} (b x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 40, normalized size = 0.56 \begin {gather*} \frac {2 b^3 x^3+6 b^2 x^2+3 b x-1}{3 x^{3/2} (b x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 55, normalized size = 0.77 \begin {gather*} \frac {{\left (2 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 3 \, b x - 1\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{4} + 4 \, b x^{3} + 4 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.27, size = 158, normalized size = 2.23 \begin {gather*} \frac {{\left (4 \, {\left (b x + 2\right )} b^{2} {\left | b \right |} - 9 \, b^{2} {\left | b \right |}\right )} \sqrt {b x + 2}}{12 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {3}{2}}} + \frac {3 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} b^{\frac {7}{2}} + 18 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac {9}{2}} + 16 \, b^{\frac {11}{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 |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.49 \begin {gather*} \frac {2 b^{3} x^{3}+6 b^{2} x^{2}+3 b x -1}{3 \left (b x +2\right )^{\frac {3}{2}} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 55, normalized size = 0.77 \begin {gather*} \frac {3 \, \sqrt {b x + 2} b}{8 \, \sqrt {x}} - \frac {{\left (b^{3} - \frac {9 \, {\left (b x + 2\right )} b^{2}}{x}\right )} x^{\frac {3}{2}}}{24 \, {\left (b x + 2\right )}^{\frac {3}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{24 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 71, normalized size = 1.00 \begin {gather*} \frac {3\,b\,x\,\sqrt {b\,x+2}-\sqrt {b\,x+2}+6\,b^2\,x^2\,\sqrt {b\,x+2}+2\,b^3\,x^3\,\sqrt {b\,x+2}}{x^{3/2}\,\left (x\,\left (3\,x\,b^2+12\,b\right )+12\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.85, size = 257, normalized size = 3.62 \begin {gather*} \frac {2 b^{\frac {27}{2}} x^{4} \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac {10 b^{\frac {25}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac {15 b^{\frac {23}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} + \frac {5 b^{\frac {21}{2}} x \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} - \frac {2 b^{\frac {19}{2}} \sqrt {1 + \frac {2}{b x}}}{3 b^{12} x^{4} + 18 b^{11} x^{3} + 36 b^{10} x^{2} + 24 b^{9} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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