Optimal. Leaf size=86 \[ -\frac {10 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}+\frac {5 \sqrt {x} \sqrt {b x+2}}{b^3}-\frac {10 x^{3/2}}{3 b^2 \sqrt {b x+2}}-\frac {2 x^{5/2}}{3 b (b x+2)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {47, 50, 54, 215} \begin {gather*} -\frac {10 x^{3/2}}{3 b^2 \sqrt {b x+2}}+\frac {5 \sqrt {x} \sqrt {b x+2}}{b^3}-\frac {10 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}-\frac {2 x^{5/2}}{3 b (b x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 215
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{(2+b x)^{5/2}} \, dx &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}+\frac {5 \int \frac {x^{3/2}}{(2+b x)^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{b^2}\\ &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \sqrt {x} \sqrt {2+b x}}{b^3}-\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{b^3}\\ &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \sqrt {x} \sqrt {2+b x}}{b^3}-\frac {10 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \sqrt {x} \sqrt {2+b x}}{b^3}-\frac {10 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.35 \begin {gather*} \frac {x^{7/2} \, _2F_1\left (\frac {5}{2},\frac {7}{2};\frac {9}{2};-\frac {b x}{2}\right )}{14 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 73, normalized size = 0.85 \begin {gather*} \frac {10 \log \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{b^{7/2}}+\frac {3 b^2 x^{5/2}+40 b x^{3/2}+60 \sqrt {x}}{3 b^3 (b x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 186, normalized size = 2.16 \begin {gather*} \left [\frac {15 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )} \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) + {\left (3 \, b^{3} x^{2} + 40 \, b^{2} x + 60 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{6} x^{2} + 4 \, b^{5} x + 4 \, b^{4}\right )}}, \frac {30 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (3 \, b^{3} x^{2} + 40 \, b^{2} x + 60 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{6} x^{2} + 4 \, b^{5} x + 4 \, b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 10.82, size = 182, normalized size = 2.12 \begin {gather*} \frac {{\left (\frac {15 \, \log \left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{b^{\frac {5}{2}}} + \frac {3 \, \sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2}}{b^{3}} + \frac {16 \, {\left (9 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} \sqrt {b} + 24 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac {3}{2}} + 28 \, b^{\frac {5}{2}}\right )}}{{\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} b^{2}}\right )} {\left | b \right |}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 136, normalized size = 1.58 \begin {gather*} \frac {\left (-\frac {5 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{b^{\frac {7}{2}}}+\frac {28 \sqrt {\left (x +\frac {2}{b}\right )^{2} b -2 x -\frac {4}{b}}}{3 \left (x +\frac {2}{b}\right ) b^{4}}-\frac {8 \sqrt {\left (x +\frac {2}{b}\right )^{2} b -2 x -\frac {4}{b}}}{3 \left (x +\frac {2}{b}\right )^{2} b^{5}}\right ) \sqrt {\left (b x +2\right ) x}}{\sqrt {b x +2}\, \sqrt {x}}+\frac {\sqrt {b x +2}\, \sqrt {x}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 105, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (2 \, b^{2} + \frac {10 \, {\left (b x + 2\right )} b}{x} - \frac {15 \, {\left (b x + 2\right )}^{2}}{x^{2}}\right )}}{3 \, {\left (\frac {{\left (b x + 2\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} - \frac {{\left (b x + 2\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}\right )}} + \frac {5 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{5/2}}{{\left (b\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.62, size = 308, normalized size = 3.58 \begin {gather*} \frac {3 b^{\frac {23}{2}} x^{15}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} + \frac {40 b^{\frac {21}{2}} x^{14}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} + \frac {60 b^{\frac {19}{2}} x^{13}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} - \frac {30 b^{10} x^{\frac {27}{2}} \sqrt {b x + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} - \frac {60 b^{9} x^{\frac {25}{2}} \sqrt {b x + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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