Optimal. Leaf size=86 \[ \frac {15 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}-\frac {15 \sqrt {x} \sqrt {b x+2}}{2 b^3}+\frac {5 x^{3/2} \sqrt {b x+2}}{2 b^2}-\frac {2 x^{5/2}}{b \sqrt {b x+2}} \]
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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} \[ \frac {5 x^{3/2} \sqrt {b x+2}}{2 b^2}-\frac {15 \sqrt {x} \sqrt {b x+2}}{2 b^3}+\frac {15 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}-\frac {2 x^{5/2}}{b \sqrt {b x+2}} \]
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)^{3/2}} \, dx &=-\frac {2 x^{5/2}}{b \sqrt {2+b x}}+\frac {5 \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx}{b}\\ &=-\frac {2 x^{5/2}}{b \sqrt {2+b x}}+\frac {5 x^{3/2} \sqrt {2+b x}}{2 b^2}-\frac {15 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{2 b^2}\\ &=-\frac {2 x^{5/2}}{b \sqrt {2+b x}}-\frac {15 \sqrt {x} \sqrt {2+b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2+b x}}{2 b^2}+\frac {15 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{2 b^3}\\ &=-\frac {2 x^{5/2}}{b \sqrt {2+b x}}-\frac {15 \sqrt {x} \sqrt {2+b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2+b x}}{2 b^2}+\frac {15 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2 x^{5/2}}{b \sqrt {2+b x}}-\frac {15 \sqrt {x} \sqrt {2+b x}}{2 b^3}+\frac {5 x^{3/2} \sqrt {2+b x}}{2 b^2}+\frac {15 \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 \[ \frac {x^{7/2} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};-\frac {b x}{2}\right )}{7 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 152, normalized size = 1.77 \[ \left [\frac {15 \, {\left (b x + 2\right )} \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) + {\left (b^{3} x^{2} - 5 \, b^{2} x - 30 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{2 \, {\left (b^{5} x + 2 \, b^{4}\right )}}, -\frac {30 \, {\left (b x + 2\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (b^{3} x^{2} - 5 \, b^{2} x - 30 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{2 \, {\left (b^{5} x + 2 \, b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 11.12, size = 119, normalized size = 1.38 \[ \frac {{\left (\sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} {\left (\frac {b x + 2}{b^{3}} - \frac {9}{b^{3}}\right )} - \frac {15 \, \log \left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{b^{\frac {5}{2}}} - \frac {64}{{\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} b^{\frac {3}{2}}}\right )} {\left | b \right |}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 106, normalized size = 1.23 \[ \frac {\left (\frac {15 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{2 b^{\frac {7}{2}}}-\frac {8 \sqrt {\left (x +\frac {2}{b}\right )^{2} b -2 x -\frac {4}{b}}}{\left (x +\frac {2}{b}\right ) b^{4}}\right ) \sqrt {\left (b x +2\right ) x}}{\sqrt {b x +2}\, \sqrt {x}}+\frac {\left (b x -7\right ) \sqrt {b x +2}\, \sqrt {x}}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 119, normalized size = 1.38 \[ -\frac {8 \, b^{2} - \frac {25 \, {\left (b x + 2\right )} b}{x} + \frac {15 \, {\left (b x + 2\right )}^{2}}{x^{2}}}{\frac {\sqrt {b x + 2} b^{5}}{\sqrt {x}} - \frac {2 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {{\left (b x + 2\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}} - \frac {15 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, b^{\frac {7}{2}}} \]
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 \[ \int \frac {x^{5/2}}{{\left (b\,x+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.10, size = 80, normalized size = 0.93 \[ \frac {x^{\frac {5}{2}}}{2 b \sqrt {b x + 2}} - \frac {5 x^{\frac {3}{2}}}{2 b^{2} \sqrt {b x + 2}} - \frac {15 \sqrt {x}}{b^{3} \sqrt {b x + 2}} + \frac {15 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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