Optimal. Leaf size=60 \[ 2 b^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 (b x+2)^{3/2}}{3 x^{3/2}}-\frac {2 b \sqrt {b x+2}}{\sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {47, 54, 215} \begin {gather*} 2 b^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 (b x+2)^{3/2}}{3 x^{3/2}}-\frac {2 b \sqrt {b x+2}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 54
Rule 215
Rubi steps
\begin {align*} \int \frac {(2+b x)^{3/2}}{x^{5/2}} \, dx &=-\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+b \int \frac {\sqrt {2+b x}}{x^{3/2}} \, dx\\ &=-\frac {2 b \sqrt {2+b x}}{\sqrt {x}}-\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx\\ &=-\frac {2 b \sqrt {2+b x}}{\sqrt {x}}-\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b \sqrt {2+b x}}{\sqrt {x}}-\frac {2 (2+b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.50 \begin {gather*} -\frac {4 \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {b x}{2}\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 55, normalized size = 0.92 \begin {gather*} -2 b^{3/2} \log \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )-\frac {4 \sqrt {b x+2} (2 b x+1)}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 108, normalized size = 1.80 \begin {gather*} \left [\frac {3 \, b^{\frac {3}{2}} x^{2} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) - 4 \, {\left (2 \, b x + 1\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, x^{2}}, -\frac {2 \, {\left (3 \, \sqrt {-b} b x^{2} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) + 2 \, {\left (2 \, b x + 1\right )} \sqrt {b x + 2} \sqrt {x}\right )}}{3 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 73, normalized size = 1.22 \begin {gather*} \frac {\sqrt {\left (b x +2\right ) x}\, b^{\frac {3}{2}} \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{\sqrt {b x +2}\, \sqrt {x}}-\frac {4 \left (2 b^{2} x^{2}+5 b x +2\right )}{3 \sqrt {b x +2}\, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 67, normalized size = 1.12 \begin {gather*} -b^{\frac {3}{2}} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + 2} b}{\sqrt {x}} - \frac {2 \, {\left (b x + 2\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{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 = -0.02 \begin {gather*} \int \frac {{\left (b\,x+2\right )}^{3/2}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.81, size = 70, normalized size = 1.17 \begin {gather*} - \frac {8 b^{\frac {3}{2}} \sqrt {1 + \frac {2}{b x}}}{3} - b^{\frac {3}{2}} \log {\left (\frac {1}{b x} \right )} + 2 b^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {2}{b x}} + 1 \right )} - \frac {4 \sqrt {b} \sqrt {1 + \frac {2}{b x}}}{3 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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