Optimal. Leaf size=67 \[ \frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {3 \sqrt {x} \sqrt {b x+2}}{2 b^2}+\frac {x^{3/2} \sqrt {b x+2}}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 67, 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 = {50, 54, 215} \[ -\frac {3 \sqrt {x} \sqrt {b x+2}}{2 b^2}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}+\frac {x^{3/2} \sqrt {b x+2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx &=\frac {x^{3/2} \sqrt {2+b x}}{2 b}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{2 b}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{2 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.76 \[ \frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}+\frac {\sqrt {x} \sqrt {b x+2} (b x-3)}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 105, normalized size = 1.57 \[ \left [\frac {{\left (b^{2} x - 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 3 \, \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{2 \, b^{3}}, \frac {{\left (b^{2} x - 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} - 6 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{2 \, b^{3}}\right ] \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 78, normalized size = 1.16 \[ \frac {\sqrt {b x +2}\, x^{\frac {3}{2}}}{2 b}-\frac {3 \sqrt {b x +2}\, \sqrt {x}}{2 b^{2}}+\frac {3 \sqrt {\left (b x +2\right ) x}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{2 \sqrt {b x +2}\, b^{\frac {5}{2}} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.86, size = 102, normalized size = 1.52 \[ \frac {\frac {5 \, \sqrt {b x + 2} b}{\sqrt {x}} - \frac {3 \, {\left (b x + 2\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}}{b^{4} - \frac {2 \, {\left (b x + 2\right )} b^{3}}{x} + \frac {{\left (b x + 2\right )}^{2} b^{2}}{x^{2}}} - \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, b^{\frac {5}{2}}} \]
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 \[ \int \frac {x^{3/2}}{\sqrt {b\,x+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.67, size = 75, normalized size = 1.12 \[ \frac {x^{\frac {5}{2}}}{2 \sqrt {b x + 2}} - \frac {x^{\frac {3}{2}}}{2 b \sqrt {b x + 2}} - \frac {3 \sqrt {x}}{b^{2} \sqrt {b x + 2}} + \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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