Optimal. Leaf size=79 \[ \frac {1}{3} \sqrt {x} (b x+2)^{5/2}+\frac {5}{6} \sqrt {x} (b x+2)^{3/2}+\frac {5}{2} \sqrt {x} \sqrt {b x+2}+\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {50, 54, 215} \[ \frac {1}{3} \sqrt {x} (b x+2)^{5/2}+\frac {5}{6} \sqrt {x} (b x+2)^{3/2}+\frac {5}{2} \sqrt {x} \sqrt {b x+2}+\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rubi steps
\begin {align*} \int \frac {(2+b x)^{5/2}}{\sqrt {x}} \, dx &=\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5}{3} \int \frac {(2+b x)^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5}{2} \int \frac {\sqrt {2+b x}}{\sqrt {x}} \, dx\\ &=\frac {5}{2} \sqrt {x} \sqrt {2+b x}+\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5}{2} \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx\\ &=\frac {5}{2} \sqrt {x} \sqrt {2+b x}+\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5}{2} \sqrt {x} \sqrt {2+b x}+\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 0.72 \[ \frac {1}{6} \sqrt {x} \sqrt {b x+2} \left (2 b^2 x^2+13 b x+33\right )+\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 123, normalized size = 1.56 \[ \left [\frac {{\left (2 \, b^{3} x^{2} + 13 \, b^{2} x + 33 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 15 \, \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{6 \, b}, \frac {{\left (2 \, b^{3} x^{2} + 13 \, b^{2} x + 33 \, b\right )} \sqrt {b x + 2} \sqrt {x} - 30 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{6 \, b}\right ] \]
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 = 84, normalized size = 1.06 \[ \frac {\left (b x +2\right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 \left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}{6}+\frac {5 \sqrt {b x +2}\, \sqrt {x}}{2}+\frac {5 \sqrt {\left (b x +2\right ) x}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {b}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.03, size = 129, normalized size = 1.63 \[ -\frac {5 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, \sqrt {b}} - \frac {\frac {15 \, \sqrt {b x + 2} b^{2}}{\sqrt {x}} - \frac {40 \, {\left (b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {33 \, {\left (b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{3} - \frac {3 \, {\left (b x + 2\right )} b^{2}}{x} + \frac {3 \, {\left (b x + 2\right )}^{2} b}{x^{2}} - \frac {{\left (b x + 2\right )}^{3}}{x^{3}}\right )}} \]
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 \[ \int \frac {{\left (b\,x+2\right )}^{5/2}}{\sqrt {x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.46, size = 97, normalized size = 1.23 \[ \frac {b^{3} x^{\frac {7}{2}}}{3 \sqrt {b x + 2}} + \frac {17 b^{2} x^{\frac {5}{2}}}{6 \sqrt {b x + 2}} + \frac {59 b x^{\frac {3}{2}}}{6 \sqrt {b x + 2}} + \frac {11 \sqrt {x}}{\sqrt {b x + 2}} + \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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