Optimal. Leaf size=79 \[ \frac {5}{2} \sqrt {x} \sqrt {2+b x}+\frac {5}{6} \sqrt {x} (2+b x)^{3/2}+\frac {1}{3} \sqrt {x} (2+b x)^{5/2}+\frac {5 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]
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Rubi [A]
time = 0.01, 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 = {52, 56, 221}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 221
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 \text {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 = 0.08, size = 63, normalized size = 0.80 \begin {gather*} \frac {1}{6} \sqrt {x} \sqrt {2+b x} \left (33+13 b x+2 b^2 x^2\right )-\frac {5 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 5.30, size = 63, normalized size = 0.80 \begin {gather*} \frac {\sqrt {b} \sqrt {x} \left (66+59 b x+17 b^2 x^2+2 b^3 x^3\right )+30 \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ] \sqrt {2+b x}}{6 \sqrt {b} \sqrt {2+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 84, normalized size = 1.06
method | result | size |
meijerg | \(-\frac {15 \left (-\frac {8 \sqrt {\pi }\, \sqrt {b}\, \sqrt {x}\, \sqrt {2}\, \left (\frac {1}{24} x^{2} b^{2}+\frac {13}{48} b x +\frac {11}{16}\right ) \sqrt {\frac {b x}{2}+1}}{15}-\frac {\sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{3}\right )}{\sqrt {b}\, \sqrt {\pi }}\) | \(63\) |
risch | \(\frac {\left (2 x^{2} b^{2}+13 b x +33\right ) \sqrt {x}\, \sqrt {b x +2}}{6}+\frac {5 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(74\) |
default | \(\frac {\left (b x +2\right )^{\frac {5}{2}} \sqrt {x}}{3}+\frac {5 \left (b x +2\right )^{\frac {3}{2}} \sqrt {x}}{6}+\frac {5 \sqrt {x}\, \sqrt {b x +2}}{2}+\frac {5 \sqrt {x \left (b x +2\right )}\, \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{2 \sqrt {b x +2}\, \sqrt {x}\, \sqrt {b}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (54) = 108\).
time = 0.35, size = 129, normalized size = 1.63 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 123, normalized size = 1.56 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.73, size = 97, normalized size = 1.23 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 141, normalized size = 1.78 \begin {gather*} \frac {b^{2} \left (2 \left (\left (\frac {\frac {1}{36}\cdot 6 \sqrt {b x+2} \sqrt {b x+2}}{b}+\frac {\frac {1}{36}\cdot 15}{b}\right ) \sqrt {b x+2} \sqrt {b x+2}+\frac {\frac {1}{36}\cdot 45}{b}\right ) \sqrt {b x+2} \sqrt {b \left (b x+2\right )-2 b}-\frac {5 \ln \left |\sqrt {b \left (b x+2\right )-2 b}-\sqrt {b} \sqrt {b x+2}\right |}{\sqrt {b}}\right )}{\left |b\right | b} \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 {{\left (b\,x+2\right )}^{5/2}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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