Optimal. Leaf size=86 \[ -\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \sqrt {x} \sqrt {2+b x}}{b^3}-\frac {10 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}} \]
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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 = {49, 52, 56, 221}
\begin {gather*} -\frac {10 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}+\frac {5 \sqrt {x} \sqrt {b x+2}}{b^3}-\frac {10 x^{3/2}}{3 b^2 \sqrt {b x+2}}-\frac {2 x^{5/2}}{3 b (b x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{(2+b x)^{5/2}} \, dx &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}+\frac {5 \int \frac {x^{3/2}}{(2+b x)^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{b^2}\\ &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \sqrt {x} \sqrt {2+b x}}{b^3}-\frac {5 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{b^3}\\ &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \sqrt {x} \sqrt {2+b x}}{b^3}-\frac {10 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2 x^{5/2}}{3 b (2+b x)^{3/2}}-\frac {10 x^{3/2}}{3 b^2 \sqrt {2+b x}}+\frac {5 \sqrt {x} \sqrt {2+b x}}{b^3}-\frac {10 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 66, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x} \left (60+40 b x+3 b^2 x^2\right )}{3 b^3 (2+b x)^{3/2}}+\frac {10 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 7.36, size = 100, normalized size = 1.16 \begin {gather*} \frac {20 \sqrt {x}}{b^3 \left (2+b x\right )^{\frac {3}{2}}}-\frac {10 x \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]}{2 b^{\frac {5}{2}}+b^{\frac {7}{2}} x}+\frac {40 x^{\frac {3}{2}}}{3 b^2 \left (2+b x\right )^{\frac {3}{2}}}+\frac {x^{\frac {5}{2}}}{b \left (2+b x\right )^{\frac {3}{2}}}-\frac {20 \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]}{2 b^{\frac {7}{2}}+b^{\frac {9}{2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 63, normalized size = 0.73
method | result | size |
meijerg | \(\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (\frac {21}{4} x^{2} b^{2}+70 b x +105\right )}{21 \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}-10 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {7}{2}} \sqrt {\pi }}\) | \(63\) |
risch | \(\frac {\sqrt {x}\, \sqrt {b x +2}}{b^{3}}+\frac {\left (-\frac {5 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {x^{2} b +2 x}\right )}{b^{\frac {7}{2}}}-\frac {8 \sqrt {\left (x +\frac {2}{b}\right )^{2} b -2 x -\frac {4}{b}}}{3 b^{5} \left (x +\frac {2}{b}\right )^{2}}+\frac {28 \sqrt {\left (x +\frac {2}{b}\right )^{2} b -2 x -\frac {4}{b}}}{3 b^{4} \left (x +\frac {2}{b}\right )}\right ) \sqrt {x \left (b x +2\right )}}{\sqrt {x}\, \sqrt {b x +2}}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 105, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (2 \, b^{2} + \frac {10 \, {\left (b x + 2\right )} b}{x} - \frac {15 \, {\left (b x + 2\right )}^{2}}{x^{2}}\right )}}{3 \, {\left (\frac {{\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}}}\right )}} + \frac {5 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 186, normalized size = 2.16 \begin {gather*} \left [\frac {15 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )} \sqrt {b} \log \left (b x - \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) + {\left (3 \, b^{3} x^{2} + 40 \, b^{2} x + 60 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{6} x^{2} + 4 \, b^{5} x + 4 \, b^{4}\right )}}, \frac {30 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (3 \, b^{3} x^{2} + 40 \, b^{2} x + 60 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{6} x^{2} + 4 \, b^{5} x + 4 \, b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 308 vs.
\(2 (82) = 164\).
time = 4.03, size = 308, normalized size = 3.58 \begin {gather*} \frac {3 b^{\frac {23}{2}} x^{15}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} + \frac {40 b^{\frac {21}{2}} x^{14}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} + \frac {60 b^{\frac {19}{2}} x^{13}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} - \frac {30 b^{10} x^{\frac {27}{2}} \sqrt {b x + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} - \frac {60 b^{9} x^{\frac {25}{2}} \sqrt {b x + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {b x + 2} + 6 b^{\frac {25}{2}} x^{\frac {25}{2}} \sqrt {b x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 121, normalized size = 1.41 \begin {gather*} 2 \left (\frac {2 \left (\left (\frac {\frac {1}{36}\cdot 9 b^{4} \sqrt {x} \sqrt {x}}{b^{5}}+\frac {\frac {1}{36}\cdot 120 b^{3}}{b^{5}}\right ) \sqrt {x} \sqrt {x}+\frac {\frac {1}{36}\cdot 180 b^{2}}{b^{5}}\right ) \sqrt {x} \sqrt {b x+2}}{\left (b x+2\right )^{2}}+\frac {5 \ln \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{b^{3} \sqrt {b}}\right ) \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 {x^{5/2}}{{\left (b\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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