Optimal. Leaf size=65 \[ -\frac {2 x^{3/2}}{3 b (2+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2+b x}}+\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 65, 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 = {49, 56, 221}
\begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {b x+2}}-\frac {2 x^{3/2}}{3 b (b x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx &=-\frac {2 x^{3/2}}{3 b (2+b x)^{3/2}}+\frac {\int \frac {\sqrt {x}}{(2+b x)^{3/2}} \, dx}{b}\\ &=-\frac {2 x^{3/2}}{3 b (2+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2+b x}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{b^2}\\ &=-\frac {2 x^{3/2}}{3 b (2+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2+b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {2 x^{3/2}}{3 b (2+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2+b x}}+\frac {2 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 58, normalized size = 0.89 \begin {gather*} -\frac {4 \sqrt {x} (3+2 b x)}{3 b^2 (2+b x)^{3/2}}-\frac {2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 5.01, size = 86, normalized size = 1.32 \begin {gather*} \frac {-4 \sqrt {x}}{b^2 \left (2+b x\right )^{\frac {3}{2}}}+\frac {2 x \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]}{2 b^{\frac {3}{2}}+b^{\frac {5}{2}} x}-\frac {8 x^{\frac {3}{2}}}{3 b \left (2+b x\right )^{\frac {3}{2}}}+\frac {4 \text {ArcSinh}\left [\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2}\right ]}{2 b^{\frac {5}{2}}+b^{\frac {7}{2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 55, normalized size = 0.85
method | result | size |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (10 b x +15\right )}{15 \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}+2 \sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {5}{2}} \sqrt {\pi }}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 69, normalized size = 1.06 \begin {gather*} -\frac {2 \, {\left (b + \frac {3 \, {\left (b x + 2\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{2}} - \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 171, normalized size = 2.63 \begin {gather*} \left [\frac {3 \, {\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 ) - 4 \, {\left (2 \, b^{2} x + 3 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{5} x^{2} + 4 \, b^{4} x + 4 \, b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\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 ) + 2 \, {\left (2 \, b^{2} x + 3 \, b\right )} \sqrt {b x + 2} \sqrt {x}\right )}}{3 \, {\left (b^{5} x^{2} + 4 \, b^{4} x + 4 \, b^{3}\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. 257 vs.
\(2 (61) = 122\).
time = 1.90, size = 257, normalized size = 3.95 \begin {gather*} - \frac {8 b^{\frac {11}{2}} x^{8}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x + 2} + 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x + 2}} - \frac {12 b^{\frac {9}{2}} x^{7}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x + 2} + 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x + 2}} + \frac {6 b^{5} x^{\frac {15}{2}} \sqrt {b x + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x + 2} + 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x + 2}} + \frac {12 b^{4} x^{\frac {13}{2}} \sqrt {b x + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x + 2} + 6 b^{\frac {13}{2}} x^{\frac {13}{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 = 97, normalized size = 1.49 \begin {gather*} 2 \left (\frac {2 \left (-\frac {\frac {1}{18}\cdot 12 b^{2} \sqrt {x} \sqrt {x}}{b^{3}}-\frac {\frac {1}{18}\cdot 18 b}{b^{3}}\right ) \sqrt {x} \sqrt {b x+2}}{\left (b x+2\right )^{2}}-\frac {\ln \left (\sqrt {b x+2}-\sqrt {b} \sqrt {x}\right )}{b^{2} \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 = -0.02 \begin {gather*} \int \frac {x^{3/2}}{{\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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