Optimal. Leaf size=45 \[ -\frac {2 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}-\frac {8 \sqrt {1-2 x}}{363 \sqrt {3+5 x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} -\frac {8 \sqrt {1-2 x}}{363 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {4}{33} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}-\frac {8 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 27, normalized size = 0.60 \begin {gather*} -\frac {2 \sqrt {1-2 x} (23+20 x)}{363 (3+5 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 34, normalized size = 0.76
method | result | size |
gosper | \(-\frac {2 \left (23+20 x \right ) \sqrt {1-2 x}}{363 \left (3+5 x \right )^{\frac {3}{2}}}\) | \(22\) |
default | \(-\frac {2 \sqrt {1-2 x}}{33 \left (3+5 x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 48, normalized size = 1.07 \begin {gather*} -\frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{33 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {8 \, \sqrt {-10 \, x^{2} - x + 3}}{363 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 33, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (20 \, x + 23\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.33, size = 102, normalized size = 2.27 \begin {gather*} \begin {cases} - \frac {8 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{1815} - \frac {2 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{825 \left (x + \frac {3}{5}\right )} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {8 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{1815} - \frac {2 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{825 \left (x + \frac {3}{5}\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (33) = 66\).
time = 1.15, size = 121, normalized size = 2.69 \begin {gather*} -\frac {1}{29040} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {36 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {9 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{1815 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 51, normalized size = 1.13 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {16\,x^2}{1815}+\frac {52\,x}{9075}-\frac {46}{9075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{5}+\frac {9\,\sqrt {1-2\,x}}{25}+x^2\,\sqrt {1-2\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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