Optimal. Leaf size=67 \[ \frac {2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x}}{363 (3+5 x)^{3/2}}-\frac {160 \sqrt {1-2 x}}{3993 \sqrt {3+5 x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {160 \sqrt {1-2 x}}{3993 \sqrt {5 x+3}}-\frac {40 \sqrt {1-2 x}}{363 (5 x+3)^{3/2}}+\frac {2}{11 (5 x+3)^{3/2} \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {20}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x}}{363 (3+5 x)^{3/2}}+\frac {80}{363} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {40 \sqrt {1-2 x}}{363 (3+5 x)^{3/2}}-\frac {160 \sqrt {1-2 x}}{3993 \sqrt {3+5 x}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 32, normalized size = 0.48 \begin {gather*} \frac {2 \left (-97+520 x+800 x^2\right )}{3993 \sqrt {1-2 x} (3+5 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 50, normalized size = 0.75
method | result | size |
gosper | \(\frac {\frac {1600}{3993} x^{2}+\frac {1040}{3993} x -\frac {194}{3993}}{\left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}\) | \(27\) |
default | \(\frac {2}{11 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}-\frac {40 \sqrt {1-2 x}}{363 \left (3+5 x \right )^{\frac {3}{2}}}-\frac {160 \sqrt {1-2 x}}{3993 \sqrt {3+5 x}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 64, normalized size = 0.96 \begin {gather*} \frac {320 \, x}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {16}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{33 \, {\left (5 \, \sqrt {-10 \, x^{2} - x + 3} x + 3 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 43, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (800 \, x^{2} + 520 \, x - 97\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3993 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, 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 = 2.95, size = 230, normalized size = 3.43 \begin {gather*} \begin {cases} - \frac {1600 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} + \frac {880 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} + \frac {242 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {1600 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} + \frac {880 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} + \frac {242 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 219615 x + 199650 \left (x + \frac {3}{5}\right )^{2} - 131769} & \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. 147 vs.
\(2 (49) = 98\).
time = 0.65, size = 147, normalized size = 2.19 \begin {gather*} -\frac {1}{63888} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {84 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {8 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{6655 \, {\left (2 \, x - 1\right )}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {21 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{3993 \, {\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 = 0.22, size = 51, normalized size = 0.76 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {64\,x^2}{3993}+\frac {208\,x}{19965}-\frac {194}{99825}\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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