Optimal. Leaf size=67 \[ \frac {2}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {40}{363 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {400 \sqrt {1-2 x}}{3993 \sqrt {3+5 x}} \]
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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 {400 \sqrt {1-2 x}}{3993 \sqrt {5 x+3}}+\frac {40}{363 \sqrt {5 x+3} \sqrt {1-2 x}}+\frac {2}{33 \sqrt {5 x+3} (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {2}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {20}{33} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {40}{363 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {200}{363} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {40}{363 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {400 \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 {282+720 x-1600 x^2}{3993 (1-2 x)^{3/2} \sqrt {3+5 x}} \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 | \(-\frac {2 \left (800 x^{2}-360 x -141\right )}{3993 \sqrt {3+5 x}\, \left (1-2 x \right )^{\frac {3}{2}}}\) | \(27\) |
default | \(\frac {2}{33 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}+\frac {40}{363 \sqrt {1-2 x}\, \sqrt {3+5 x}}-\frac {400 \sqrt {1-2 x}}{3993 \sqrt {3+5 x}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 64, normalized size = 0.96 \begin {gather*} \frac {800 \, x}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {40}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{33 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \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 = 0.50, size = 43, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (800 \, x^{2} - 360 \, x - 141\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3993 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 = 2.57, size = 230, normalized size = 3.43 \begin {gather*} \begin {cases} - \frac {8000 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} + \frac {13200 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} - \frac {3630 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {8000 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{2}}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} + \frac {13200 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} - \frac {3630 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{- 878460 x + 399300 \left (x + \frac {3}{5}\right )^{2} - 43923} & \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. 100 vs.
\(2 (49) = 98\).
time = 1.37, size = 100, normalized size = 1.49 \begin {gather*} -\frac {5 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{2662 \, \sqrt {5 \, x + 3}} - \frac {8 \, {\left (5 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{19965 \, {\left (2 \, x - 1\right )}^{2}} + \frac {10 \, \sqrt {10} \sqrt {5 \, x + 3}}{1331 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.44, size = 52, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {5\,x+3}\,\left (-\frac {160\,x^2}{3993}+\frac {24\,x}{1331}+\frac {47}{6655}\right )}{\frac {x\,\sqrt {1-2\,x}}{10}-\frac {3\,\sqrt {1-2\,x}}{10}+x^2\,\sqrt {1-2\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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