Optimal. Leaf size=66 \[ \frac {9317}{48 (1-2 x)^{3/2}}-\frac {8349}{8 \sqrt {1-2 x}}-\frac {2805}{4} \sqrt {1-2 x}+\frac {1675}{24} (1-2 x)^{3/2}-\frac {75}{16} (1-2 x)^{5/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78}
\begin {gather*} -\frac {75}{16} (1-2 x)^{5/2}+\frac {1675}{24} (1-2 x)^{3/2}-\frac {2805}{4} \sqrt {1-2 x}-\frac {8349}{8 \sqrt {1-2 x}}+\frac {9317}{48 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(2+3 x) (3+5 x)^3}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {9317}{16 (1-2 x)^{5/2}}-\frac {8349}{8 (1-2 x)^{3/2}}+\frac {2805}{4 \sqrt {1-2 x}}-\frac {1675}{8} \sqrt {1-2 x}+\frac {375}{16} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {9317}{48 (1-2 x)^{3/2}}-\frac {8349}{8 \sqrt {1-2 x}}-\frac {2805}{4} \sqrt {1-2 x}+\frac {1675}{24} (1-2 x)^{3/2}-\frac {75}{16} (1-2 x)^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {4457-13533 x+6240 x^2+1225 x^3+225 x^4}{3 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 47, normalized size = 0.71
method | result | size |
gosper | \(-\frac {225 x^{4}+1225 x^{3}+6240 x^{2}-13533 x +4457}{3 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(30\) |
trager | \(-\frac {\left (225 x^{4}+1225 x^{3}+6240 x^{2}-13533 x +4457\right ) \sqrt {1-2 x}}{3 \left (-1+2 x \right )^{2}}\) | \(37\) |
risch | \(\frac {225 x^{4}+1225 x^{3}+6240 x^{2}-13533 x +4457}{3 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(37\) |
derivativedivides | \(\frac {9317}{48 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {1675 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {75 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {8349}{8 \sqrt {1-2 x}}-\frac {2805 \sqrt {1-2 x}}{4}\) | \(47\) |
default | \(\frac {9317}{48 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {1675 \left (1-2 x \right )^{\frac {3}{2}}}{24}-\frac {75 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {8349}{8 \sqrt {1-2 x}}-\frac {2805 \sqrt {1-2 x}}{4}\) | \(47\) |
meijerg | \(-\frac {36 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {117 \sqrt {\pi }-\frac {117 \sqrt {\pi }\, \left (-24 x +8\right )}{8 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {285 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{2 \sqrt {\pi }}+\frac {\frac {1850 \sqrt {\pi }}{3}-\frac {925 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{192 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {125 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}\) | \(165\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 42, normalized size = 0.64 \begin {gather*} -\frac {75}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {1675}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {2805}{4} \, \sqrt {-2 \, x + 1} + \frac {121 \, {\left (828 \, x - 337\right )}}{48 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.70, size = 41, normalized size = 0.62 \begin {gather*} -\frac {{\left (225 \, x^{4} + 1225 \, x^{3} + 6240 \, x^{2} - 13533 \, x + 4457\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.03, size = 58, normalized size = 0.88 \begin {gather*} - \frac {75 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} + \frac {1675 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {2805 \sqrt {1 - 2 x}}{4} - \frac {8349}{8 \sqrt {1 - 2 x}} + \frac {9317}{48 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 56, normalized size = 0.85 \begin {gather*} -\frac {75}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {1675}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {2805}{4} \, \sqrt {-2 \, x + 1} - \frac {121 \, {\left (828 \, x - 337\right )}}{48 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 41, normalized size = 0.62 \begin {gather*} \frac {\frac {8349\,x}{4}-\frac {40777}{48}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2805\,\sqrt {1-2\,x}}{4}+\frac {1675\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {75\,{\left (1-2\,x\right )}^{5/2}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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