Optimal. Leaf size=53 \[ \frac {847}{8 \sqrt {1-2 x}}+\frac {1133}{8} \sqrt {1-2 x}-\frac {505}{24} (1-2 x)^{3/2}+\frac {15}{8} (1-2 x)^{5/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 53, 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 {15}{8} (1-2 x)^{5/2}-\frac {505}{24} (1-2 x)^{3/2}+\frac {1133}{8} \sqrt {1-2 x}+\frac {847}{8 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(2+3 x) (3+5 x)^2}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {847}{8 (1-2 x)^{3/2}}-\frac {1133}{8 \sqrt {1-2 x}}+\frac {505}{8} \sqrt {1-2 x}-\frac {75}{8} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {847}{8 \sqrt {1-2 x}}+\frac {1133}{8} \sqrt {1-2 x}-\frac {505}{24} (1-2 x)^{3/2}+\frac {15}{8} (1-2 x)^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} \frac {685-631 x-185 x^2-45 x^3}{3 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 38, normalized size = 0.72
method | result | size |
gosper | \(-\frac {45 x^{3}+185 x^{2}+631 x -685}{3 \sqrt {1-2 x}}\) | \(25\) |
risch | \(-\frac {45 x^{3}+185 x^{2}+631 x -685}{3 \sqrt {1-2 x}}\) | \(25\) |
trager | \(\frac {\left (45 x^{3}+185 x^{2}+631 x -685\right ) \sqrt {1-2 x}}{-3+6 x}\) | \(32\) |
derivativedivides | \(-\frac {505 \left (1-2 x \right )^{\frac {3}{2}}}{24}+\frac {15 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {847}{8 \sqrt {1-2 x}}+\frac {1133 \sqrt {1-2 x}}{8}\) | \(38\) |
default | \(-\frac {505 \left (1-2 x \right )^{\frac {3}{2}}}{24}+\frac {15 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {847}{8 \sqrt {1-2 x}}+\frac {1133 \sqrt {1-2 x}}{8}\) | \(38\) |
meijerg | \(-\frac {18 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-87 \sqrt {\pi }+\frac {87 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {35 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-30 \sqrt {\pi }+\frac {15 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 37, normalized size = 0.70 \begin {gather*} \frac {15}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {505}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1133}{8} \, \sqrt {-2 \, x + 1} + \frac {847}{8 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.95, size = 31, normalized size = 0.58 \begin {gather*} \frac {{\left (45 \, x^{3} + 185 \, x^{2} + 631 \, x - 685\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.59, size = 46, normalized size = 0.87 \begin {gather*} \frac {15 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {505 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} + \frac {1133 \sqrt {1 - 2 x}}{8} + \frac {847}{8 \sqrt {1 - 2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.22, size = 44, normalized size = 0.83 \begin {gather*} \frac {15}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {505}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1133}{8} \, \sqrt {-2 \, x + 1} + \frac {847}{8 \, \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 = 37, normalized size = 0.70 \begin {gather*} \frac {847}{8\,\sqrt {1-2\,x}}+\frac {1133\,\sqrt {1-2\,x}}{8}-\frac {505\,{\left (1-2\,x\right )}^{3/2}}{24}+\frac {15\,{\left (1-2\,x\right )}^{5/2}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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