Optimal. Leaf size=79 \[ \frac {26411}{32 \sqrt {1-2 x}}+\frac {57281}{32} \sqrt {1-2 x}-\frac {8281}{16} (1-2 x)^{3/2}+\frac {10773}{80} (1-2 x)^{5/2}-\frac {4671}{224} (1-2 x)^{7/2}+\frac {45}{32} (1-2 x)^{9/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 79, 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 {45}{32} (1-2 x)^{9/2}-\frac {4671}{224} (1-2 x)^{7/2}+\frac {10773}{80} (1-2 x)^{5/2}-\frac {8281}{16} (1-2 x)^{3/2}+\frac {57281}{32} \sqrt {1-2 x}+\frac {26411}{32 \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)^4 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {26411}{32 (1-2 x)^{3/2}}-\frac {57281}{32 \sqrt {1-2 x}}+\frac {24843}{16} \sqrt {1-2 x}-\frac {10773}{16} (1-2 x)^{3/2}+\frac {4671}{32} (1-2 x)^{5/2}-\frac {405}{32} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {26411}{32 \sqrt {1-2 x}}+\frac {57281}{32} \sqrt {1-2 x}-\frac {8281}{16} (1-2 x)^{3/2}+\frac {10773}{80} (1-2 x)^{5/2}-\frac {4671}{224} (1-2 x)^{7/2}+\frac {45}{32} (1-2 x)^{9/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} \frac {77456-75776 x-31448 x^2-18288 x^3-7740 x^4-1575 x^5}{35 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 56, normalized size = 0.71
method | result | size |
gosper | \(-\frac {1575 x^{5}+7740 x^{4}+18288 x^{3}+31448 x^{2}+75776 x -77456}{35 \sqrt {1-2 x}}\) | \(35\) |
risch | \(-\frac {1575 x^{5}+7740 x^{4}+18288 x^{3}+31448 x^{2}+75776 x -77456}{35 \sqrt {1-2 x}}\) | \(35\) |
trager | \(\frac {\left (1575 x^{5}+7740 x^{4}+18288 x^{3}+31448 x^{2}+75776 x -77456\right ) \sqrt {1-2 x}}{-35+70 x}\) | \(42\) |
derivativedivides | \(-\frac {8281 \left (1-2 x \right )^{\frac {3}{2}}}{16}+\frac {10773 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {4671 \left (1-2 x \right )^{\frac {7}{2}}}{224}+\frac {45 \left (1-2 x \right )^{\frac {9}{2}}}{32}+\frac {26411}{32 \sqrt {1-2 x}}+\frac {57281 \sqrt {1-2 x}}{32}\) | \(56\) |
default | \(-\frac {8281 \left (1-2 x \right )^{\frac {3}{2}}}{16}+\frac {10773 \left (1-2 x \right )^{\frac {5}{2}}}{80}-\frac {4671 \left (1-2 x \right )^{\frac {7}{2}}}{224}+\frac {45 \left (1-2 x \right )^{\frac {9}{2}}}{32}+\frac {26411}{32 \sqrt {1-2 x}}+\frac {57281 \sqrt {1-2 x}}{32}\) | \(56\) |
meijerg | \(-\frac {48 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-368 \sqrt {\pi }+\frac {46 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {282 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-\frac {3456 \sqrt {\pi }}{5}+\frac {27 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{5 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {1323 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {360 \sqrt {\pi }}{7}+\frac {45 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{896 \sqrt {1-2 x}}}{\sqrt {\pi }}\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 55, normalized size = 0.70 \begin {gather*} \frac {45}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {4671}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {10773}{80} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {8281}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {57281}{32} \, \sqrt {-2 \, x + 1} + \frac {26411}{32 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 41, normalized size = 0.52 \begin {gather*} \frac {{\left (1575 \, x^{5} + 7740 \, x^{4} + 18288 \, x^{3} + 31448 \, x^{2} + 75776 \, x - 77456\right )} \sqrt {-2 \, x + 1}}{35 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.59, size = 70, normalized size = 0.89 \begin {gather*} \frac {45 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {4671 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} + \frac {10773 \left (1 - 2 x\right )^{\frac {5}{2}}}{80} - \frac {8281 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} + \frac {57281 \sqrt {1 - 2 x}}{32} + \frac {26411}{32 \sqrt {1 - 2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.14, size = 76, normalized size = 0.96 \begin {gather*} \frac {45}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {4671}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {10773}{80} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {8281}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {57281}{32} \, \sqrt {-2 \, x + 1} + \frac {26411}{32 \, \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {26411}{32\,\sqrt {1-2\,x}}+\frac {57281\,\sqrt {1-2\,x}}{32}-\frac {8281\,{\left (1-2\,x\right )}^{3/2}}{16}+\frac {10773\,{\left (1-2\,x\right )}^{5/2}}{80}-\frac {4671\,{\left (1-2\,x\right )}^{7/2}}{224}+\frac {45\,{\left (1-2\,x\right )}^{9/2}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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