Optimal. Leaf size=92 \[ \frac {184877}{192 (1-2 x)^{3/2}}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645}{64} \sqrt {1-2 x}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {135}{64} (1-2 x)^{9/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 92, 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 {135}{64} (1-2 x)^{9/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {519645}{64} \sqrt {1-2 x}-\frac {60025}{8 \sqrt {1-2 x}}+\frac {184877}{192 (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)^5 (3+5 x)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {184877}{64 (1-2 x)^{5/2}}-\frac {60025}{8 (1-2 x)^{3/2}}+\frac {519645}{64 \sqrt {1-2 x}}-\frac {37485}{8} \sqrt {1-2 x}+\frac {97335}{64} (1-2 x)^{3/2}-\frac {1053}{4} (1-2 x)^{5/2}+\frac {1215}{64} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {184877}{192 (1-2 x)^{3/2}}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645}{64} \sqrt {1-2 x}+\frac {12495}{8} (1-2 x)^{3/2}-\frac {19467}{64} (1-2 x)^{5/2}+\frac {1053}{28} (1-2 x)^{7/2}-\frac {135}{64} (1-2 x)^{9/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {280696-844104 x+412812 x^2+114084 x^3+49653 x^4+16767 x^5+2835 x^6}{21 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 65, normalized size = 0.71
method | result | size |
gosper | \(-\frac {2835 x^{6}+16767 x^{5}+49653 x^{4}+114084 x^{3}+412812 x^{2}-844104 x +280696}{21 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(40\) |
trager | \(-\frac {\left (2835 x^{6}+16767 x^{5}+49653 x^{4}+114084 x^{3}+412812 x^{2}-844104 x +280696\right ) \sqrt {1-2 x}}{21 \left (-1+2 x \right )^{2}}\) | \(47\) |
risch | \(\frac {2835 x^{6}+16767 x^{5}+49653 x^{4}+114084 x^{3}+412812 x^{2}-844104 x +280696}{21 \left (-1+2 x \right ) \sqrt {1-2 x}}\) | \(47\) |
derivativedivides | \(\frac {184877}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {12495 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {19467 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {1053 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {135 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645 \sqrt {1-2 x}}{64}\) | \(65\) |
default | \(\frac {184877}{192 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {12495 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {19467 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {1053 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {135 \left (1-2 x \right )^{\frac {9}{2}}}{64}-\frac {60025}{8 \sqrt {1-2 x}}-\frac {519645 \sqrt {1-2 x}}{64}\) | \(65\) |
meijerg | \(-\frac {64 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {880 \sqrt {\pi }}{3}-\frac {110 \sqrt {\pi }\, \left (-24 x +8\right )}{3 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {560 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {4560 \sqrt {\pi }-\frac {285 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{8 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {1305 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{4 \sqrt {\pi }}+\frac {\frac {12744 \sqrt {\pi }}{7}-\frac {1593 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{896 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {405 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{32 \sqrt {\pi }}\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 60, normalized size = 0.65 \begin {gather*} -\frac {135}{64} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1053}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {19467}{64} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {12495}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {519645}{64} \, \sqrt {-2 \, x + 1} + \frac {2401 \, {\left (1200 \, x - 523\right )}}{192 \, {\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 = 1.36, size = 51, normalized size = 0.55 \begin {gather*} -\frac {{\left (2835 \, x^{6} + 16767 \, x^{5} + 49653 \, x^{4} + 114084 \, x^{3} + 412812 \, x^{2} - 844104 \, x + 280696\right )} \sqrt {-2 \, x + 1}}{21 \, {\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 = 20.41, size = 82, normalized size = 0.89 \begin {gather*} - \frac {135 \left (1 - 2 x\right )^{\frac {9}{2}}}{64} + \frac {1053 \left (1 - 2 x\right )^{\frac {7}{2}}}{28} - \frac {19467 \left (1 - 2 x\right )^{\frac {5}{2}}}{64} + \frac {12495 \left (1 - 2 x\right )^{\frac {3}{2}}}{8} - \frac {519645 \sqrt {1 - 2 x}}{64} - \frac {60025}{8 \sqrt {1 - 2 x}} + \frac {184877}{192 \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 = 2.73, size = 88, normalized size = 0.96 \begin {gather*} -\frac {135}{64} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1053}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {19467}{64} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {12495}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {519645}{64} \, \sqrt {-2 \, x + 1} - \frac {2401 \, {\left (1200 \, x - 523\right )}}{192 \, {\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 = 0.03, size = 59, normalized size = 0.64 \begin {gather*} \frac {\frac {60025\,x}{4}-\frac {1255723}{192}}{{\left (1-2\,x\right )}^{3/2}}-\frac {519645\,\sqrt {1-2\,x}}{64}+\frac {12495\,{\left (1-2\,x\right )}^{3/2}}{8}-\frac {19467\,{\left (1-2\,x\right )}^{5/2}}{64}+\frac {1053\,{\left (1-2\,x\right )}^{7/2}}{28}-\frac {135\,{\left (1-2\,x\right )}^{9/2}}{64} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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