Optimal. Leaf size=113 \[ -\frac {3}{40} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3-\frac {259}{800} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (77820 x+187559)}{128000}+\frac {10866247 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 153, 147, 54, 216} \[ -\frac {3}{40} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3-\frac {259}{800} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (77820 x+187559)}{128000}+\frac {10866247 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 100
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {1}{40} \int \frac {\left (-238-\frac {777 x}{2}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {\int \frac {(2+3 x) \left (\frac {41769}{2}+\frac {136185 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1200}\\ &=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{256000}\\ &=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{128000 \sqrt {5}}\\ &=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{128000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.73 \[ -\frac {\sqrt {1-2 x} \left (30 \sqrt {2 x-1} \sqrt {5 x+3} \left (86400 x^3+297120 x^2+462540 x+518491\right )+10866247 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{1280000 \sqrt {2 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 72, normalized size = 0.64 \[ -\frac {3}{128000} \, {\left (86400 \, x^{3} + 297120 \, x^{2} + 462540 \, x + 518491\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {10866247}{2560000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 63, normalized size = 0.56 \[ -\frac {1}{6400000} \, \sqrt {5} {\left (6 \, {\left (12 \, {\left (8 \, {\left (180 \, x + 403\right )} {\left (5 \, x + 3\right )} + 16609\right )} {\left (5 \, x + 3\right )} + 1646339\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 54331235 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 104, normalized size = 0.92 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-5184000 \sqrt {-10 x^{2}-x +3}\, x^{3}-17827200 \sqrt {-10 x^{2}-x +3}\, x^{2}-27752400 \sqrt {-10 x^{2}-x +3}\, x +10866247 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-31109460 \sqrt {-10 x^{2}-x +3}\right )}{2560000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 75, normalized size = 0.66 \[ -\frac {81}{40} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {5571}{800} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {69381}{6400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {10866247}{2560000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {1555473}{128000} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.93, size = 708, normalized size = 6.27 \[ \frac {10866247\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{640000}-\frac {\frac {6770247\,\left (\sqrt {1-2\,x}-1\right )}{195312500\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {33291573\,{\left (\sqrt {1-2\,x}-1\right )}^3}{78125000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {883182573\,{\left (\sqrt {1-2\,x}-1\right )}^5}{156250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {451883391\,{\left (\sqrt {1-2\,x}-1\right )}^7}{62500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {451883391\,{\left (\sqrt {1-2\,x}-1\right )}^9}{25000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {883182573\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{10000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {33291573\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{800000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}-\frac {6770247\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{320000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {49152\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {258048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1032192\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {16147968\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {258048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {16128\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {768\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}}{\frac {1024\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {224\,{\left (\sqrt {1-2\,x}-1\right )}^8}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {448\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {112\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {16\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{16}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {256}{390625}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 x + 2\right )^{4}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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