Optimal. Leaf size=142 \[ \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {357 (3 x+2)^3}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {5281 \sqrt {1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (55300905 x+33035947)}{8784600 \sqrt {5 x+3}}+\frac {2997 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 150, 143, 54, 216} \[ \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {357 (3 x+2)^3}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {5281 \sqrt {1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (55300905 x+33035947)}{8784600 \sqrt {5 x+3}}+\frac {2997 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 143
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^3 \left (141+\frac {507 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-6537-\frac {48861 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\left (-\frac {1453983}{4}-\frac {5027355 x}{8}\right ) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997}{400} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{200 \sqrt {5}}\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}}\\ \end {align*}
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Mathematica [C] time = 1.37, size = 257, normalized size = 1.81 \[ -\frac {37 \left (-1320000 (3 x+2)^3 (1-2 x)^{7/2} \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};-\frac {5}{11} (2 x-1)\right )-1050000 (x+3) \left (6 x^2+x-2\right )^2 (1-2 x)^{5/2} \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-\frac {5}{11} (2 x-1)\right )+77 \sqrt {55} \left (\sqrt {10-20 x} \sqrt {5 x+3} \left (43200 x^5+28080 x^4-400032 x^3+1229303 x^2+2053496 x+1669914\right )-27951 \left (108 x^3+513 x^2+1296 x+374\right ) \sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )\right )}{614922000 \sqrt {22} (1-2 x)^3}+\frac {1183 \left (19573 x^3+62232 x^2+52044 x+13040\right )}{878460 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {3 (3 x+2)^4}{10 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.06, size = 121, normalized size = 0.85 \[ -\frac {131637231 \, \sqrt {10} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (213465780 \, x^{4} - 1247811640 \, x^{3} - 1260430251 \, x^{2} + 19593966 \, x + 168318961\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{175692000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.51, size = 191, normalized size = 1.35 \[ -\frac {1}{439230000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {4092 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {2997}{2000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (10673289 \, \sqrt {5} {\left (5 \, x + 3\right )} - 440040554 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 7233942969 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{5490375000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1023 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{27451875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 182, normalized size = 1.28 \[ \frac {\sqrt {-2 x +1}\, \left (13163723100 \sqrt {10}\, x^{4} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-4269315600 \sqrt {-10 x^{2}-x +3}\, x^{4}+2632744620 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+24956232800 \sqrt {-10 x^{2}-x +3}\, x^{3}-7766596629 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+25208605020 \sqrt {-10 x^{2}-x +3}\, x^{2}-789823386 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-391879320 \sqrt {-10 x^{2}-x +3}\, x +1184735079 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3366379220 \sqrt {-10 x^{2}-x +3}\right )}{175692000 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 197, normalized size = 1.39 \[ -\frac {243 \, x^{4}}{10 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {999}{5856400} \, x {\left (\frac {7220 \, x}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {361}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {2997}{4000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {360639}{2928200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {5842159 \, x}{878460 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3429 \, x^{2}}{25 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {947293}{21961500 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3016649 \, x}{90750 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1851167}{90750 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^5}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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