Optimal. Leaf size=222 \[ -\frac {310399 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{382725}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2108 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{1701}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {64628 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{8505}-\frac {310399 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{76545}-\frac {25111 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725} \]
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Rubi [A] time = 0.08, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ -\frac {40}{81} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2108 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{1701}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {64628 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{8505}-\frac {310399 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{76545}-\frac {310399 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}-\frac {25111 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4}{405} \int \frac {\left (\frac {1625}{4}-\frac {13175 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\left (\frac {1409075}{8}-\frac {1211775 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42525}\\ &=\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (\frac {2027475}{8}-\frac {23279925 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{637875}\\ &=-\frac {310399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{76545}+\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\frac {53475825}{16}+\frac {1883325 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5740875}\\ &=-\frac {310399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{76545}+\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {25111 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{382725}+\frac {3414389 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{765450}\\ &=-\frac {310399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{76545}+\frac {64628 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{8505}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2108 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{1701}-\frac {40}{81} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {25111 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}-\frac {310399 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{382725}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 112, normalized size = 0.50 \[ \frac {10192945 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {30 \sqrt {1-2 x} \sqrt {5 x+3} \left (567000 x^4-386100 x^3-259650 x^2+245751 x+21964\right )}{\sqrt {3 x+2}}+50222 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{2296350} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 155, normalized size = 0.70 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \left (-170100000 x^{6}+98820000 x^{5}+140508000 x^{4}-100684800 x^{3}-37330230 x^{2}+21458670 x +50222 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+10192945 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1976760\right )}{2296350 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \]
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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