Optimal. Leaf size=220 \[ -\frac {24369 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{109375}-\frac {6 \sqrt {1-2 x} (3 x+2)^{7/2}}{\sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac {622}{175} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}+\frac {3872 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{4375}+\frac {4801 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{21875}-\frac {25643 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375} \]
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Rubi [A] time = 0.08, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac {6 \sqrt {1-2 x} (3 x+2)^{7/2}}{\sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac {622}{175} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}+\frac {3872 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{4375}+\frac {4801 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{21875}-\frac {24369 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {25643 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {9}{2}-30 x\right ) \sqrt {1-2 x} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (\frac {1545}{2}-\frac {4665 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {4 \int \frac {(2+3 x)^{3/2} \left (-\frac {15705}{4}+14520 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2625}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {4 \int \frac {\left (\frac {29625}{2}-\frac {216045 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{65625}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4801 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {4 \int \frac {-\frac {2042685}{8}-\frac {1153935 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{984375}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4801 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {25643 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{109375}+\frac {73107 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{218750}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x} (2+3 x)^{7/2}}{\sqrt {3+5 x}}+\frac {4801 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {3872 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}+\frac {622}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {25643 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {24369 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 112, normalized size = 0.51 \[ \frac {168035 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {10 \sqrt {1-2 x} \sqrt {3 x+2} \left (-202500 x^4-189000 x^3+174525 x^2+216050 x+52067\right )}{(5 x+3)^{3/2}}+51286 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{656250} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 234, normalized size = 1.06 \[ -\frac {\left (12150000 x^{6}+13365000 x^{5}-12631500 x^{4}-18488250 x^{3}-1794020 x^{2}+256430 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+840175 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+3800330 x +153858 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+504105 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1041340\right ) \sqrt {3 x +2}\, \sqrt {-2 x +1}}{656250 \left (6 x^{2}+x -2\right ) \left (5 x +3\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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