Optimal. Leaf size=129 \[ -\frac {34154 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{16875}-\frac {4}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {1088 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{3375}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875} \]
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Rubi [A] time = 0.04, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {102, 154, 158, 113, 119} \[ -\frac {4}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {1088 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{3375}-\frac {34154 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875} \]
Antiderivative was successfully verified.
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Rule 102
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx &=-\frac {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{75} \int \frac {\left (\frac {41}{2}-272 x\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3375}-\frac {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4 \int \frac {\frac {5653}{4}-\frac {26597 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3375}\\ &=-\frac {1088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3375}-\frac {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {53194 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{16875}+\frac {187847 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{16875}\\ &=-\frac {1088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3375}-\frac {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875}-\frac {34154 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 97, normalized size = 0.75 \[ \frac {616735 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+60 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} (90 x-317)-53194 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{50625} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{15 \, x^{2} + 19 \, x + 6}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} \sqrt {3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 145, normalized size = 1.12 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-162000 x^{4}+446400 x^{3}+475260 x^{2}-100740 x -53194 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+616735 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-114120\right )}{50625 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} \sqrt {3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{\sqrt {3\,x+2}\,\sqrt {5\,x+3}} \,d x \]
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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