Optimal. Leaf size=218 \[ -\frac {43624697 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{283500 \sqrt {33}}-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {329683 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{34650}-\frac {43624697 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{623700}-\frac {725140729 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{141750 \sqrt {33}} \]
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Rubi [A] time = 0.08, antiderivative size = 218, 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 = {101, 154, 158, 113, 119} \[ -\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {329683 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{34650}-\frac {43624697 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{623700}-\frac {43624697 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{283500 \sqrt {33}}-\frac {725140729 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{141750 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {1}{11} \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2} \left (\frac {215}{2}+170 x\right )}{\sqrt {1-2 x}} \, dx\\ &=-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{495} \int \frac {\left (-15200-\frac {47385 x}{2}\right ) \sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {\int \frac {(3+5 x)^{3/2} \left (\frac {6483095}{4}+\frac {4945245 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{17325}\\ &=-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {\int \frac {\left (-\frac {212630445}{2}-\frac {654370455 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{259875}\\ &=-\frac {43624697 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{623700}-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {\int \frac {\frac {27544681245}{8}+\frac {10877110935 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2338875}\\ &=-\frac {43624697 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{623700}-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {43624697 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{567000}+\frac {725140729 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1559250}\\ &=-\frac {43624697 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{623700}-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {725140729 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{141750 \sqrt {33}}-\frac {43624697 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{283500 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 110, normalized size = 0.50 \[ \frac {2900562916 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (292189583 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (12757500 x^4+48384000 x^3+81985950 x^2+86822370 x+75000749\right )\right )}{9355500 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2 \, x - 1}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 160, normalized size = 0.73 \[ \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-11481750000 x^{7}-52348275000 x^{6}-104493240000 x^{5}-122253448500 x^{4}-101481939900 x^{3}-18760348110 x^{2}+31378183890 x -2900562916 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1460947915 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+13500134820\right )}{561330000 x^{3}+430353000 x^{2}-130977000 x -112266000} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]
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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