Optimal. Leaf size=222 \[ \frac {1400888 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{132055}-\frac {46585232 \sqrt {1-2 x} \sqrt {3 x+2}}{290521 \sqrt {5 x+3}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14928 \sqrt {1-2 x}}{18865 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {138 \sqrt {1-2 x}}{2695 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055} \]
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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 = {104, 152, 158, 113, 119} \[ -\frac {46585232 \sqrt {1-2 x} \sqrt {3 x+2}}{290521 \sqrt {5 x+3}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14928 \sqrt {1-2 x}}{18865 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {138 \sqrt {1-2 x}}{2695 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {1400888 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055} \]
Antiderivative was successfully verified.
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Rule 104
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2}{77} \int \frac {-\frac {163}{2}-105 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {4 \int \frac {-1291+\frac {1725 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{2695}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {301413}{4}+83970 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{56595}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {12741465}{4}+\frac {7879995 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{396165}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}+\frac {32 \int \frac {-\frac {331786305}{8}-\frac {131020965 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4357815}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}-\frac {2101332 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{132055}-\frac {139755696 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1452605}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}+\frac {1400888 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 109, normalized size = 0.49 \[ \frac {2 \left (\frac {6289006320 x^4+9225477612 x^3+1919527182 x^2-2283681406 x-884250959}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}-2 \sqrt {2} \left (11646308 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5867645 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )\right )}{1452605} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8100 \, x^{8} + 23220 \, x^{7} + 21141 \, x^{6} + 690 \, x^{5} - 9791 \, x^{4} - 4696 \, x^{3} + 424 \, x^{2} + 768 \, x + 144}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 314, normalized size = 1.41 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-6289006320 x^{4}-9225477612 x^{3}+209633544 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-105617610 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1919527182 x^{2}+279511392 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-140823480 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2283681406 x +93170464 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-46941160 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+884250959\right )}{1452605 \left (3 x +2\right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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