Optimal. Leaf size=249 \[ \frac {145418632 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{10168235 \sqrt {33}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {3 x+2}}{67110351 \sqrt {5 x+3}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {499564 \sqrt {1-2 x}}{1452605 (3 x+2)^{3/2} \sqrt {5 x+3}}-\frac {2206 \sqrt {1-2 x}}{207515 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {1616}{17787 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4839325048 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}} \]
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Rubi [A] time = 0.10, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {4839325048 \sqrt {1-2 x} \sqrt {3 x+2}}{67110351 \sqrt {5 x+3}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {499564 \sqrt {1-2 x}}{1452605 (3 x+2)^{3/2} \sqrt {5 x+3}}-\frac {2206 \sqrt {1-2 x}}{207515 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {1616}{17787 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {145418632 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {4839325048 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}} \]
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)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2}{231} \int \frac {-\frac {269}{2}-135 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {55457}{4}+21210 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {8 \int \frac {\frac {429823}{4}+\frac {82725 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{622545}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {32051607}{8}-\frac {16860285 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{13073445}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {32 \int \frac {\frac {661979505}{4}-\frac {817979805 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{91514115}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}-\frac {64 \int \frac {\frac {34464999645}{16}+\frac {27221203395 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1006655265}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}-\frac {72709316 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10168235}-\frac {4839325048 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{111850585}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}+\frac {4839325048 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {145418632 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 115, normalized size = 0.46 \[ \frac {2 \left (-2 \sqrt {2} \left (1209831262 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-609979405 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )-\frac {1306617762960 x^5+1263428429256 x^4-559512908172 x^3-673871013766 x^2+53503915182 x+91855922241}{(1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}}\right )}{335551755} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{16200 \, x^{9} + 38340 \, x^{8} + 19062 \, x^{7} - 19761 \, x^{6} - 20272 \, x^{5} + 399 \, x^{4} + 5544 \, x^{3} + 1112 \, x^{2} - 480 \, x - 144}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 406, normalized size = 1.63 \[ \frac {2 \sqrt {-2 x +1}\, \left (-1306617762960 x^{5}-1263428429256 x^{4}+43553925432 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-21959258580 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+559512908172 x^{3}+36294937860 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-18299382150 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+673871013766 x^{2}-9678650096 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4879835240 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-53503915182 x -9678650096 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4879835240 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-91855922241\right )}{335551755 \left (3 x +2\right )^{\frac {5}{2}} \left (2 x -1\right )^{2} \sqrt {5 x +3}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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