Optimal. Leaf size=218 \[ -\frac {1282376 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{41503 \sqrt {33}}+\frac {213119320 \sqrt {1-2 x} \sqrt {3 x+2}}{1369599 \sqrt {5 x+3}}-\frac {3205940 \sqrt {1-2 x} \sqrt {3 x+2}}{124509 (5 x+3)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac {42623864 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}} \]
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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 = {104, 152, 158, 113, 119} \[ \frac {213119320 \sqrt {1-2 x} \sqrt {3 x+2}}{1369599 \sqrt {5 x+3}}-\frac {3205940 \sqrt {1-2 x} \sqrt {3 x+2}}{124509 (5 x+3)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac {1282376 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}-\frac {42623864 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}} \]
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)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {167}{2}-105 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {4 \int \frac {-1137+\frac {2025 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{1617}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {8 \int \frac {-\frac {285195}{4}+81540 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{11319}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {16 \int \frac {-\frac {5827965}{2}+\frac {7213365 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{373527}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {213119320 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}-\frac {32 \int \frac {-\frac {303580485}{8}-\frac {239759235 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4108797}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {213119320 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}+\frac {641188 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{41503}+\frac {42623864 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{456533}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {14496 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {3205940 \sqrt {1-2 x} \sqrt {2+3 x}}{124509 (3+5 x)^{3/2}}+\frac {213119320 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}-\frac {42623864 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}-\frac {1282376 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 109, normalized size = 0.50 \[ \frac {2 \left (2 \sqrt {2} \left (10655966 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5366165 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {-9590369400 x^4-13428808080 x^3-2415287594 x^2+3336610202 x+1213551469}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )}{1369599} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{13500 \, x^{8} + 37800 \, x^{7} + 33255 \, x^{6} + 121 \, x^{5} - 15709 \, x^{4} - 7145 \, x^{3} + 774 \, x^{2} + 1188 \, x + 216}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 311, normalized size = 1.43 \[ \frac {2 \sqrt {-2 x +1}\, \left (9590369400 x^{4}+13428808080 x^{3}-319678980 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+160984950 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2415287594 x^{2}-404926708 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+203914270 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-3336610202 x -127871592 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+64393980 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1213551469\right )}{1369599 \left (3 x +2\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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