Optimal. Leaf size=187 \[ -\frac {7388 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{65219 \sqrt {33}}+\frac {598660 \sqrt {1-2 x} \sqrt {3 x+2}}{2152227 \sqrt {5 x+3}}-\frac {18470 \sqrt {1-2 x} \sqrt {3 x+2}}{195657 (5 x+3)^{3/2}}+\frac {368 \sqrt {3 x+2}}{5929 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {4 \sqrt {3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {119732 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}} \]
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Rubi [A] time = 0.07, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {598660 \sqrt {1-2 x} \sqrt {3 x+2}}{2152227 \sqrt {5 x+3}}-\frac {18470 \sqrt {1-2 x} \sqrt {3 x+2}}{195657 (5 x+3)^{3/2}}+\frac {368 \sqrt {3 x+2}}{5929 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {4 \sqrt {3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {7388 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}}-\frac {119732 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}} \]
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} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {201}{2}-75 x}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {4 \int \frac {\frac {20445}{4}+6210 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {19965}{2}-\frac {83115 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{586971}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}+\frac {598660 \sqrt {1-2 x} \sqrt {2+3 x}}{2152227 \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {1799235}{8}+\frac {1346985 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6456681}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}+\frac {598660 \sqrt {1-2 x} \sqrt {2+3 x}}{2152227 \sqrt {3+5 x}}+\frac {3694 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{65219}+\frac {119732 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{717409}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}+\frac {598660 \sqrt {1-2 x} \sqrt {2+3 x}}{2152227 \sqrt {3+5 x}}-\frac {119732 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}}-\frac {7388 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 103, normalized size = 0.55 \[ \frac {2 \left (\sqrt {2} \left (1085 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+59866 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )+\frac {\sqrt {3 x+2} \left (5986600 x^3-2800980 x^2-1822554 x+881831\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}\right )}{2152227} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3000 \, x^{7} + 2900 \, x^{6} - 2010 \, x^{5} - 2277 \, x^{4} + 425 \, x^{3} + 603 \, x^{2} - 27 \, x - 54}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 311, normalized size = 1.66 \[ -\frac {2 \left (-17959800 x^{4}-3570260 x^{3}+598660 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+10850 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+11069622 x^{2}+59866 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1085 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+999615 x -179598 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-3255 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1763662\right ) \sqrt {-2 x +1}}{2152227 \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )^{2} \sqrt {3 x +2}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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