Optimal. Leaf size=160 \[ -\frac {34154 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{46305}+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac {53194 \sqrt {1-2 x} \sqrt {5 x+3}}{46305 \sqrt {3 x+2}}+\frac {544 \sqrt {1-2 x} \sqrt {5 x+3}}{6615 (3 x+2)^{3/2}}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305} \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac {53194 \sqrt {1-2 x} \sqrt {5 x+3}}{46305 \sqrt {3 x+2}}+\frac {544 \sqrt {1-2 x} \sqrt {5 x+3}}{6615 (3 x+2)^{3/2}}-\frac {34154 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305} \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 150
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx &=\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {2}{105} \int \frac {\left (-243-\frac {865 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx\\ &=\frac {544 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {4 \int \frac {-\frac {49871}{4}-\frac {88105 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{6615}\\ &=\frac {544 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}-\frac {53194 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {8 \int \frac {-\frac {28265}{8}+\frac {132985 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{46305}\\ &=\frac {544 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}-\frac {53194 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}-\frac {53194 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{46305}+\frac {187847 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{46305}\\ &=\frac {544 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}-\frac {53194 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^{5/2}}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {34154 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 99, normalized size = 0.62 \[ \frac {\sqrt {2} \left (616735 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-53194 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )-\frac {6 \sqrt {1-2 x} \sqrt {5 x+3} \left (239373 x^2+311247 x+101257\right )}{(3 x+2)^{5/2}}}{138915} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 314, normalized size = 1.96 \[ -\frac {\left (14362380 x^{4}+20111058 x^{3}-478746 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5550615 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+3634188 x^{2}-638328 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+7400820 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-4994904 x -212776 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2466940 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1822626\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{138915 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}} \,d x \]
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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