Optimal. Leaf size=156 \[ -\frac {1}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {4}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {3}{5} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ -\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {4}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {3}{5} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2}{45} \int \frac {\left (-\frac {45}{2}-\frac {405 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{405} \int \frac {\frac {5265}{4}+\frac {3645 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {9}{5} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {3}{5} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {1}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A] time = 0.25, size = 112, normalized size = 0.72 \[ \frac {15 \sqrt {2} (3 x+2) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+10 \sqrt {1-2 x} x \sqrt {3 x+2} \sqrt {5 x+3} (10 x+7)+18 \sqrt {2} (3 x+2) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{90 x+60} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 144, normalized size = 0.92 \[ -\frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {3 x +2}\, \left (-1000 x^{4}-800 x^{3}+230 x^{2}+210 x +18 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+15 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )\right )}{30 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \]
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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