Optimal. Leaf size=129 \[ -\frac {4}{9} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {124 \sqrt {1-2 x} \sqrt {5 x+3}}{9 \sqrt {3 x+2}}+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}-\frac {124}{9} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ \frac {124 \sqrt {1-2 x} \sqrt {5 x+3}}{9 \sqrt {3 x+2}}+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}-\frac {4}{9} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {124}{9} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {14 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {53-29 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {124 \sqrt {1-2 x} \sqrt {3+5 x}}{9 \sqrt {2+3 x}}+\frac {4}{63} \int \frac {\frac {1379}{2}+1085 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {124 \sqrt {1-2 x} \sqrt {3+5 x}}{9 \sqrt {2+3 x}}+\frac {22}{9} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {124}{9} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {124 \sqrt {1-2 x} \sqrt {3+5 x}}{9 \sqrt {2+3 x}}-\frac {124}{9} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {4}{9} \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 = 0.30, size = 97, normalized size = 0.75 \[ \frac {2}{27} \left (-29 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (186 x+131)}{(3 x+2)^{3/2}}+62 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 219, normalized size = 1.70 \[ \frac {2 \left (5580 x^{3}+4488 x^{2}-186 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+87 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1281 x -124 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+58 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1179\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{27 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]
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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