Optimal. Leaf size=129 \[ -\frac {41}{189} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 \sqrt {3 x+2}}-\frac {205}{189} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {974}{189} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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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, 154, 158, 113, 119} \[ \frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 \sqrt {3 x+2}}-\frac {205}{189} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {41}{189} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {974}{189} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx &=\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}-\frac {2}{21} \int \frac {\left (-45-\frac {205 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {205}{189} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}+\frac {2}{189} \int \frac {\frac {6295}{4}+2435 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {205}{189} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}+\frac {451}{378} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {974}{189} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {205}{189} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 \sqrt {2+3 x}}-\frac {974}{189} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {41}{189} \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 = 0.19, size = 97, normalized size = 0.75 \[ \frac {-595 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-\frac {6 \sqrt {1-2 x} \sqrt {5 x+3} (525 x+356)}{\sqrt {3 x+2}}+1948 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{1134} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 140, normalized size = 1.09 \[ \frac {\sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \left (-31500 x^{3}-24510 x^{2}+7314 x -1948 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+595 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+6408\right )}{34020 x^{3}+26082 x^{2}-7938 x -6804} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}} \,d x \]
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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