Optimal. Leaf size=160 \[ \frac {5057 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1250}+\frac {7 \sqrt {5 x+3} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}+\frac {312}{275} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {14517 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{2750}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250} \]
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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 = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {98, 154, 158, 113, 119} \[ \frac {7 \sqrt {5 x+3} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}+\frac {312}{275} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {14517 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{2750}+\frac {5057 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250} \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(2+3 x)^{3/2} \left (\frac {381}{2}+312 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {1}{275} \int \frac {\left (-13425-\frac {43551 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {\int \frac {\frac {1915857}{4}+\frac {1513107 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4125}\\ &=\frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {15171 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2500}-\frac {504369 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{13750}\\ &=\frac {14517 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2750}+\frac {312}{275} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {7 (2+3 x)^{5/2} \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {168123 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}+\frac {5057 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 110, normalized size = 0.69 \[ \frac {169365 \sqrt {2-4 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-10 \sqrt {3 x+2} \sqrt {5 x+3} \left (2970 x^2+11154 x-27757\right )-336246 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{27500 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 145, normalized size = 0.91 \[ \frac {\sqrt {3 x +2}\, \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (445500 x^{4}+2237400 x^{3}-1866090 x^{2}-4604590 x +336246 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-169365 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1665420\right )}{825000 x^{3}+632500 x^{2}-192500 x -165000} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \]
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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