Optimal. Leaf size=158 \[ -\frac {68 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715}+\frac {1752 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 \sqrt {3 x+2}}+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{245 (3 x+2)^{3/2}}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}-\frac {584 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \]
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Rubi [A] time = 0.05, antiderivative size = 158, 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 = {99, 152, 158, 113, 119} \[ \frac {1752 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 \sqrt {3 x+2}}+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{245 (3 x+2)^{3/2}}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}-\frac {68 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {584 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \]
Antiderivative was successfully verified.
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Rule 99
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {2}{35} \int \frac {\frac {29}{2}+15 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {18 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {4}{735} \int \frac {174-\frac {135 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {18 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {1752 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}+\frac {8 \int \frac {\frac {8445}{4}+3285 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5145}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {18 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {1752 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}+\frac {374 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1715}+\frac {1752 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1715}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {18 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {1752 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}-\frac {584 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {68 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 98, normalized size = 0.62 \[ \frac {2 \left (\sqrt {2} \left (292 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-105 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3} \left (7884 x^2+10701 x+3581\right )}{(3 x+2)^{5/2}}\right )}{1715} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16}, x\right ) \]
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 \[ \int \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 314, normalized size = 1.99 \[ \frac {2 \left (78840 x^{4}+114894 x^{3}-2628 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+945 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+22859 x^{2}-3504 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1260 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-28522 x -1168 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+420 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-10743\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{1715 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {\sqrt {5\,x+3}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}} \,d x \]
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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