Optimal. Leaf size=129 \[ -\frac {598 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5625}+\frac {2}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {194 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1125}-\frac {2797 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625} \]
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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 = {101, 154, 158, 113, 119} \[ \frac {2}{25} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {194 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1125}-\frac {598 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625}-\frac {2797 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx &=\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2}{25} \int \frac {\left (-\frac {67}{2}-\frac {97 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1125}+\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {4 \int \frac {-584-\frac {2797 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1125}\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1125}+\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2797 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5625}+\frac {3289 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5625}\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1125}+\frac {2}{25} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2797 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625}-\frac {598 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5625}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 97, normalized size = 0.75 \[ \frac {7070 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+60 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} (71-45 x)+2797 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{16875} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3}}, x\right ) \]
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 \[ \int \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 145, normalized size = 1.12 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (81000 x^{4}-65700 x^{3}-116880 x^{2}+13620 x +2797 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+7070 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+25560\right )}{16875 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}}{\sqrt {5\,x+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 2}}{\sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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