Optimal. Leaf size=160 \[ -\frac {2281 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{65625}+\frac {2}{35} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {62}{875} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {487 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{13125}-\frac {46159 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625} \]
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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 = {101, 154, 158, 113, 119} \[ \frac {2}{35} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {62}{875} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {487 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{13125}-\frac {2281 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625}-\frac {46159 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625} \]
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} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx &=\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {2}{35} \int \frac {\left (-\frac {69}{2}-\frac {93 x}{2}\right ) \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {4 \int \frac {\left (-\frac {1425}{2}-\frac {1461 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2625}\\ &=-\frac {487 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13125}+\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {4 \int \frac {\frac {181227}{8}+\frac {138477 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{39375}\\ &=-\frac {487 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13125}+\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {25091 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{131250}+\frac {46159 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{65625}\\ &=-\frac {487 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13125}+\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {46159 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625}-\frac {2281 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 97, normalized size = 0.61 \[ \frac {-17045 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (-4500 x^2+2040 x+2873\right )+92318 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{196875 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (6 \, x^{2} + x - 2\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 150, normalized size = 0.94 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-4050000 x^{5}-1269000 x^{4}+4938300 x^{3}+2363970 x^{2}-970530 x -92318 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+17045 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-517140\right )}{11812500 x^{3}+9056250 x^{2}-2756250 x -2362500} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}}{\sqrt {5\,x+3}} \,d x \]
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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