Optimal. Leaf size=160 \[ -\frac {2657 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{118125}+\frac {2}{35} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {194 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{2625}-\frac {2657 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{23625}-\frac {118898 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{118125} \]
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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} \[ \frac {2}{35} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {194 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{2625}-\frac {2657 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{23625}-\frac {2657 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{118125}-\frac {118898 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{118125} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x} \, dx &=\frac {2}{35} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2}{35} \int \frac {\left (-\frac {67}{2}-\frac {97 x}{2}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2625}+\frac {2}{35} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4 \int \frac {\left (-\frac {1203}{2}-\frac {2657 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2625}\\ &=-\frac {2657 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{23625}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2625}+\frac {2}{35} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {4 \int \frac {\frac {148523}{8}+\frac {59449 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{23625}\\ &=-\frac {2657 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{23625}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2625}+\frac {2}{35} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {29227 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{236250}+\frac {118898 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{118125}\\ &=-\frac {2657 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{23625}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2625}+\frac {2}{35} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {118898 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{118125}-\frac {2657 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{118125}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 97, normalized size = 0.61 \[ \frac {-150115 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (-13500 x^2+7380 x+6631\right )+237796 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{354375 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 150, normalized size = 0.94 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-12150000 x^{5}-2673000 x^{4}+13895100 x^{3}+5455590 x^{2}-2720910 x -237796 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+150115 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1193580\right )}{21262500 x^{3}+16301250 x^{2}-4961250 x -4252500} \]
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 \[ \int \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}\,\sqrt {5\,x+3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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