Optimal. Leaf size=160 \[ -\frac {823 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{13125}+\frac {2}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac {27}{875} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {823 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2625}-\frac {55019 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26250} \]
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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} \[ \frac {2}{35} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}-\frac {27}{875} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {823 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2625}-\frac {823 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{13125}-\frac {55019 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26250} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x} \, dx &=\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {2}{35} \int \frac {\left (-\frac {25}{2}-\frac {27 x}{2}\right ) \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {27}{875} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {2}{875} \int \frac {\sqrt {3+5 x} \left (\frac {3229}{4}+\frac {2469 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {823 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}-\frac {27}{875} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {2 \int \frac {-\frac {52233}{2}-\frac {165057 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{7875}\\ &=-\frac {823 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}-\frac {27}{875} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {9053 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26250}+\frac {55019 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{26250}\\ &=-\frac {823 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2625}-\frac {27}{875} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {55019 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26250}-\frac {823 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{13125}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 97, normalized size = 0.61 \[ \frac {-27860 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (2250 x^2+2445 x-166\right )+55019 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{39375 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 150, normalized size = 0.94 \[ \frac {\sqrt {3 x +2}\, \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (2025000 x^{5}+3753000 x^{4}+1065150 x^{3}-1032990 x^{2}-405240 x -55019 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+27860 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+29880\right )}{2362500 x^{3}+1811250 x^{2}-551250 x -472500} \]
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 \[ \int \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}\,{d x} \]
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 \[ \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3} \,d x \]
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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