Optimal. Leaf size=191 \[ -\frac {84134 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1771875}+\frac {2}{45} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}+\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}{1575}-\frac {347 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{39375}-\frac {84134 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{354375}-\frac {5684677 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750} \]
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Rubi [A] time = 0.07, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {2}{45} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}+\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}{1575}-\frac {347 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{39375}-\frac {84134 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{354375}-\frac {84134 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875}-\frac {5684677 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750} \]
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} (2+3 x)^{3/2} \sqrt {3+5 x} \, dx &=\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {2}{45} \int \left (-\frac {69}{2}-\frac {93 x}{2}\right ) \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \, dx\\ &=\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {4 \int \frac {\left (-765-\frac {1041 x}{4}\right ) \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{4725}\\ &=-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {4 \int \frac {\sqrt {3+5 x} \left (\frac {334107}{8}+\frac {126201 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{118125}\\ &=-\frac {84134 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{354375}-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {4 \int \frac {-\frac {10787703}{8}-\frac {17054031 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1063125}\\ &=-\frac {84134 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{354375}-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {462737 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1771875}+\frac {5684677 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3543750}\\ &=-\frac {84134 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{354375}-\frac {347 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{39375}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {5684677 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750}-\frac {84134 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 105, normalized size = 0.55 \[ \frac {5684677 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (581651 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (472500 x^3+153000 x^2-359685 x-84697\right )\right )}{5315625 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (6 \, x^{2} + x - 2\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 155, normalized size = 0.81 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-425250000 x^{6}-463725000 x^{5}+317371500 x^{4}+441589950 x^{3}+10447080 x^{2}-82529670 x -5684677 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2908255 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-15245460\right )}{318937500 x^{3}+244518750 x^{2}-74418750 x -63787500} \]
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 \[ \int \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3} \,d x \]
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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