Optimal. Leaf size=98 \[ \frac {8}{25} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {22 \sqrt {1-2 x} \sqrt {3 x+2}}{5 \sqrt {5 x+3}}+\frac {62}{25} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {98, 158, 113, 119} \[ -\frac {22 \sqrt {1-2 x} \sqrt {3 x+2}}{5 \sqrt {5 x+3}}+\frac {8}{25} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {62}{25} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx &=-\frac {22 \sqrt {1-2 x} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {2}{5} \int \frac {23+31 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {22 \sqrt {1-2 x} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {44}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {62}{25} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {22 \sqrt {1-2 x} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}+\frac {62}{25} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {8}{25} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 92, normalized size = 0.94 \[ \frac {1}{75} \left (-70 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-\frac {330 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}-62 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 135, normalized size = 1.38 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \left (-990 x^{2}-165 x +31 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+35 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+330\right )}{75 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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