Optimal. Leaf size=63 \[ \frac {6 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}-2 \sqrt {\frac {5}{7}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {104, 21, 113} \[ \frac {6 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}-2 \sqrt {\frac {5}{7}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right ) \]
Antiderivative was successfully verified.
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Rule 21
Rule 104
Rule 113
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {2}{7} \int \frac {10+15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {10}{7} \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}-2 \sqrt {\frac {5}{7}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 62, normalized size = 0.98 \[ \frac {2}{7} \left (\frac {3 \sqrt {1-2 x} \sqrt {5 x+3}}{\sqrt {3 x+2}}+\sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 92, normalized size = 1.46 \[ -\frac {2 \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-30 x^{2}-3 x +\sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+9\right )}{7 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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