Optimal. Leaf size=117 \[ \frac {4 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{\sqrt {33}}-\frac {20 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}+4 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 152, 158, 113, 119} \[ -\frac {20 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}+\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{\sqrt {33}}+4 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 99
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {2 \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}}-2 \int \frac {-8+5 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}+\frac {4}{11} \int \frac {-\frac {209}{2}-165 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}-2 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-12 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}+4 \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{\sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 128, normalized size = 1.09 \[ \frac {2 \sqrt {2} \left (15 x^2+19 x+6\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-4 \sqrt {2} \left (15 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} (30 x+19)}{(3 x+2) (5 x+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36}, x\right ) \]
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 \[ \int \frac {\sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 135, normalized size = 1.15 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-60 x^{2}-8 x +2 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-\sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+19\right )}{30 x^{3}+23 x^{2}-7 x -6} \]
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 \[ \int \frac {\sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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