Optimal. Leaf size=160 \[ \frac {192}{49} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {31940 \sqrt {1-2 x} \sqrt {3 x+2}}{539 \sqrt {5 x+3}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.06, 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 = {104, 152, 158, 113, 119} \[ -\frac {31940 \sqrt {1-2 x} \sqrt {3 x+2}}{539 \sqrt {5 x+3}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {192}{49} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 104
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2}{21} \int \frac {42-45 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {4}{147} \int \frac {\frac {3495}{2}-1080 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}-\frac {8 \int \frac {\frac {45495}{2}+\frac {71865 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}-\frac {288}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {19164}{539} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {192}{49} \sqrt {\frac {3}{11}} 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 = 0.16, size = 100, normalized size = 0.62 \[ \frac {2}{539} \left (-2 \sqrt {2} \left (1597 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-805 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )-\frac {\sqrt {1-2 x} \left (143730 x^2+186888 x+60635\right )}{(3 x+2)^{3/2} \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 219, normalized size = 1.37 \[ -\frac {2 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (287460 x^{3}+230046 x^{2}-9582 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4830 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-65618 x -6388 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+3220 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-60635\right )}{539 \left (3 x +2\right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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