Optimal. Leaf size=125 \[ -\frac {4 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{11 \sqrt {33}}+\frac {620 \sqrt {1-2 x} \sqrt {3 x+2}}{363 \sqrt {5 x+3}}-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}-\frac {124 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}} \]
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Rubi [A] time = 0.04, antiderivative size = 125, 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 = {104, 152, 158, 113, 119} \[ \frac {620 \sqrt {1-2 x} \sqrt {3 x+2}}{363 \sqrt {5 x+3}}-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}-\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}}-\frac {124 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}} \]
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} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{33 (3+5 x)^{3/2}}-\frac {2}{33} \int \frac {22-15 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{33 (3+5 x)^{3/2}}+\frac {620 \sqrt {1-2 x} \sqrt {2+3 x}}{363 \sqrt {3+5 x}}+\frac {4}{363} \int \frac {\frac {591}{2}+465 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{33 (3+5 x)^{3/2}}+\frac {620 \sqrt {1-2 x} \sqrt {2+3 x}}{363 \sqrt {3+5 x}}+\frac {2}{11} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {124}{121} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{33 (3+5 x)^{3/2}}+\frac {620 \sqrt {1-2 x} \sqrt {2+3 x}}{363 \sqrt {3+5 x}}-\frac {124 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}}-\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 97, normalized size = 0.78 \[ \frac {2}{363} \left (-29 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {25 \sqrt {1-2 x} \sqrt {3 x+2} (62 x+35)}{(5 x+3)^{3/2}}+62 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{750 \, x^{5} + 1475 \, x^{4} + 785 \, x^{3} - 153 \, x^{2} - 243 \, x - 54}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 219, normalized size = 1.75 \[ \frac {2 \left (9300 x^{3}+6800 x^{2}-310 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+145 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2225 x -186 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+87 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1750\right ) \sqrt {3 x +2}\, \sqrt {-2 x +1}}{363 \left (6 x^{2}+x -2\right ) \left (5 x +3\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {1-2\,x}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {1 - 2 x} \sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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