Optimal. Leaf size=143 \[ -\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}-\frac {963 \sqrt {1-2 x} (5 x+3)^{7/2}}{4000}-\frac {78167 \sqrt {1-2 x} (5 x+3)^{5/2}}{48000}-\frac {859837 \sqrt {1-2 x} (5 x+3)^{3/2}}{76800}-\frac {9458207 \sqrt {1-2 x} \sqrt {5 x+3}}{102400}+\frac {104040277 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}-\frac {963 \sqrt {1-2 x} (5 x+3)^{7/2}}{4000}-\frac {78167 \sqrt {1-2 x} (5 x+3)^{5/2}}{48000}-\frac {859837 \sqrt {1-2 x} (5 x+3)^{3/2}}{76800}-\frac {9458207 \sqrt {1-2 x} \sqrt {5 x+3}}{102400}+\frac {104040277 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 90
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}-\frac {1}{50} \int \frac {\left (-314-\frac {963 x}{2}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {78167 \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx}{8000}\\ &=-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {859837 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{19200}\\ &=-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {9458207 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{51200}\\ &=-\frac {9458207 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {104040277 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{204800}\\ &=-\frac {9458207 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {104040277 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{102400 \sqrt {5}}\\ &=-\frac {9458207 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {963 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x) (3+5 x)^{7/2}+\frac {104040277 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 88, normalized size = 0.62 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (6912000 x^4+26294400 x^3+44906720 x^2+48658820 x+46187289\right )+312120831 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{3072000 \sqrt {2 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 77, normalized size = 0.54 \[ -\frac {1}{307200} \, {\left (6912000 \, x^{4} + 26294400 \, x^{3} + 44906720 \, x^{2} + 48658820 \, x + 46187289\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {104040277}{2048000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 72, normalized size = 0.50 \[ -\frac {1}{15360000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, {\left (240 \, x + 481\right )} {\left (5 \, x + 3\right )} + 78167\right )} {\left (5 \, x + 3\right )} + 4299185\right )} {\left (5 \, x + 3\right )} + 141873105\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 1560604155 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 121, normalized size = 0.85 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-138240000 \sqrt {-10 x^{2}-x +3}\, x^{4}-525888000 \sqrt {-10 x^{2}-x +3}\, x^{3}-898134400 \sqrt {-10 x^{2}-x +3}\, x^{2}-973176400 \sqrt {-10 x^{2}-x +3}\, x +312120831 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-923745780 \sqrt {-10 x^{2}-x +3}\right )}{6144000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 92, normalized size = 0.64 \[ -\frac {45}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {2739}{32} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {280667}{1920} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {2432941}{15360} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {104040277}{2048000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {15395763}{102400} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]
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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