Optimal. Leaf size=84 \[ \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (38770 x+24439)}{99825 (5 x+3)^{3/2}}-\frac {27 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25 \sqrt {10}} \]
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Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 145, 54, 216} \[ \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (38770 x+24439)}{99825 (5 x+3)^{3/2}}-\frac {27 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 145
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1}{11} \int \frac {(2+3 x) \left (19+\frac {99 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (24439+38770 x)}{99825 (3+5 x)^{3/2}}-\frac {27}{50} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (24439+38770 x)}{99825 (3+5 x)^{3/2}}-\frac {27 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25 \sqrt {5}}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (24439+38770 x)}{99825 (3+5 x)^{3/2}}-\frac {27 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25 \sqrt {10}}\\ \end {align*}
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Mathematica [C] time = 0.91, size = 189, normalized size = 2.25 \[ \frac {343 \left (\frac {160 (2 x-1) (3 x+2)^3 \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};-\frac {5}{11} (2 x-1)\right )}{79233}-\frac {200 (x+3) \left (6 x^2+x-2\right )^2 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-\frac {5}{11} (2 x-1)\right )}{124509}+\frac {\sqrt {10-20 x} \sqrt {5 x+3} \left (43200 x^5+28080 x^4-400032 x^3+1229303 x^2+2053496 x+1669914\right )-27951 \left (108 x^3+513 x^2+1296 x+374\right ) \sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{154350 \sqrt {55} (1-2 x)^{5/2}}\right )}{121 \sqrt {22-44 x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.52, size = 101, normalized size = 1.20 \[ \frac {107811 \, \sqrt {10} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (649265 \, x^{2} + 772408 \, x + 229661\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1996500 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.40, size = 165, normalized size = 1.96 \[ -\frac {1}{7986000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2460 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {27}{250} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {343 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{6655 \, {\left (2 \, x - 1\right )}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {615 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{499125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 134, normalized size = 1.60 \[ -\frac {\sqrt {-2 x +1}\, \left (5390550 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3773385 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+12985300 \sqrt {-10 x^{2}-x +3}\, x^{2}-1293732 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+15448160 \sqrt {-10 x^{2}-x +3}\, x -970299 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+4593220 \sqrt {-10 x^{2}-x +3}\right )}{1996500 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 78, normalized size = 0.93 \[ -\frac {27}{500} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {129853 \, x}{99825 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {382849}{499125 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2}{4125 \, {\left (5 \, \sqrt {-10 \, x^{2} - x + 3} x + 3 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^3}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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