Optimal. Leaf size=161 \[ \frac {(5 x+3)^{3/2} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {33}{20} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^3+\frac {10377 \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{1600}+\frac {9 \sqrt {1-2 x} (5 x+3)^{3/2} (2253560 x+4772357)}{256000}+\frac {1018114917 \sqrt {1-2 x} \sqrt {5 x+3}}{1024000}-\frac {11199264087 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1024000 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 153, 147, 50, 54, 216} \[ \frac {(5 x+3)^{3/2} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {33}{20} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^3+\frac {10377 \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{1600}+\frac {9 \sqrt {1-2 x} (5 x+3)^{3/2} (2253560 x+4772357)}{256000}+\frac {1018114917 \sqrt {1-2 x} \sqrt {5 x+3}}{1024000}-\frac {11199264087 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1024000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 97
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^3 \sqrt {3+5 x} \left (51+\frac {165 x}{2}\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1}{50} \int \frac {\left (-8070-\frac {51885 x}{4}\right ) (2+3 x)^2 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {\int \frac {(2+3 x) \sqrt {3+5 x} \left (\frac {3983295}{4}+\frac {12676275 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{2000}\\ &=\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac {1018114917 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{512000}\\ &=\frac {1018114917 \sqrt {1-2 x} \sqrt {3+5 x}}{1024000}+\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac {11199264087 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2048000}\\ &=\frac {1018114917 \sqrt {1-2 x} \sqrt {3+5 x}}{1024000}+\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac {11199264087 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1024000 \sqrt {5}}\\ &=\frac {1018114917 \sqrt {1-2 x} \sqrt {3+5 x}}{1024000}+\frac {10377 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac {11199264087 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1024000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 93, normalized size = 0.58 \[ \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (41472000 x^5+200966400 x^4+461171520 x^3+732415080 x^2+1206337246 x-1702927233\right )-11199264087 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{10240000 \sqrt {-(1-2 x)^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.83, size = 96, normalized size = 0.60 \[ \frac {11199264087 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (41472000 \, x^{5} + 200966400 \, x^{4} + 461171520 \, x^{3} + 732415080 \, x^{2} + 1206337246 \, x - 1702927233\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20480000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 110, normalized size = 0.68 \[ -\frac {11199264087}{10240000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (12 \, {\left (24 \, {\left (12 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} + 443 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 44497 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 10283927 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1696858195 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 55996320435 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{128000000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 157, normalized size = 0.98 \[ -\frac {\left (-829440000 \sqrt {-10 x^{2}-x +3}\, x^{5}-4019328000 \sqrt {-10 x^{2}-x +3}\, x^{4}-9223430400 \sqrt {-10 x^{2}-x +3}\, x^{3}-14648301600 \sqrt {-10 x^{2}-x +3}\, x^{2}+22398528174 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-24126744920 \sqrt {-10 x^{2}-x +3}\, x -11199264087 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+34058544660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{20480000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.29, size = 198, normalized size = 1.23 \[ \frac {81}{400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} - \frac {6669}{640} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {12607994487}{20480000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1760913}{25600} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) - \frac {359469}{12800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {14553}{64} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x - \frac {2420847}{51200} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {305613}{1280} \, \sqrt {10 \, x^{2} - 21 \, x + 8} + \frac {540891153}{1024000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2401 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {1029 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (2 \, x - 1\right )}} - \frac {79233 \, \sqrt {-10 \, x^{2} - x + 3}}{64 \, {\left (2 \, x - 1\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]
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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