Optimal. Leaf size=154 \[ \frac {321709971 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {9748787 \sqrt {1-2 x} (3+5 x)^{3/2}}{51200}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac {3538809681 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 154, 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 = {99, 158, 152,
52, 56, 222} \begin {gather*} -\frac {3538809681 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{204800 \sqrt {10}}+\frac {(5 x+3)^{5/2} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {33}{20} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac {9748787 \sqrt {1-2 x} (5 x+3)^{3/2}}{51200}+\frac {9 \sqrt {1-2 x} (5 x+3)^{5/2} (13820 x+27937)}{6400}+\frac {321709971 \sqrt {1-2 x} \sqrt {5 x+3}}{204800} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 99
Rule 152
Rule 158
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^2 (3+5 x)^{3/2} \left (52+\frac {165 x}{2}\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {33}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {1}{50} \int \frac {\left (-\frac {16505}{2}-\frac {51825 x}{4}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {33}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac {9748787 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{12800}\\ &=\frac {9748787 \sqrt {1-2 x} (3+5 x)^{3/2}}{51200}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac {321709971 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{102400}\\ &=\frac {321709971 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {9748787 \sqrt {1-2 x} (3+5 x)^{3/2}}{51200}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac {3538809681 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{409600}\\ &=\frac {321709971 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {9748787 \sqrt {1-2 x} (3+5 x)^{3/2}}{51200}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac {3538809681 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{204800 \sqrt {5}}\\ &=\frac {321709971 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {9748787 \sqrt {1-2 x} (3+5 x)^{3/2}}{51200}+\frac {33}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{5/2} (27937+13820 x)}{6400}-\frac {3538809681 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 83, normalized size = 0.54 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (-538018839+381820658 x+233394520 x^2+148751040 x^3+65836800 x^4+13824000 x^5\right )+3538809681 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2048000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 157, normalized size = 1.02
method | result | size |
default | \(-\frac {\left (-276480000 x^{5} \sqrt {-10 x^{2}-x +3}-1316736000 x^{4} \sqrt {-10 x^{2}-x +3}-2975020800 x^{3} \sqrt {-10 x^{2}-x +3}+7077619362 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -4667890400 x^{2} \sqrt {-10 x^{2}-x +3}-3538809681 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-7636413160 x \sqrt {-10 x^{2}-x +3}+10760376780 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{4096000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, size = 126, normalized size = 0.82 \begin {gather*} -\frac {675 \, x^{6}}{2 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {57915 \, x^{5}}{32 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {588291 \, x^{4}}{128 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {40330643 \, x^{3}}{5120 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {52185737 \, x^{2}}{4096 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3538809681}{4096000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1544632221 \, x}{204800 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1614056517}{204800 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.22, size = 96, normalized size = 0.62 \begin {gather*} \frac {3538809681 \, \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 (13824000 \, x^{5} + 65836800 \, x^{4} + 148751040 \, x^{3} + 233394520 \, x^{2} + 381820658 \, x - 538018839\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4096000 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.67, size = 110, normalized size = 0.71 \begin {gather*} -\frac {3538809681}{2048000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (24 \, {\left (36 \, {\left (16 \, \sqrt {5} {\left (5 \, x + 3\right )} + 141 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 42197 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9748787 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 536183285 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 17694048405 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{25600000 \, {\left (2 \, x - 1\right )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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