Optimal. Leaf size=183 \[ \frac {9738340821 \sqrt {1-2 x} \sqrt {3+5 x}}{1638400}+\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {107121749031 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1638400 \sqrt {10}} \]
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Rubi [A]
time = 0.04, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {107121749031 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1638400 \sqrt {10}}+\frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {13}{8} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac {999}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac {295101237 \sqrt {1-2 x} (5 x+3)^{3/2}}{409600}+\frac {\sqrt {1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac {9738340821 \sqrt {1-2 x} \sqrt {5 x+3}}{1638400} \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)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^3 (3+5 x)^{3/2} \left (61+\frac {195 x}{2}\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {1}{60} \int \frac {\left (-11805-\frac {74925 x}{4}\right ) (2+3 x)^2 (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {\int \frac {(2+3 x) (3+5 x)^{3/2} \left (\frac {7494225}{4}+\frac {23531625 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{3000}\\ &=\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {295101237 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{102400}\\ &=\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {9738340821 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{819200}\\ &=\frac {9738340821 \sqrt {1-2 x} \sqrt {3+5 x}}{1638400}+\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {107121749031 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3276800}\\ &=\frac {9738340821 \sqrt {1-2 x} \sqrt {3+5 x}}{1638400}+\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {107121749031 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1638400 \sqrt {5}}\\ &=\frac {9738340821 \sqrt {1-2 x} \sqrt {3+5 x}}{1638400}+\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {107121749031 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1638400 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 88, normalized size = 0.48 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (-16267424049+11734056318 x+7755469800 x^2+5945485120 x^3+3687379200 x^4+1479168000 x^5+276480000 x^6\right )+107121749031 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{16384000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 174, normalized size = 0.95
method | result | size |
default | \(-\frac {\left (-5529600000 \sqrt {-10 x^{2}-x +3}\, x^{6}-29583360000 x^{5} \sqrt {-10 x^{2}-x +3}-73747584000 x^{4} \sqrt {-10 x^{2}-x +3}-118909702400 x^{3} \sqrt {-10 x^{2}-x +3}+214243498062 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -155109396000 x^{2} \sqrt {-10 x^{2}-x +3}-107121749031 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-234681126360 x \sqrt {-10 x^{2}-x +3}+325348480980 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{32768000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 143, normalized size = 0.78 \begin {gather*} -\frac {3375 \, x^{7}}{4 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {80325 \, x^{6}}{16 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3574125 \, x^{5}}{256 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {25493477 \, x^{4}}{1024 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1415345109 \, x^{3}}{40960 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {8193669099 \, x^{2}}{163840 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {107121749031}{32768000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {46134951291 \, x}{1638400 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {48802272147}{1638400 \, \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 = 0.88, size = 101, normalized size = 0.55 \begin {gather*} \frac {107121749031 \, \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 (276480000 \, x^{6} + 1479168000 \, x^{5} + 3687379200 \, x^{4} + 5945485120 \, x^{3} + 7755469800 \, x^{2} + 11734056318 \, x - 16267424049\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{32768000 \, {\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 = 1.35, size = 123, normalized size = 0.67 \begin {gather*} -\frac {107121749031}{16384000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 35 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4299 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 3832457 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 295101237 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 16230568035 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 535608745155 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{204800000 \, {\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 )}^4\,{\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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