Optimal. Leaf size=164 \[ -\frac {1626211523 \sqrt {1-2 x} \sqrt {3+5 x}}{1126400}-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}} \]
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Rubi [A]
time = 0.04, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {99, 155, 158,
152, 52, 56, 222} \begin {gather*} \frac {1626211523 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400 \sqrt {10}}+\frac {(5 x+3)^{3/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac {123 (5 x+3)^{3/2} (3 x+2)^3}{22 \sqrt {1-2 x}}-\frac {3315}{352} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2} (10798680 x+22868329)}{281600}-\frac {1626211523 \sqrt {1-2 x} \sqrt {5 x+3}}{1126400} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 99
Rule 152
Rule 155
Rule 158
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^3 \sqrt {3+5 x} \left (51+\frac {165 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-7734-\frac {49725 x}{4}\right ) (2+3 x)^2 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {(2+3 x) \sqrt {3+5 x} \left (\frac {3817455}{4}+\frac {12148515 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{1320}\\ &=-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{563200}\\ &=-\frac {1626211523 \sqrt {1-2 x} \sqrt {3+5 x}}{1126400}-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{204800}\\ &=-\frac {1626211523 \sqrt {1-2 x} \sqrt {3+5 x}}{1126400}-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{102400 \sqrt {5}}\\ &=-\frac {1626211523 \sqrt {1-2 x} \sqrt {3+5 x}}{1126400}-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 88, normalized size = 0.54 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (739060191-2034703904 x+633940524 x^2+236669040 x^3+83548800 x^4+15552000 x^5\right )+4878634569 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{3072000 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 171, normalized size = 1.04
method | result | size |
default | \(\frac {\left (-311040000 x^{5} \sqrt {-10 x^{2}-x +3}-1670976000 x^{4} \sqrt {-10 x^{2}-x +3}+19514538276 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-4733380800 x^{3} \sqrt {-10 x^{2}-x +3}-19514538276 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -12678810480 x^{2} \sqrt {-10 x^{2}-x +3}+4878634569 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+40694078080 x \sqrt {-10 x^{2}-x +3}-14781203820 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{6144000 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.55, size = 241, normalized size = 1.47 \begin {gather*} \frac {81}{64} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1666460963}{2048000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {251559}{12800} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) + \frac {10161}{1280} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {2079}{32} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x + \frac {29403}{5120} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {43659}{640} \, \sqrt {10 \, x^{2} - 21 \, x + 8} - \frac {34897797}{102400} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2401 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{96 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {1029 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1323 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (2 \, x - 1\right )}} + \frac {26411 \, \sqrt {-10 \, x^{2} - x + 3}}{192 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {491519 \, \sqrt {-10 \, x^{2} - x + 3}}{192 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 106, normalized size = 0.65 \begin {gather*} -\frac {4878634569 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, 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 (15552000 \, x^{5} + 83548800 \, x^{4} + 236669040 \, x^{3} + 633940524 \, x^{2} - 2034703904 \, x + 739060191\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6144000 \, {\left (4 \, x^{2} - 4 \, 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 = 1.17, size = 110, normalized size = 0.67 \begin {gather*} \frac {1626211523}{1024000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9 \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} + 427 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 42657 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9855815 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 3252423046 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 53664980259 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{38400000 \, {\left (2 \, x - 1\right )}^{2}} \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 )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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