Optimal. Leaf size=132 \[ \frac {13246251 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {145708761 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}} \]
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Rubi [A]
time = 0.02, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {145708761 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}}+\frac {(5 x+3)^{3/2} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {27}{16} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2+\frac {9 \sqrt {1-2 x} (5 x+3)^{3/2} (29320 x+62091)}{12800}+\frac {13246251 \sqrt {1-2 x} \sqrt {5 x+3}}{51200} \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)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^2 \sqrt {3+5 x} \left (42+\frac {135 x}{2}\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1}{40} \int \frac {\left (-\frac {10365}{2}-\frac {32985 x}{4}\right ) (2+3 x) \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {13246251 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{25600}\\ &=\frac {13246251 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {145708761 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{102400}\\ &=\frac {13246251 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {145708761 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{51200 \sqrt {5}}\\ &=\frac {13246251 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {145708761 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 82, normalized size = 0.62 \begin {gather*} \frac {-\frac {5 \sqrt {3+5 x} \left (-22217679+15218818 x+8057880 x^2+3729600 x^3+864000 x^4\right )}{\sqrt {1-2 x}}+145708761 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{256000} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 140, normalized size = 1.06
method | result | size |
default | \(-\frac {\left (-17280000 x^{4} \sqrt {-10 x^{2}-x +3}-74592000 x^{3} \sqrt {-10 x^{2}-x +3}+291417522 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -161157600 x^{2} \sqrt {-10 x^{2}-x +3}-145708761 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-304376360 x \sqrt {-10 x^{2}-x +3}+444353580 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1024000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.60, size = 184, normalized size = 1.39 \begin {gather*} -\frac {27}{32} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {155771121}{1024000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {251559}{25600} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) - \frac {2547}{640} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {2079}{64} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x - \frac {9801}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {43659}{1280} \, \sqrt {10 \, x^{2} - 21 \, x + 8} + \frac {5811399}{51200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (2 \, x - 1\right )}} - \frac {11319 \, \sqrt {-10 \, x^{2} - x + 3}}{32 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 91, normalized size = 0.69 \begin {gather*} \frac {145708761 \, \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 (864000 \, x^{4} + 3729600 \, x^{3} + 8057880 \, x^{2} + 15218818 \, x - 22217679\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1024000 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.22, size = 97, normalized size = 0.73 \begin {gather*} -\frac {145708761}{512000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (36 \, {\left (8 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 115 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 8919 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4415417 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 145708761 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1280000 \, {\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 )}^{3/2}}{{\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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