Optimal. Leaf size=110 \[ \frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3200 \sqrt {10}} \]
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Rubi [A]
time = 0.02, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {99, 158, 152,
56, 222} \begin {gather*} -\frac {1463447 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200 \sqrt {10}}+\frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {7}{4} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {\sqrt {1-2 x} \sqrt {5 x+3} (73380 x+176833)}{3200} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 99
Rule 152
Rule 158
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^2 \left (32+\frac {105 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{30} \int \frac {\left (-\frac {5625}{2}-\frac {18345 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{6400}\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{3200 \sqrt {5}}\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3200 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 73, normalized size = 0.66 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (-224833+142686 x+57960 x^2+14400 x^3\right )+1463447 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{32000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 123, normalized size = 1.12
method | result | size |
default | \(-\frac {\left (-288000 x^{3} \sqrt {-10 x^{2}-x +3}+2926894 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1159200 x^{2} \sqrt {-10 x^{2}-x +3}-1463447 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2853720 x \sqrt {-10 x^{2}-x +3}+4496660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{64000 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 79, normalized size = 0.72 \begin {gather*} -\frac {1463447}{64000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1593}{160} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {89793}{3200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{8 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 86, normalized size = 0.78 \begin {gather*} \frac {1463447 \, \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 (14400 \, x^{3} + 57960 \, x^{2} + 142686 \, x - 224833\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{64000 \, {\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} \sqrt {5 x + 3}}{\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 = 0.76, size = 84, normalized size = 0.76 \begin {gather*} -\frac {1463447}{32000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (18 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 89 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4927 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1463447 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{80000 \, {\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\,\sqrt {5\,x+3}}{{\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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