Optimal. Leaf size=94 \[ -\frac {519}{88} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac {21 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {519 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8 \sqrt {10}} \]
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Rubi [A]
time = 0.02, antiderivative size = 94, 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 = {91, 79, 52, 56,
222} \begin {gather*} \frac {519 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8 \sqrt {10}}-\frac {21 (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}+\frac {49 (5 x+3)^{3/2}}{66 (1-2 x)^{3/2}}-\frac {519}{88} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 79
Rule 91
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2 \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac {49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {\sqrt {3+5 x} \left (\frac {1089}{2}+297 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=\frac {49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac {21 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {519}{44} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {519}{88} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac {21 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {519}{16} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {519}{88} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac {21 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {519 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8 \sqrt {5}}\\ &=-\frac {519}{88} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac {21 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {519 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 73, normalized size = 0.78 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (2481-7712 x+1188 x^2\right )+17127 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2640 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 120, normalized size = 1.28
method | result | size |
default | \(\frac {\left (68508 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-68508 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -23760 x^{2} \sqrt {-10 x^{2}-x +3}+17127 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+154240 x \sqrt {-10 x^{2}-x +3}-49620 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{5280 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A]
time = 0.47, size = 91, normalized size = 0.97 \begin {gather*} -\frac {17127 \, \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 (1188 \, x^{2} - 7712 \, x + 2481\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{5280 \, {\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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{2} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 71, normalized size = 0.76 \begin {gather*} \frac {519}{80} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (297 \, \sqrt {5} {\left (5 \, x + 3\right )} - 11422 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 188397 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{33000 \, {\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 )}^2\,\sqrt {5\,x+3}}{{\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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