Optimal. Leaf size=67 \[ -\frac {2}{55 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {82 \sqrt {3+5 x}}{1815 (1-2 x)^{3/2}}+\frac {164 \sqrt {3+5 x}}{3993 \sqrt {1-2 x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37}
\begin {gather*} \frac {164 \sqrt {5 x+3}}{3993 \sqrt {1-2 x}}+\frac {82 \sqrt {5 x+3}}{1815 (1-2 x)^{3/2}}-\frac {2}{55 (1-2 x)^{3/2} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rubi steps
\begin {align*} \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx &=-\frac {2}{55 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {41}{55} \int \frac {1}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2}{55 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {82 \sqrt {3+5 x}}{1815 (1-2 x)^{3/2}}+\frac {82}{363} \int \frac {1}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2}{55 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {82 \sqrt {3+5 x}}{1815 (1-2 x)^{3/2}}+\frac {164 \sqrt {3+5 x}}{3993 \sqrt {1-2 x}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 32, normalized size = 0.48 \begin {gather*} \frac {888+738 x-1640 x^2}{3993 (1-2 x)^{3/2} \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 34, normalized size = 0.51
method | result | size |
gosper | \(-\frac {2 \left (820 x^{2}-369 x -444\right )}{3993 \sqrt {3+5 x}\, \left (1-2 x \right )^{\frac {3}{2}}}\) | \(27\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (820 x^{2}-369 x -444\right )}{3993 \sqrt {3+5 x}\, \left (-1+2 x \right )^{2}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 64, normalized size = 0.96 \begin {gather*} \frac {820 \, x}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {41}{3993 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {7}{33 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 43, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (820 \, x^{2} - 369 \, x - 444\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3993 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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 {3 x + 2}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (49) = 98\).
time = 1.71, size = 100, normalized size = 1.49 \begin {gather*} -\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{2662 \, \sqrt {5 \, x + 3}} - \frac {2 \, {\left (152 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1221 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{99825 \, {\left (2 \, x - 1\right )}^{2}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{1331 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.47, size = 52, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {5\,x+3}\,\left (-\frac {164\,x^2}{3993}+\frac {123\,x}{6655}+\frac {148}{6655}\right )}{\frac {x\,\sqrt {1-2\,x}}{10}-\frac {3\,\sqrt {1-2\,x}}{10}+x^2\,\sqrt {1-2\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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