Optimal. Leaf size=89 \[ \frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {14}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1649 \sqrt {1-2 x}}{7986 (3+5 x)^{3/2}}-\frac {3298 \sqrt {1-2 x}}{43923 \sqrt {3+5 x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {91, 79, 47, 37}
\begin {gather*} -\frac {3298 \sqrt {1-2 x}}{43923 \sqrt {5 x+3}}-\frac {1649 \sqrt {1-2 x}}{7986 (5 x+3)^{3/2}}+\frac {14}{121 (5 x+3)^{3/2} \sqrt {1-2 x}}+\frac {49}{66 (5 x+3)^{3/2} (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rule 91
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{66} \int \frac {-\frac {381}{2}+297 x}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {14}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {1649}{484} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {14}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1649 \sqrt {1-2 x}}{7986 (3+5 x)^{3/2}}+\frac {1649 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{3993}\\ &=\frac {49}{66 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {14}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {1649 \sqrt {1-2 x}}{7986 (3+5 x)^{3/2}}-\frac {3298 \sqrt {1-2 x}}{43923 \sqrt {3+5 x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 37, normalized size = 0.42 \begin {gather*} \frac {18728+49200 x-9894 x^2-65960 x^3}{43923 (1-2 x)^{3/2} (3+5 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 39, normalized size = 0.44
method | result | size |
gosper | \(-\frac {2 \left (32980 x^{3}+4947 x^{2}-24600 x -9364\right )}{43923 \left (3+5 x \right )^{\frac {3}{2}} \left (1-2 x \right )^{\frac {3}{2}}}\) | \(32\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (32980 x^{3}+4947 x^{2}-24600 x -9364\right )}{43923 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 59, normalized size = 0.66 \begin {gather*} \frac {6596 \, x}{43923 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1649}{219615 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1229 \, x}{1815 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {733}{1815 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 53, normalized size = 0.60 \begin {gather*} -\frac {2 \, {\left (32980 \, x^{3} + 4947 \, x^{2} - 24600 \, x - 9364\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{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. 160 vs.
\(2 (65) = 130\).
time = 1.99, size = 160, normalized size = 1.80 \begin {gather*} -\frac {1}{3513840} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {1716 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {14 \, {\left (164 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1287 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1098075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {429 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{219615 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.52, size = 69, normalized size = 0.78 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (-\frac {6596\,x^3}{219615}-\frac {1649\,x^2}{366025}+\frac {328\,x}{14641}+\frac {9364}{1098075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{25}+\frac {9\,\sqrt {1-2\,x}}{50}-\frac {7\,x^2\,\sqrt {1-2\,x}}{10}-x^3\,\sqrt {1-2\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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