Optimal. Leaf size=100 \[ \frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {45}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {38, 41, 222}
\begin {gather*} 6 \sqrt {6} (1-2 x)^{5/2} x (2 x+1)^{5/2}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1}+\frac {45}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 222
Rubi steps
\begin {align*} \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx &=6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+5 \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx\\ &=15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {45}{2} \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx\\ &=\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {135}{2} \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx\\ &=\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {135}{2} \int \frac {1}{\sqrt {6-24 x^2}} \, dx\\ &=\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {45}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x)\\ \end {align*}
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Mathematica [A]
time = 1.11, size = 193, normalized size = 1.93 \begin {gather*} -\frac {3 \sqrt {3-6 x} x \left (33-104 x^2+128 x^4\right ) \left (8119+45112 x+91052 x^2+80768 x^3+30160 x^4+3712 x^5+64 x^6-\sqrt {2+4 x} \left (5741+26158 x+41096 x^2+26224 x^3+6160 x^4+352 x^5\right )\right )}{2 \left (11482+52316 x+82192 x^2+52448 x^3+12320 x^4+704 x^5-\sqrt {2+4 x} \left (8119+28874 x+33304 x^2+14160 x^3+1840 x^4+32 x^5\right )\right )}+45 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {-\sqrt {2}+\sqrt {1+2 x}}{\sqrt {1-2 x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.18, size = 134, normalized size = 1.34
method | result | size |
risch | \(-\frac {3 x \left (128 x^{4}-104 x^{2}+33\right ) \left (2 x -1\right ) \left (1+2 x \right ) \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \sqrt {6}}{4 \sqrt {-\left (2 x -1\right ) \left (1+2 x \right )}\, \sqrt {3-6 x}\, \sqrt {2+4 x}}+\frac {45 \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{8 \sqrt {2+4 x}\, \sqrt {3-6 x}}\) | \(107\) |
default | \(\frac {\left (3-6 x \right )^{\frac {5}{2}} \left (2+4 x \right )^{\frac {7}{2}}}{24}+\frac {\left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {7}{2}}}{8}+\frac {9 \sqrt {3-6 x}\, \left (2+4 x \right )^{\frac {7}{2}}}{32}-\frac {3 \left (2+4 x \right )^{\frac {5}{2}} \sqrt {3-6 x}}{16}-\frac {15 \left (2+4 x \right )^{\frac {3}{2}} \sqrt {3-6 x}}{16}-\frac {45 \sqrt {3-6 x}\, \sqrt {2+4 x}}{8}+\frac {45 \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{8 \sqrt {2+4 x}\, \sqrt {3-6 x}}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 46, normalized size = 0.46 \begin {gather*} \frac {1}{6} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {5}{2}} x + \frac {5}{4} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}} x + \frac {45}{4} \, \sqrt {-24 \, x^{2} + 6} x + \frac {45}{8} \, \sqrt {6} \arcsin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 65, normalized size = 0.65 \begin {gather*} \frac {3}{4} \, {\left (128 \, x^{5} - 104 \, x^{3} + 33 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3} - \frac {45}{8} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {\sqrt {3} \sqrt {2} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)]
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 227 vs.
\(2 (70) = 140\).
time = 0.04, size = 690, normalized size = 6.90 \begin {gather*} \sqrt {3} \sqrt {2} \left (1152 \left (2 \left (\left (\left (\left (\left (\frac {31}{3840}-\frac {1}{768} \sqrt {-2 x+1} \sqrt {-2 x+1}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}-\frac {107}{5120}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}+\frac {451}{15360}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}-\frac {149}{6144}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}+\frac {27}{2048}\right ) \sqrt {-2 x+1} \sqrt {2 x+1}+\frac {5}{512} \arcsin \left (\frac {\sqrt {-2 x+1}}{\sqrt {2}}\right )\right )-576 \left (2 \left (\left (\left (\left (-\frac {21}{1280}+\frac {1}{320} \sqrt {-2 x+1} \sqrt {-2 x+1}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}+\frac {133}{3840}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}-\frac {59}{1536}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}+\frac {13}{512}\right ) \sqrt {-2 x+1} \sqrt {2 x+1}+\frac {3}{128} \arcsin \left (\frac {\sqrt {-2 x+1}}{\sqrt {2}}\right )\right )-576 \left (2 \left (\left (\left (\frac {13}{384}-\frac {1}{128} \sqrt {-2 x+1} \sqrt {-2 x+1}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}-\frac {43}{768}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}+\frac {13}{256}\right ) \sqrt {-2 x+1} \sqrt {2 x+1}+\frac {3}{64} \arcsin \left (\frac {\sqrt {-2 x+1}}{\sqrt {2}}\right )\right )+288 \left (2 \left (\left (\frac {1}{48} \sqrt {-2 x+1} \sqrt {-2 x+1}-\frac {7}{96}\right ) \sqrt {-2 x+1} \sqrt {-2 x+1}+\frac {3}{32}\right ) \sqrt {-2 x+1} \sqrt {2 x+1}+\frac {\arcsin \left (\frac {\sqrt {-2 x+1}}{\sqrt {2}}\right )}{8}\right )-36 \left (2 \left (\frac {1}{8} \sqrt {-2 x+1} \sqrt {-2 x+1}-\frac {3}{8}\right ) \sqrt {-2 x+1} \sqrt {2 x+1}-\frac {\arcsin \left (\frac {\sqrt {-2 x+1}}{\sqrt {2}}\right )}{2}\right )-36 \left (\frac {1}{2} \sqrt {-2 x+1} \sqrt {2 x+1}+\arcsin \left (\frac {\sqrt {-2 x+1}}{\sqrt {2}}\right )\right )\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 {\left (4\,x+2\right )}^{5/2}\,{\left (3-6\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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