Optimal. Leaf size=180 \[ -\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}} \]
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Rubi [A]
time = 0.04, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210}
\begin {gather*} -\frac {3674891 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}}+\frac {3 (5 x+3)^{3/2} (1-2 x)^{7/2}}{35 (3 x+2)^5}+\frac {251 (5 x+3)^{3/2} (1-2 x)^{5/2}}{280 (3 x+2)^4}+\frac {2761 (5 x+3)^{3/2} (1-2 x)^{3/2}}{336 (3 x+2)^3}+\frac {30371 (5 x+3)^{3/2} \sqrt {1-2 x}}{448 (3 x+2)^2}-\frac {334081 \sqrt {5 x+3} \sqrt {1-2 x}}{6272 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 210
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251}{70} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761}{112} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371}{224} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {334081}{896} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {3674891 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{12544}\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac {3674891 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{6272}\\ &=-\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{6272 (2+3 x)}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{3/2}}{35 (2+3 x)^5}+\frac {251 (1-2 x)^{5/2} (3+5 x)^{3/2}}{280 (2+3 x)^4}+\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{336 (2+3 x)^3}+\frac {30371 \sqrt {1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}-\frac {3674891 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 84, normalized size = 0.47 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (82697568+487066088 x+1076423732 x^2+1058136330 x^3+390269835 x^4\right )}{(2+3 x)^5}-55123365 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{658560} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs.
\(2(141)=282\).
time = 0.11, size = 298, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (390269835 x^{4}+1058136330 x^{3}+1076423732 x^{2}+487066088 x +82697568\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{94080 \left (2+3 x \right )^{5} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {3674891 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{87808 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(134\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (13394977695 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+44649925650 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+59533234200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+5463777690 x^{4} \sqrt {-10 x^{2}-x +3}+39688822800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+14813908620 x^{3} \sqrt {-10 x^{2}-x +3}+13229607600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +15069932248 x^{2} \sqrt {-10 x^{2}-x +3}+1763947680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6818925232 x \sqrt {-10 x^{2}-x +3}+1157765952 \sqrt {-10 x^{2}-x +3}\right )}{1317120 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(298\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 198, normalized size = 1.10 \begin {gather*} \frac {3674891}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {151855}{4704} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{15 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {73 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{40 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2573 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{336 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {91113 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3136 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1123727 \, \sqrt {-10 \, x^{2} - x + 3}}{18816 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 131, normalized size = 0.73 \begin {gather*} -\frac {55123365 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (390269835 \, x^{4} + 1058136330 \, x^{3} + 1076423732 \, x^{2} + 487066088 \, x + 82697568\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1317120 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 426 vs.
\(2 (141) = 282\).
time = 1.30, size = 426, normalized size = 2.37 \begin {gather*} \frac {3674891}{878080} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {14641 \, \sqrt {10} {\left (753 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 1524880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 503767040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 77139328000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {4628359680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {18513438720000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9408 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \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 (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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