Optimal. Leaf size=173 \[ -\frac {181304825 \sqrt {1-2 x}}{12096 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}+\frac {22957 \sqrt {1-2 x}}{288 (2+3 x)^2 \sqrt {3+5 x}}+\frac {3997345 \sqrt {1-2 x}}{4032 (2+3 x) \sqrt {3+5 x}}+\frac {46095555 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}} \]
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Rubi [A]
time = 0.04, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {100, 154, 156,
157, 12, 95, 210} \begin {gather*} \frac {46095555 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}}+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}+\frac {3997345 \sqrt {1-2 x}}{4032 (3 x+2) \sqrt {5 x+3}}+\frac {22957 \sqrt {1-2 x}}{288 (3 x+2)^2 \sqrt {5 x+3}}+\frac {2051 \sqrt {1-2 x}}{216 (3 x+2)^3 \sqrt {5 x+3}}-\frac {181304825 \sqrt {1-2 x}}{12096 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 154
Rule 156
Rule 157
Rule 210
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {1}{12} \int \frac {\left (\frac {425}{2}-194 x\right ) \sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}-\frac {1}{108} \int \frac {-\frac {81763}{4}+29601 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}+\frac {22957 \sqrt {1-2 x}}{288 (2+3 x)^2 \sqrt {3+5 x}}-\frac {\int \frac {-\frac {15125495}{8}+2410485 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx}{1512}\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}+\frac {22957 \sqrt {1-2 x}}{288 (2+3 x)^2 \sqrt {3+5 x}}+\frac {3997345 \sqrt {1-2 x}}{4032 (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {-\frac {1784763365}{16}+\frac {419721225 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{10584}\\ &=-\frac {181304825 \sqrt {1-2 x}}{12096 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}+\frac {22957 \sqrt {1-2 x}}{288 (2+3 x)^2 \sqrt {3+5 x}}+\frac {3997345 \sqrt {1-2 x}}{4032 (2+3 x) \sqrt {3+5 x}}+\frac {\int -\frac {95832658845}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{58212}\\ &=-\frac {181304825 \sqrt {1-2 x}}{12096 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}+\frac {22957 \sqrt {1-2 x}}{288 (2+3 x)^2 \sqrt {3+5 x}}+\frac {3997345 \sqrt {1-2 x}}{4032 (2+3 x) \sqrt {3+5 x}}-\frac {46095555}{896} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {181304825 \sqrt {1-2 x}}{12096 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}+\frac {22957 \sqrt {1-2 x}}{288 (2+3 x)^2 \sqrt {3+5 x}}+\frac {3997345 \sqrt {1-2 x}}{4032 (2+3 x) \sqrt {3+5 x}}-\frac {46095555}{448} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {181304825 \sqrt {1-2 x}}{12096 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}+\frac {22957 \sqrt {1-2 x}}{288 (2+3 x)^2 \sqrt {3+5 x}}+\frac {3997345 \sqrt {1-2 x}}{4032 (2+3 x) \sqrt {3+5 x}}+\frac {46095555 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 84, normalized size = 0.49 \begin {gather*} \frac {-\frac {7 \sqrt {1-2 x} \left (103735088+628209228 x+1426133132 x^2+1438446565 x^3+543914475 x^4\right )}{(2+3 x)^4 \sqrt {3+5 x}}+46095555 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136} \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.
\(2(134)=268\).
time = 0.12, size = 298, normalized size = 1.72
method | result | size |
default | \(-\frac {\left (18668699775 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+60984419265 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+79653119040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+7614802650 x^{4} \sqrt {-10 x^{2}-x +3}+51995786040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+20138251910 x^{3} \sqrt {-10 x^{2}-x +3}+16963164240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +19965863848 x^{2} \sqrt {-10 x^{2}-x +3}+2212586640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8794929192 x \sqrt {-10 x^{2}-x +3}+1452291232 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{6272 \left (2+3 x \right )^{4} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(298\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 296 vs.
\(2 (134) = 268\).
time = 0.51, size = 296, normalized size = 1.71 \begin {gather*} -\frac {46095555}{6272} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {181304825 \, x}{6048 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {189299515}{12096 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{108 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {13181}{648 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {466361}{2592 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {1301839}{576 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \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 = 0.51, size = 131, normalized size = 0.76 \begin {gather*} \frac {46095555 \, \sqrt {7} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\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 (543914475 \, x^{4} + 1438446565 \, x^{3} + 1426133132 \, x^{2} + 628209228 \, x + 103735088\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6272 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\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. 438 vs.
\(2 (134) = 268\).
time = 0.86, size = 438, normalized size = 2.53 \begin {gather*} -\frac {9219111}{12544} \, \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 {605}{2} \, \sqrt {10} {\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 )} - \frac {605 \, {\left (77025 \, \sqrt {10} {\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} + 51138136 \, \sqrt {10} {\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} + 12067876800 \, \sqrt {10} {\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} + 984130112000 \, \sqrt {10} {\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 )}\right )}}{224 \, {\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 )}^{4}} \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}}{{\left (3\,x+2\right )}^5\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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