Optimal. Leaf size=209 \[ -\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}-\frac {391280725 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \]
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Rubi [A]
time = 0.05, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 154, 156,
12, 95, 210} \begin {gather*} -\frac {391280725 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}}-\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}+\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^5}+\frac {2770202075 \sqrt {5 x+3} \sqrt {1-2 x}}{14224896 (3 x+2)}+\frac {26486645 \sqrt {5 x+3} \sqrt {1-2 x}}{1016064 (3 x+2)^2}+\frac {151621 \sqrt {5 x+3} \sqrt {1-2 x}}{36288 (3 x+2)^3}+\frac {647 \sqrt {5 x+3} \sqrt {1-2 x}}{864 (3 x+2)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 99
Rule 154
Rule 156
Rule 210
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^7} \, dx &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}-\frac {1}{270} \int \frac {\sqrt {1-2 x} \left (-\frac {2235}{4}+375 x\right )}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {\int \frac {\frac {385905}{8}-\frac {139575 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{3240}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {\int \frac {\frac {71784825}{16}-\frac {11371575 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{68040}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {\int \frac {\frac {8553681375}{32}-\frac {1986498375 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{952560}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}+\frac {\int \frac {475406080875}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6667920}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}+\frac {391280725 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}+\frac {391280725 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616}\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{12 (2+3 x)^5}+\frac {647 \sqrt {1-2 x} \sqrt {3+5 x}}{864 (2+3 x)^4}+\frac {151621 \sqrt {1-2 x} \sqrt {3+5 x}}{36288 (2+3 x)^3}+\frac {26486645 \sqrt {1-2 x} \sqrt {3+5 x}}{1016064 (2+3 x)^2}+\frac {2770202075 \sqrt {1-2 x} \sqrt {3+5 x}}{14224896 (2+3 x)}-\frac {391280725 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 89, normalized size = 0.43 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (3522190656+26026519504 x+76960600672 x^2+113834022672 x^3+84218501340 x^4+24931818675 x^5\right )}{(2+3 x)^6}-1173842175 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3687936} \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. \(345\) vs.
\(2(164)=328\).
time = 0.14, size = 346, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (24931818675 x^{5}+84218501340 x^{4}+113834022672 x^{3}+76960600672 x^{2}+26026519504 x +3522190656\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{526848 \left (2+3 x \right )^{6} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {391280725 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (855730945575 \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) \sqrt {7}\, x^{6}+3422923782300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+5704872970500 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+349045461450 x^{5} \sqrt {-10 x^{2}-x +3}+5070998196000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+1179059018760 x^{4} \sqrt {-10 x^{2}-x +3}+2535499098000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1593676317408 x^{3} \sqrt {-10 x^{2}-x +3}+676133092800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1077448409408 x^{2} \sqrt {-10 x^{2}-x +3}+75125899200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+364371273056 x \sqrt {-10 x^{2}-x +3}+49310669184 \sqrt {-10 x^{2}-x +3}\right )}{7375872 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 244, normalized size = 1.17 \begin {gather*} \frac {391280725}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {16168625}{131712} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{18 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {19 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{12 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {4673 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{672 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {821945 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28224 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {9701175 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{87808 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {119647825 \, \sqrt {-10 \, x^{2} - x + 3}}{526848 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 146, normalized size = 0.70 \begin {gather*} -\frac {1173842175 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 (24931818675 \, x^{5} + 84218501340 \, x^{4} + 113834022672 \, x^{3} + 76960600672 \, x^{2} + 26026519504 \, x + 3522190656\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7375872 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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. 484 vs.
\(2 (164) = 328\).
time = 1.83, size = 484, normalized size = 2.32 \begin {gather*} \frac {78256145}{4917248} \, \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 {366025 \, \sqrt {10} {\left (3207 \, {\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 )}^{11} - 8960840 \, {\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} - 4031723136 \, {\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} - 929280844800 \, {\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} - 111701434880000 \, {\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 {5519365017600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {22077460070400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{263424 \, {\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 )}^{6}} \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 = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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