Optimal. Leaf size=209 \[ -\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {47279 \sqrt {1-2 x} \sqrt {3+5 x}}{1270080 (2+3 x)^3}+\frac {1057139 \sqrt {1-2 x} \sqrt {3+5 x}}{7112448 (2+3 x)^2}+\frac {106751933 \sqrt {1-2 x} \sqrt {3+5 x}}{99574272 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}-\frac {15036307 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312 \sqrt {7}} \]
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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 {15036307 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1229312 \sqrt {7}}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{18 (3 x+2)^6}-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{1260 (3 x+2)^5}+\frac {106751933 \sqrt {1-2 x} \sqrt {5 x+3}}{99574272 (3 x+2)}+\frac {1057139 \sqrt {1-2 x} \sqrt {5 x+3}}{7112448 (3 x+2)^2}+\frac {47279 \sqrt {1-2 x} \sqrt {5 x+3}}{1270080 (3 x+2)^3}-\frac {6533 \sqrt {1-2 x} \sqrt {5 x+3}}{211680 (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 {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx\\ &=-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac {\int \frac {\left (-\frac {387}{4}-2595 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{1890}\\ &=-\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac {\int \frac {-\frac {822687}{8}-\frac {432615 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{158760}\\ &=-\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {47279 \sqrt {1-2 x} \sqrt {3+5 x}}{1270080 (2+3 x)^3}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac {\int \frac {\frac {10523625}{16}-\frac {4964295 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{3333960}\\ &=-\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {47279 \sqrt {1-2 x} \sqrt {3+5 x}}{1270080 (2+3 x)^3}+\frac {1057139 \sqrt {1-2 x} \sqrt {3+5 x}}{7112448 (2+3 x)^2}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac {\int \frac {\frac {2256323055}{32}-\frac {554997975 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{46675440}\\ &=-\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {47279 \sqrt {1-2 x} \sqrt {3+5 x}}{1270080 (2+3 x)^3}+\frac {1057139 \sqrt {1-2 x} \sqrt {3+5 x}}{7112448 (2+3 x)^2}+\frac {106751933 \sqrt {1-2 x} \sqrt {3+5 x}}{99574272 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac {\int \frac {127883791035}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{326728080}\\ &=-\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {47279 \sqrt {1-2 x} \sqrt {3+5 x}}{1270080 (2+3 x)^3}+\frac {1057139 \sqrt {1-2 x} \sqrt {3+5 x}}{7112448 (2+3 x)^2}+\frac {106751933 \sqrt {1-2 x} \sqrt {3+5 x}}{99574272 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac {15036307 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2458624}\\ &=-\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {47279 \sqrt {1-2 x} \sqrt {3+5 x}}{1270080 (2+3 x)^3}+\frac {1057139 \sqrt {1-2 x} \sqrt {3+5 x}}{7112448 (2+3 x)^2}+\frac {106751933 \sqrt {1-2 x} \sqrt {3+5 x}}{99574272 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}+\frac {15036307 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1229312}\\ &=-\frac {6533 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {47279 \sqrt {1-2 x} \sqrt {3+5 x}}{1270080 (2+3 x)^3}+\frac {1057139 \sqrt {1-2 x} \sqrt {3+5 x}}{7112448 (2+3 x)^2}+\frac {106751933 \sqrt {1-2 x} \sqrt {3+5 x}}{99574272 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1260 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{18 (2+3 x)^6}-\frac {15036307 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 89, normalized size = 0.43 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (665270208+4978384240 x+14818971424 x^2+21960917808 x^3+16234789140 x^4+4803836985 x^5\right )}{(2+3 x)^6}-225544605 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{129077760} \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 | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (4803836985 x^{5}+16234789140 x^{4}+21960917808 x^{3}+14818971424 x^{2}+4978384240 x +665270208\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{18439680 \left (2+3 x \right )^{6} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {15036307 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{17210368 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (164422017045 \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) \sqrt {7}\, x^{6}+657688068180 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+1096146780300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+67253717790 x^{5} \sqrt {-10 x^{2}-x +3}+974352693600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+227287047960 x^{4} \sqrt {-10 x^{2}-x +3}+487176346800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+307452849312 x^{3} \sqrt {-10 x^{2}-x +3}+129913692480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +207465599936 x^{2} \sqrt {-10 x^{2}-x +3}+14434854720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+69697379360 x \sqrt {-10 x^{2}-x +3}+9313782912 \sqrt {-10 x^{2}-x +3}\right )}{258155520 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 244, normalized size = 1.17 \begin {gather*} \frac {15036307}{17210368} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {621335}{921984} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{126 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} - \frac {169 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2940 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {547 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{23520 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {31055 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{197568 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {372801 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {4597879 \, \sqrt {-10 \, x^{2} - x + 3}}{3687936 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 146, normalized size = 0.70 \begin {gather*} -\frac {225544605 \, \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 (4803836985 \, x^{5} + 16234789140 \, x^{4} + 21960917808 \, x^{3} + 14818971424 \, x^{2} + 4978384240 \, x + 665270208\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{258155520 \, {\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 = 2.07, size = 484, normalized size = 2.32 \begin {gather*} \frac {15036307}{172103680} \, \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 (3081 \, {\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} + 4888520 \, {\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} + 3188465280 \, {\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} - 599903001600 \, {\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} - 103716175360000 \, {\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 {5302514380800000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {21210057523200000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1843968 \, {\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 {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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