Optimal. Leaf size=180 \[ -\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}-\frac {5591773 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}} \]
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Rubi [A]
time = 0.04, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {99, 156, 12, 95,
210} \begin {gather*} -\frac {5591773 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}}+\frac {1466281 \sqrt {1-2 x} \sqrt {5 x+3}}{131712 (3 x+2)}+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{9408 (3 x+2)^2}+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{1680 (3 x+2)^3}+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{840 (3 x+2)^4}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 99
Rule 156
Rule 210
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^6} \, dx &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {-\frac {1}{2}-10 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {1}{420} \int \frac {\frac {1341}{4}-555 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {\int \frac {\frac {265125}{8}-42315 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{8820}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {\int \frac {\frac {31687635}{16}-\frac {7362075 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{123480}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {\int \frac {1761408495}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{864360}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {5591773 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {5591773 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{840 (2+3 x)^4}+\frac {403 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {14023 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {1466281 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}-\frac {5591773 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 84, normalized size = 0.47 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (41933792+247045192 x+546004068 x^2+536695650 x^3+197947935 x^4\right )}{(2+3 x)^5}-27958865 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1536640} \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 = 0.12, size = 298, normalized size = 1.66
method | result | size |
risch | \(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (197947935 x^{4}+536695650 x^{3}+546004068 x^{2}+247045192 x +41933792\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{219520 \left (2+3 x \right )^{5} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {5591773 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{614656 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(134\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (6794004195 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+22646680650 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+30195574200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+2771271090 x^{4} \sqrt {-10 x^{2}-x +3}+20130382800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+7513739100 x^{3} \sqrt {-10 x^{2}-x +3}+6710127600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +7644056952 x^{2} \sqrt {-10 x^{2}-x +3}+894683680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3458632688 x \sqrt {-10 x^{2}-x +3}+587073088 \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(298\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 198, normalized size = 1.10 \begin {gather*} \frac {5591773}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {231065}{32928} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{280 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {1305 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{784 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {138639 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1709881 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 131, normalized size = 0.73 \begin {gather*} -\frac {27958865 \, \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 (197947935 \, x^{4} + 536695650 \, x^{3} + 546004068 \, x^{2} + 247045192 \, x + 41933792\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3073280 \, {\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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{6}}\, dx \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 = 0.82, size = 426, normalized size = 2.37 \begin {gather*} \frac {5591773}{6146560} \, \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 {121 \, \sqrt {10} {\left (46213 \, {\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} - 85961680 \, {\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} - 30665564160 \, {\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} - 4732042112000 \, {\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 {284050977280000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1136203909120000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{21952 \, {\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 [B]
time = 20.39, size = 1745, normalized size = 9.69 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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