Optimal. Leaf size=180 \[ \frac {1466281 \sqrt {1-2 x} \sqrt {5 x+3}}{131712 (3 x+2)}+\frac {14023 \sqrt {1-2 x} \sqrt {5 x+3}}{9408 (3 x+2)^2}+\frac {403 \sqrt {1-2 x} \sqrt {5 x+3}}{1680 (3 x+2)^3}+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{840 (3 x+2)^4}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}-\frac {5591773 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]
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Rubi [A] time = 0.07, 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 = {97, 151, 12, 93, 204} \begin {gather*} \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}-\frac {5591773 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 97
Rule 151
Rule 204
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 \operatorname {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 = 0.14, size = 135, normalized size = 0.75 \begin {gather*} \frac {1}{35} \left (\frac {5 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (4398843 x^2+6119462 x+2067760\right )}{(3 x+2)^3}-5591773 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{43904}+\frac {111 (1-2 x)^{3/2} (5 x+3)^{3/2}}{8 (3 x+2)^4}+\frac {3 (1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 138, normalized size = 0.77 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {231065 (1-2 x)^4}{(5 x+3)^4}-\frac {10745210 (1-2 x)^3}{(5 x+3)^3}-\frac {95829888 (1-2 x)^2}{(5 x+3)^2}-\frac {369690790 (1-2 x)}{5 x+3}-554787065\right )}{219520 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^5}-\frac {5591773 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.71, 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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giac [B] time = 2.77, 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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maple [B] time = 0.02, size = 298, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (6794004195 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+22646680650 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2771271090 \sqrt {-10 x^{2}-x +3}\, x^{4}+30195574200 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+7513739100 \sqrt {-10 x^{2}-x +3}\, x^{3}+20130382800 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+7644056952 \sqrt {-10 x^{2}-x +3}\, x^{2}+6710127600 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3458632688 \sqrt {-10 x^{2}-x +3}\, x +894683680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+587073088 \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, 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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mupad [B] time = 20.39, size = 1745, normalized size = 9.69
result too large to display
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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