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 = 0.0658313, antiderivative size = 180, normalized size of antiderivative = 1., 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} \[ \frac{1466281 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{14023 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{840 (3 x+2)^4}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{5591773 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 151
Rule 12
Rule 93
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*}
Mathematica [A] time = 0.134089, size = 135, normalized size = 0.75 \[ \frac{1}{35} \left (\frac{5 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (4398843 x^2+6119462 x+2067760\right )}{(3 x+2)^3}-5591773 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{43904}+\frac{111 (1-2 x)^{3/2} (5 x+3)^{3/2}}{8 (3 x+2)^4}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^5}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{3073280\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6794004195\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+22646680650\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+30195574200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+2771271090\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+20130382800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+7513739100\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+6710127600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+7644056952\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+894683680\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3458632688\,x\sqrt{-10\,{x}^{2}-x+3}+587073088\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.17888, size = 267, normalized size = 1.48 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8903, size = 439, normalized size = 2.44 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{1 - 2 x} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 4.41085, size = 582, normalized size = 3.23 \begin{align*} \frac{121}{6146560} \, \sqrt{5}{\left (46213 \, \sqrt{70} \sqrt{2}{\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{280 \, \sqrt{2}{\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 )}}{{\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}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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