Integrand size = 35, antiderivative size = 263 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^4} \, dx=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}-\frac {16830401 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{77322924900 \sqrt {5-2 x}}+\frac {24957247 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{4956597750 \sqrt {66} \sqrt {-5+2 x}}+\frac {15664616449 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{15980071146000 \sqrt {11} \sqrt {-5+2 x}} \]
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Time = 0.27 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {166, 1618, 1621, 174, 552, 551, 164, 115, 114, 122, 120} \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^4} \, dx=\frac {24957247 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{4956597750 \sqrt {66} \sqrt {2 x-5}}-\frac {16830401 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{77322924900 \sqrt {5-2 x}}+\frac {15664616449 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{15980071146000 \sqrt {11} \sqrt {2 x-5}}+\frac {16830401 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{30929169960 (5 x+7)}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1668420 (5 x+7)^2}-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{15 (5 x+7)^3} \]
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Rule 114
Rule 115
Rule 120
Rule 122
Rule 164
Rule 166
Rule 174
Rule 551
Rule 552
Rule 1618
Rule 1621
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {1}{30} \int \frac {-21+140 x-72 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^3} \, dx \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {\int \frac {-401471+855020 x-214872 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx}{3336840} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}+\frac {\int \frac {-2850617379+1003030560 x+1211788872 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{185575019760} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}+\frac {\int \frac {-\frac {3467369304}{25}+\frac {1211788872 x}{5}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{185575019760}-\frac {15664616449 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{1546458498000} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}+\frac {16830401 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{25774308300}+\frac {24957247 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{9913195500}+\frac {15664616449 \text {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {-\frac {11}{3}-\frac {2 x^2}{3}}} \, dx,x,\sqrt {2-3 x}\right )}{773229249000} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}+\frac {\left (24957247 \sqrt {5-2 x}\right ) \int \frac {1}{\sqrt {2-3 x} \sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{4956597750 \sqrt {22} \sqrt {-5+2 x}}+\frac {\left (15664616449 \sqrt {5-2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {1+\frac {2 x^2}{11}}} \, dx,x,\sqrt {2-3 x}\right )}{257743083000 \sqrt {33} \sqrt {-5+2 x}}+\frac {\left (16830401 \sqrt {-5+2 x}\right ) \int \frac {\sqrt {\frac {15}{11}-\frac {6 x}{11}}}{\sqrt {2-3 x} \sqrt {\frac {3}{11}+\frac {12 x}{11}}} \, dx}{25774308300 \sqrt {5-2 x}} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{15 (7+5 x)^3}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1668420 (7+5 x)^2}+\frac {16830401 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{30929169960 (7+5 x)}-\frac {16830401 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{77322924900 \sqrt {5-2 x}}+\frac {24957247 \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{4956597750 \sqrt {66} \sqrt {-5+2 x}}+\frac {15664616449 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{15980071146000 \sqrt {11} \sqrt {-5+2 x}} \\ \end{align*}
Time = 5.90 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^4} \, dx=\frac {\sqrt {-5+2 x} \left (\frac {17050 \sqrt {2-3 x} \sqrt {1+4 x} \left (-75460017+2007981640 x+420760025 x^2\right )}{(7+5 x)^3}+\frac {\sqrt {11} \left (-114783334820 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )+120693246492 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )-46993849347 \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{\sqrt {5-2 x}}\right )}{527342347818000} \]
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Time = 1.65 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.14
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {\sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{15 \left (7+5 x \right )^{3}}+\frac {8953 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{1668420 \left (7+5 x \right )^{2}}+\frac {16830401 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{30929169960 \left (7+5 x \right )}-\frac {48157907 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{7796728260750 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {16830401 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{1559345652150 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {15664616449 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{1075948499983500 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(299\) |
risch | \(-\frac {\left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (420760025 x^{2}+2007981640 x -75460017\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{30929169960 \left (7+5 x \right )^{3} \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {48157907 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{23390184782250 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {16830401 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{4678036956450 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {15664616449 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{5800765825998000 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\) | \(316\) |
default | \(-\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (274048323500 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{3}-2661307158125 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{3}-3916154112250 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x^{3}+1151002958700 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-11177490064125 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-16447847271450 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x^{2}+1611404142180 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -15648486089775 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -23026986180030 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x +751988599684 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-7302626841895 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-10745926884014 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )-63871371795000 x^{5}-118520111883250 x^{4}+844601251369975 x^{3}-326733155441000 x^{2}-116981861971975 x +4772846075250\right )}{195626999997000 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (7+5 x \right )^{3}}\) | \(566\) |
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^4} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{4}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^4} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1}}{\left (5 x + 7\right )^{4}}\, dx \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^4} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{4}} \,d x } \]
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\[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^4} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^4} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}}{{\left (5\,x+7\right )}^4} \,d x \]
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