Optimal. Leaf size=340 \[ -\frac {18159 \sqrt {x^4+3 x^2+4} x}{33392128 \left (x^2+2\right )}+\frac {51875 \sqrt {x^4+3 x^2+4} x}{33392128 \left (5 x^2+7\right )}+\frac {625 \sqrt {x^4+3 x^2+4} x}{54208 \left (5 x^2+7\right )^2}+\frac {\left (139 x^2+548\right ) x}{596288 \sqrt {x^4+3 x^2+4}}-\frac {529425 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{133568512}+\frac {843 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{379456 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {18159 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{16696064 \sqrt {2} \sqrt {x^4+3 x^2+4}}-\frac {3000075 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{934979584 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
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Rubi [A] time = 0.87, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1228, 1178, 1197, 1103, 1195, 1223, 1696, 1714, 1708, 1706, 1216} \[ -\frac {18159 \sqrt {x^4+3 x^2+4} x}{33392128 \left (x^2+2\right )}+\frac {51875 \sqrt {x^4+3 x^2+4} x}{33392128 \left (5 x^2+7\right )}+\frac {625 \sqrt {x^4+3 x^2+4} x}{54208 \left (5 x^2+7\right )^2}+\frac {\left (139 x^2+548\right ) x}{596288 \sqrt {x^4+3 x^2+4}}-\frac {529425 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{133568512}+\frac {843 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{379456 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {18159 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{16696064 \sqrt {2} \sqrt {x^4+3 x^2+4}}-\frac {3000075 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{934979584 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1178
Rule 1195
Rule 1197
Rule 1216
Rule 1223
Rule 1228
Rule 1696
Rule 1706
Rule 1708
Rule 1714
Rubi steps
\begin {align*} \int \frac {1}{\left (7+5 x^2\right )^3 \left (4+3 x^2+x^4\right )^{3/2}} \, dx &=\int \left (\frac {388+215 x^2}{85184 \left (4+3 x^2+x^4\right )^{3/2}}+\frac {25}{44 \left (7+5 x^2\right )^3 \sqrt {4+3 x^2+x^4}}-\frac {25}{1936 \left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}}-\frac {1075}{85184 \left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}}\right ) \, dx\\ &=\frac {\int \frac {388+215 x^2}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx}{85184}-\frac {1075 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{85184}-\frac {25 \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}} \, dx}{1936}+\frac {25}{44} \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {x \left (548+139 x^2\right )}{596288 \sqrt {4+3 x^2+x^4}}+\frac {625 x \sqrt {4+3 x^2+x^4}}{54208 \left (7+5 x^2\right )^2}-\frac {625 x \sqrt {4+3 x^2+x^4}}{1192576 \left (7+5 x^2\right )}+\frac {\int \frac {524-556 x^2}{\sqrt {4+3 x^2+x^4}} \, dx}{2385152}+\frac {25 \int \frac {12+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{1192576}-\frac {25 \int \frac {-76-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}} \, dx}{54208}+\frac {1075 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{255552}-\frac {5375 \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{127776}\\ &=\frac {x \left (548+139 x^2\right )}{596288 \sqrt {4+3 x^2+x^4}}+\frac {625 x \sqrt {4+3 x^2+x^4}}{54208 \left (7+5 x^2\right )^2}+\frac {51875 x \sqrt {4+3 x^2+x^4}}{33392128 \left (7+5 x^2\right )}-\frac {1075 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{340736}+\frac {1075 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{511104 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {18275 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{7155456 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {25 \int \frac {-4412-4690 x^2-2775 x^4}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{33392128}+\frac {5 \int \frac {410+425 x^2}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{1192576}-\frac {125 \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx}{596288}-\frac {21 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{85184}+\frac {139 \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx}{298144}\\ &=\frac {x \left (548+139 x^2\right )}{596288 \sqrt {4+3 x^2+x^4}}-\frac {153 x \sqrt {4+3 x^2+x^4}}{1192576 \left (2+x^2\right )}+\frac {625 x \sqrt {4+3 x^2+x^4}}{54208 \left (7+5 x^2\right )^2}+\frac {51875 x \sqrt {4+3 x^2+x^4}}{33392128 \left (7+5 x^2\right )}-\frac {1075 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{340736}+\frac {153 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{596288 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {23 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{11616 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {18275 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{7155456 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {5 \int \frac {-60910-31775 x^2}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{33392128}+\frac {25 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{40656}+\frac {13875 \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx}{16696064}-\frac {4625 \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{1788864}\\ &=\frac {x \left (548+139 x^2\right )}{596288 \sqrt {4+3 x^2+x^4}}-\frac {18159 x \sqrt {4+3 x^2+x^4}}{33392128 \left (2+x^2\right )}+\frac {625 x \sqrt {4+3 x^2+x^4}}{54208 \left (7+5 x^2\right )^2}+\frac {51875 x \sqrt {4+3 x^2+x^4}}{33392128 \left (7+5 x^2\right )}-\frac {15975 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{4770304}+\frac {18159 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{16696064 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {31 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{13552 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {90525 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{33392128 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {25 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{189728}-\frac {136875 \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{16696064}\\ &=\frac {x \left (548+139 x^2\right )}{596288 \sqrt {4+3 x^2+x^4}}-\frac {18159 x \sqrt {4+3 x^2+x^4}}{33392128 \left (2+x^2\right )}+\frac {625 x \sqrt {4+3 x^2+x^4}}{54208 \left (7+5 x^2\right )^2}+\frac {51875 x \sqrt {4+3 x^2+x^4}}{33392128 \left (7+5 x^2\right )}-\frac {529425 \sqrt {\frac {5}{77}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{133568512}+\frac {18159 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{16696064 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {843 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{379456 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {3000075 \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{934979584 \sqrt {2} \sqrt {4+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.75, size = 320, normalized size = 0.94 \[ \frac {28 x \left (453975 x^6+2838330 x^4+5811451 x^2+4496212\right )+3 i \sqrt {6+2 i \sqrt {7}} \sqrt {1-\frac {2 i x^2}{\sqrt {7}-3 i}} \sqrt {1+\frac {2 i x^2}{\sqrt {7}+3 i}} \left (5 x^2+7\right )^2 \left (7 i \left (6053 \sqrt {7}+23633 i\right ) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+42371 \left (3-i \sqrt {7}\right ) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+352950 \Pi \left (\frac {5}{14} \left (3+i \sqrt {7}\right );i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )\right )}{934979584 \left (5 x^2+7\right )^2 \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 3 \, x^{2} + 4}}{125 \, x^{14} + 1275 \, x^{12} + 6010 \, x^{10} + 16678 \, x^{8} + 29153 \, x^{6} + 31871 \, x^{4} + 19992 \, x^{2} + 5488}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 457, normalized size = 1.34 \[ \frac {625 \sqrt {x^{4}+3 x^{2}+4}\, x}{54208 \left (5 x^{2}+7\right )^{2}}+\frac {51875 \sqrt {x^{4}+3 x^{2}+4}\, x}{33392128 \left (5 x^{2}+7\right )}-\frac {18159 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{1043504 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}+\frac {1173 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{1192576 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}+\frac {18159 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{1043504 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}-\frac {529425 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticPi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{233744896 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {2 \left (-\frac {139}{1192576} x^{3}-\frac {137}{298144} x \right )}{\sqrt {x^{4}+3 x^{2}+4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\left (5\,x^2+7\right )}^3\,{\left (x^4+3\,x^2+4\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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