Optimal. Leaf size=440 \[ \frac {9 \sqrt {x^4+3 x^2+4} x}{1960 \left (x^2+2\right )}+\frac {167 \sqrt {x^4+3 x^2+4} x}{9800 \left (5 x^2+7\right )}+\frac {11 \sqrt {x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )^2}+\frac {1347 \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{7840 \sqrt {385}}-\frac {22 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{13125 \sqrt {x^4+3 x^2+4}}-\frac {817 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{91875 \sqrt {2} \sqrt {x^4+3 x^2+4}}-\frac {6 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{875 \sqrt {x^4+3 x^2+4}}+\frac {111 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{24500 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {7633 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{274400 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
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Rubi [A] time = 0.80, antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1228, 1103, 1139, 1195, 1223, 1696, 1714, 1708, 1706, 1216} \[ \frac {9 \sqrt {x^4+3 x^2+4} x}{1960 \left (x^2+2\right )}+\frac {167 \sqrt {x^4+3 x^2+4} x}{9800 \left (5 x^2+7\right )}+\frac {11 \sqrt {x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )^2}+\frac {1347 \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )}{7840 \sqrt {385}}-\frac {22 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{13125 \sqrt {x^4+3 x^2+4}}-\frac {817 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{91875 \sqrt {2} \sqrt {x^4+3 x^2+4}}-\frac {6 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{875 \sqrt {x^4+3 x^2+4}}+\frac {111 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{24500 \sqrt {2} \sqrt {x^4+3 x^2+4}}+\frac {7633 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{274400 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1139
Rule 1195
Rule 1216
Rule 1223
Rule 1228
Rule 1696
Rule 1706
Rule 1708
Rule 1714
Rubi steps
\begin {align*} \int \frac {\left (4+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx &=\int \left (\frac {9}{625 \sqrt {4+3 x^2+x^4}}+\frac {x^2}{125 \sqrt {4+3 x^2+x^4}}+\frac {1936}{625 \left (7+5 x^2\right )^3 \sqrt {4+3 x^2+x^4}}+\frac {88}{625 \left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}}+\frac {89}{625 \left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}}\right ) \, dx\\ &=\frac {1}{125} \int \frac {x^2}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {9}{625} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {88}{625} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}} \, dx+\frac {89}{625} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx+\frac {1936}{625} \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {11 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )^2}+\frac {x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac {9 \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 )}{1250 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {\int \frac {12+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{4375}-\frac {11 \int \frac {-76-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}} \, dx}{4375}+\frac {2}{125} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx-\frac {2}{125} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx-\frac {89 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{1875}+\frac {178}{375} \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {x \sqrt {4+3 x^2+x^4}}{125 \left (2+x^2\right )}+\frac {11 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )^2}+\frac {167 x \sqrt {4+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}+\frac {89 \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{500 \sqrt {385}}-\frac {\sqrt {2} \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 )}{125 \sqrt {4+3 x^2+x^4}}-\frac {8 \sqrt {2} \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 )}{1875 \sqrt {4+3 x^2+x^4}}+\frac {1513 \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 )}{52500 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {\int \frac {-4412-4690 x^2-2775 x^4}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{245000}-\frac {\int \frac {410+425 x^2}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{21875}+\frac {2}{875} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {6 x \sqrt {4+3 x^2+x^4}}{875 \left (2+x^2\right )}+\frac {11 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )^2}+\frac {167 x \sqrt {4+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}+\frac {89 \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{500 \sqrt {385}}-\frac {6 \sqrt {2} \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 )}{875 \sqrt {4+3 x^2+x^4}}-\frac {8 \sqrt {2} \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 )}{1875 \sqrt {4+3 x^2+x^4}}+\frac {1513 \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 )}{52500 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {\int \frac {-60910-31775 x^2}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{1225000}+\frac {111 \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx}{24500}-\frac {88 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{13125}+\frac {74 \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{2625}\\ &=\frac {9 x \sqrt {4+3 x^2+x^4}}{1960 \left (2+x^2\right )}+\frac {11 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )^2}+\frac {167 x \sqrt {4+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}+\frac {3}{175} \sqrt {\frac {11}{35}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )+\frac {111 \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 )}{24500 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {6 \sqrt {2} \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 )}{875 \sqrt {4+3 x^2+x^4}}-\frac {26 \sqrt {2} \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 )}{4375 \sqrt {4+3 x^2+x^4}}+\frac {187 \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 )}{6125 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {22 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{30625}-\frac {219 \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{4900}\\ &=\frac {9 x \sqrt {4+3 x^2+x^4}}{1960 \left (2+x^2\right )}+\frac {11 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )^2}+\frac {167 x \sqrt {4+3 x^2+x^4}}{9800 \left (7+5 x^2\right )}+\frac {3}{175} \sqrt {\frac {11}{35}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )-\frac {657 \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )}{39200 \sqrt {385}}+\frac {111 \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 )}{24500 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {6 \sqrt {2} \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 )}{875 \sqrt {4+3 x^2+x^4}}-\frac {11 \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 )}{30625 \sqrt {2} \sqrt {4+3 x^2+x^4}}-\frac {26 \sqrt {2} \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 )}{4375 \sqrt {4+3 x^2+x^4}}+\frac {7633 \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 )}{274400 \sqrt {2} \sqrt {4+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.69, size = 309, normalized size = 0.70 \[ \frac {\frac {140 x \left (167 x^2+357\right ) \left (x^4+3 x^2+4\right )}{\left (5 x^2+7\right )^2}-i \sqrt {6+2 i \sqrt {7}} \sqrt {1-\frac {2 i x^2}{\sqrt {7}-3 i}} \sqrt {1+\frac {2 i x^2}{\sqrt {7}+3 i}} \left (7 \left (103+45 i \sqrt {7}\right ) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+315 \left (3-i \sqrt {7}\right ) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+2694 \Pi \left (\frac {5}{14} \left (3+i \sqrt {7}\right );i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )\right )}{274400 \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}, x\right ) \]
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 \[ \int \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 434, normalized size = 0.99 \[ \frac {11 \sqrt {x^{4}+3 x^{2}+4}\, x}{175 \left (5 x^{2}+7\right )^{2}}+\frac {167 \sqrt {x^{4}+3 x^{2}+4}\, x}{9800 \left (5 x^{2}+7\right )}+\frac {36 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{245 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}+\frac {17 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{350 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {36 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{245 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}+\frac {1347 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticPi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{68600 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}} \]
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 \[ \int \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (x^4+3\,x^2+4\right )}^{3/2}}{{\left (5\,x^2+7\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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