Optimal. Leaf size=305 \[ \frac {4 \sqrt {x^4+3 x^2+4} x}{175 \left (x^2+2\right )}+\frac {22 \sqrt {x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )}+\frac {1}{75} \sqrt {x^4+3 x^2+4} x+\frac {13}{350} \sqrt {\frac {11}{35}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )+\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{175 \sqrt {x^4+3 x^2+4}}-\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{175 \sqrt {x^4+3 x^2+4}}+\frac {2431 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{36750 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
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Rubi [A] time = 0.53, antiderivative size = 372, normalized size of antiderivative = 1.22, number of steps used = 19, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1228, 1103, 1139, 1195, 1122, 1197, 1223, 1714, 1708, 1706, 1216} \[ \frac {4 \sqrt {x^4+3 x^2+4} x}{175 \left (x^2+2\right )}+\frac {22 \sqrt {x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )}+\frac {1}{75} \sqrt {x^4+3 x^2+4} x+\frac {13}{350} \sqrt {\frac {11}{35}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {x^4+3 x^2+4}}\right )+\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{175 \sqrt {x^4+3 x^2+4}}-\frac {4 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{175 \sqrt {x^4+3 x^2+4}}+\frac {187 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{13125 \sqrt {x^4+3 x^2+4}}+\frac {6919 \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac {9}{280};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{183750 \sqrt {2} \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1122
Rule 1139
Rule 1195
Rule 1197
Rule 1216
Rule 1223
Rule 1228
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 )^2} \, dx &=\int \left (\frac {152}{625 \sqrt {4+3 x^2+x^4}}+\frac {16 x^2}{125 \sqrt {4+3 x^2+x^4}}+\frac {x^4}{25 \sqrt {4+3 x^2+x^4}}+\frac {1936}{625 \left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}}+\frac {88}{625 \left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}}\right ) \, dx\\ &=\frac {1}{25} \int \frac {x^4}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {16}{125} \int \frac {x^2}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {88}{625} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx+\frac {152}{625} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {1936}{625} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {1}{75} x \sqrt {4+3 x^2+x^4}+\frac {22 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac {38 \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 )}{625 \sqrt {4+3 x^2+x^4}}-\frac {22 \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 {1}{75} \int \frac {4+6 x^2}{\sqrt {4+3 x^2+x^4}} \, dx-\frac {88 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{1875}+\frac {32}{125} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx-\frac {32}{125} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {176}{375} \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {1}{75} x \sqrt {4+3 x^2+x^4}+\frac {16 x \sqrt {4+3 x^2+x^4}}{125 \left (2+x^2\right )}+\frac {22 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac {2}{125} \sqrt {\frac {11}{35}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )-\frac {16 \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 {212 \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 {187 \sqrt {2} \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 )}{13125 \sqrt {4+3 x^2+x^4}}-\frac {22 \int \frac {410+425 x^2}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{21875}+\frac {44}{875} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx+\frac {4}{25} \int \frac {1-\frac {x^2}{2}}{\sqrt {4+3 x^2+x^4}} \, dx-\frac {16}{75} \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx\\ &=\frac {1}{75} x \sqrt {4+3 x^2+x^4}+\frac {4 x \sqrt {4+3 x^2+x^4}}{175 \left (2+x^2\right )}+\frac {22 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac {2}{125} \sqrt {\frac {11}{35}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )-\frac {4 \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 )}{175 \sqrt {4+3 x^2+x^4}}+\frac {112 \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 {187 \sqrt {2} \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 )}{13125 \sqrt {4+3 x^2+x^4}}-\frac {1936 \int \frac {1}{\sqrt {4+3 x^2+x^4}} \, dx}{13125}+\frac {1628 \int \frac {1+\frac {x^2}{2}}{\left (7+5 x^2\right ) \sqrt {4+3 x^2+x^4}} \, dx}{2625}\\ &=\frac {1}{75} x \sqrt {4+3 x^2+x^4}+\frac {4 x \sqrt {4+3 x^2+x^4}}{175 \left (2+x^2\right )}+\frac {22 x \sqrt {4+3 x^2+x^4}}{175 \left (7+5 x^2\right )}+\frac {13}{350} \sqrt {\frac {11}{35}} \tan ^{-1}\left (\frac {2 \sqrt {\frac {11}{35}} x}{\sqrt {4+3 x^2+x^4}}\right )-\frac {4 \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 )}{175 \sqrt {4+3 x^2+x^4}}+\frac {4 \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 )}{175 \sqrt {4+3 x^2+x^4}}+\frac {6919 \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 )}{183750 \sqrt {2} \sqrt {4+3 x^2+x^4}}+\frac {187 \sqrt {2} \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 )}{13125 \sqrt {4+3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.57, size = 309, normalized size = 1.01 \[ \frac {\frac {175 x \left (7 x^2+23\right ) \left (x^4+3 x^2+4\right )}{5 x^2+7}-i \sqrt {6+2 i \sqrt {7}} \sqrt {1-\frac {2 i x^2}{\sqrt {7}-3 i}} \sqrt {1+\frac {2 i x^2}{\sqrt {7}+3 i}} \left (7 \left (158+15 i \sqrt {7}\right ) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+105 \left (3-i \sqrt {7}\right ) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+429 \Pi \left (\frac {5}{14} \left (3+i \sqrt {7}\right );i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )\right )}{18375 \sqrt {x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{25 \, x^{4} + 70 \, x^{2} + 49}, x\right ) \]
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 \[ \int \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 425, normalized size = 1.39 \[ \frac {22 \sqrt {x^{4}+3 x^{2}+4}\, x}{175 \left (5 x^{2}+7\right )}+\frac {\sqrt {x^{4}+3 x^{2}+4}\, x}{75}+\frac {128 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticE \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{175 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}+\frac {232 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{375 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {128 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticF \left (\frac {\sqrt {-6+2 i \sqrt {7}}\, x}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{175 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (i \sqrt {7}+3\right )}+\frac {286 \sqrt {\frac {3 x^{2}}{8}-\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \sqrt {\frac {3 x^{2}}{8}+\frac {i \sqrt {7}\, x^{2}}{8}+1}\, \EllipticPi \left (\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, x , -\frac {5}{7 \left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right )}, \frac {\sqrt {-\frac {3}{8}-\frac {i \sqrt {7}}{8}}}{\sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}}\right )}{6125 \sqrt {-\frac {3}{8}+\frac {i \sqrt {7}}{8}}\, \sqrt {x^{4}+3 x^{2}+4}} \]
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 \[ \int \frac {{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (x^4+3\,x^2+4\right )}^{3/2}}{{\left (5\,x^2+7\right )}^2} \,d x \]
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 \[ \int \frac {\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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