Optimal. Leaf size=102 \[ \frac {563 \sqrt {-x^4+x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac {17 \sqrt {-x^4+x^2+2} x}{350 \left (5 x^2+7\right )^2}-\frac {1251 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{24500}+\frac {191 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{9800}+\frac {9879 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{343000} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.50, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {1228, 1095, 419, 1132, 493, 424, 1223, 1696, 1716, 1180, 524, 1212, 537} \[ \frac {563 \sqrt {-x^4+x^2+2} x}{9800 \left (5 x^2+7\right )}-\frac {17 \sqrt {-x^4+x^2+2} x}{350 \left (5 x^2+7\right )^2}-\frac {1251 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{24500}+\frac {191 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{9800}+\frac {9879 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{343000} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 419
Rule 424
Rule 493
Rule 524
Rule 537
Rule 1095
Rule 1132
Rule 1180
Rule 1212
Rule 1223
Rule 1228
Rule 1696
Rule 1716
Rubi steps
\begin {align*} \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^3} \, dx &=\int \left (-\frac {31}{625 \sqrt {2+x^2-x^4}}+\frac {x^2}{125 \sqrt {2+x^2-x^4}}+\frac {1156}{625 \left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}}-\frac {1292}{625 \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}}+\frac {429}{625 \left (7+5 x^2\right ) \sqrt {2+x^2-x^4}}\right ) \, dx\\ &=\frac {1}{125} \int \frac {x^2}{\sqrt {2+x^2-x^4}} \, dx-\frac {31}{625} \int \frac {1}{\sqrt {2+x^2-x^4}} \, dx+\frac {429}{625} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx+\frac {1156}{625} \int \frac {1}{\left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4}} \, dx-\frac {1292}{625} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx\\ &=-\frac {17 x \sqrt {2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac {19 x \sqrt {2+x^2-x^4}}{175 \left (7+5 x^2\right )}+\frac {17 \int \frac {186-190 x^2+25 x^4}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx}{8750}-\frac {19 \int \frac {118-70 x^2-25 x^4}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx}{4375}+\frac {2}{125} \int \frac {x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx-\frac {62}{625} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+\frac {858}{625} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2} \left (7+5 x^2\right )} \, dx\\ &=-\frac {17 x \sqrt {2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac {563 x \sqrt {2+x^2-x^4}}{9800 \left (7+5 x^2\right )}-\frac {31}{625} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {429 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{4375}+\frac {\int \frac {37698-32690 x^2-12525 x^4}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx}{245000}+\frac {19 \int \frac {175+125 x^2}{\sqrt {2+x^2-x^4}} \, dx}{109375}+\frac {1}{125} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx-\frac {2}{125} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx-\frac {3173 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx}{4375}\\ &=-\frac {17 x \sqrt {2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac {563 x \sqrt {2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac {1}{125} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {36}{625} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {429 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{4375}-\frac {\int \frac {75775+62625 x^2}{\sqrt {2+x^2-x^4}} \, dx}{6125000}+\frac {38 \int \frac {175+125 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{109375}+\frac {11783 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx}{49000}-\frac {6346 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2} \left (7+5 x^2\right )} \, dx}{4375}\\ &=-\frac {17 x \sqrt {2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac {563 x \sqrt {2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac {1}{125} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {36}{625} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {34 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{6125}-\frac {\int \frac {75775+62625 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{3062500}+\frac {76 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{4375}+\frac {19}{875} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx+\frac {11783 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2} \left (7+5 x^2\right )} \, dx}{24500}\\ &=-\frac {17 x \sqrt {2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac {563 x \sqrt {2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac {26}{875} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {214 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{4375}+\frac {9879 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{343000}-\frac {263 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{61250}-\frac {501 \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx}{49000}\\ &=-\frac {17 x \sqrt {2+x^2-x^4}}{350 \left (7+5 x^2\right )^2}+\frac {563 x \sqrt {2+x^2-x^4}}{9800 \left (7+5 x^2\right )}+\frac {191 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{9800}-\frac {1251 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{24500}+\frac {9879 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{343000}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.41, size = 244, normalized size = 2.39 \[ \frac {-197050 x^7-45500 x^5+636650 x^3-2541 i \sqrt {2} \left (5 x^2+7\right )^2 \sqrt {-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+13370 i \sqrt {2} \left (5 x^2+7\right )^2 \sqrt {-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )-246975 i \sqrt {2} \sqrt {-x^4+x^2+2} x^4 \Pi \left (\frac {5}{7};i \sinh ^{-1}(x)|-\frac {1}{2}\right )-691530 i \sqrt {2} \sqrt {-x^4+x^2+2} x^2 \Pi \left (\frac {5}{7};i \sinh ^{-1}(x)|-\frac {1}{2}\right )-484071 i \sqrt {2} \sqrt {-x^4+x^2+2} \Pi \left (\frac {5}{7};i \sinh ^{-1}(x)|-\frac {1}{2}\right )+485100 x}{686000 \left (5 x^2+7\right )^2 \sqrt {-x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 189, normalized size = 1.85 \[ -\frac {17 \sqrt {-x^{4}+x^{2}+2}\, x}{350 \left (5 x^{2}+7\right )^{2}}+\frac {563 \sqrt {-x^{4}+x^{2}+2}\, x}{9800 \left (5 x^{2}+7\right )}+\frac {191 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{19600 \sqrt {-x^{4}+x^{2}+2}}-\frac {1251 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{49000 \sqrt {-x^{4}+x^{2}+2}}+\frac {9879 \sqrt {2}\, \sqrt {-\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {\sqrt {2}\, x}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{343000 \sqrt {-x^{4}+x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (-x^4+x^2+2\right )}^{3/2}}{{\left (5\,x^2+7\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________