Optimal. Leaf size=479 \[ -\frac {\left (-\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\left (b^2-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3 \sqrt {e}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 e^{3/2}}-\frac {b x \sqrt {d+e x^2}}{2 c^2 e}+\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 c e^{5/2}}-\frac {3 d x \sqrt {d+e x^2}}{8 c e^2}+\frac {x^3 \sqrt {d+e x^2}}{4 c e} \]
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Rubi [A] time = 1.86, antiderivative size = 479, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1303, 217, 206, 321, 1692, 377, 205} \[ -\frac {\left (-\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\left (b^2-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3 \sqrt {e}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 e^{3/2}}-\frac {b x \sqrt {d+e x^2}}{2 c^2 e}+\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 c e^{5/2}}-\frac {3 d x \sqrt {d+e x^2}}{8 c e^2}+\frac {x^3 \sqrt {d+e x^2}}{4 c e} \]
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 217
Rule 321
Rule 377
Rule 1303
Rule 1692
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac {b^2-a c}{c^3 \sqrt {d+e x^2}}-\frac {b x^2}{c^2 \sqrt {d+e x^2}}+\frac {x^4}{c \sqrt {d+e x^2}}-\frac {a \left (b^2-a c\right )+b \left (b^2-2 a c\right ) x^2}{c^3 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=-\frac {\int \frac {a \left (b^2-a c\right )+b \left (b^2-2 a c\right ) x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{c^3}-\frac {b \int \frac {x^2}{\sqrt {d+e x^2}} \, dx}{c^2}+\frac {\int \frac {x^4}{\sqrt {d+e x^2}} \, dx}{c}+\frac {\left (b^2-a c\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{c^3}\\ &=-\frac {b x \sqrt {d+e x^2}}{2 c^2 e}+\frac {x^3 \sqrt {d+e x^2}}{4 c e}-\frac {\int \left (\frac {b \left (b^2-2 a c\right )+\frac {-b^4+4 a b^2 c-2 a^2 c^2}{\sqrt {b^2-4 a c}}}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}+\frac {b \left (b^2-2 a c\right )-\frac {-b^4+4 a b^2 c-2 a^2 c^2}{\sqrt {b^2-4 a c}}}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}\right ) \, dx}{c^3}+\frac {\left (b^2-a c\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c^3}+\frac {(b d) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{2 c^2 e}-\frac {(3 d) \int \frac {x^2}{\sqrt {d+e x^2}} \, dx}{4 c e}\\ &=-\frac {3 d x \sqrt {d+e x^2}}{8 c e^2}-\frac {b x \sqrt {d+e x^2}}{2 c^2 e}+\frac {x^3 \sqrt {d+e x^2}}{4 c e}+\frac {\left (b^2-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3 \sqrt {e}}-\frac {\left (b^3-2 a b c-\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{c^3}-\frac {\left (b^3-2 a b c+\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{c^3}+\frac {\left (3 d^2\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{8 c e^2}+\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{2 c^2 e}\\ &=-\frac {3 d x \sqrt {d+e x^2}}{8 c e^2}-\frac {b x \sqrt {d+e x^2}}{2 c^2 e}+\frac {x^3 \sqrt {d+e x^2}}{4 c e}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 e^{3/2}}+\frac {\left (b^2-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3 \sqrt {e}}-\frac {\left (b^3-2 a b c-\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c^3}-\frac {\left (b^3-2 a b c+\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c^3}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{8 c e^2}\\ &=-\frac {3 d x \sqrt {d+e x^2}}{8 c e^2}-\frac {b x \sqrt {d+e x^2}}{2 c^2 e}+\frac {x^3 \sqrt {d+e x^2}}{4 c e}-\frac {\left (b^3-2 a b c-\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b^3-2 a b c+\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 c e^{5/2}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 e^{3/2}}+\frac {\left (b^2-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3 \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 1.87, size = 461, normalized size = 0.96 \[ \frac {-\frac {8 \left (-\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac {x \sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}-\frac {8 \left (\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\right ) \tan ^{-1}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {8 \left (b^2-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}+\frac {4 b c d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}-\frac {4 b c x \sqrt {d+e x^2}}{e}+\frac {3 c^2 d \left (d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\sqrt {e} x \sqrt {d+e x^2}\right )}{e^{5/2}}+\frac {2 c^2 x^3 \sqrt {d+e x^2}}{e}}{8 c^3} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.03, size = 105, normalized size = 0.22 \[ \frac {1}{8} \, \sqrt {x^{2} e + d} {\left (\frac {2 \, x^{2} e^{\left (-1\right )}}{c} - \frac {{\left (3 \, c^{5} d e + 4 \, b c^{4} e^{2}\right )} e^{\left (-3\right )}}{c^{6}}\right )} x - \frac {{\left (3 \, c^{2} d^{2} + 4 \, b c d e + 8 \, b^{2} e^{2} - 8 \, a c e^{2}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2}\right )}{16 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 377, normalized size = 0.79 \[ \frac {\sqrt {e \,x^{2}+d}\, x^{3}}{4 c e}-\frac {a \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{c^{2} \sqrt {e}}+\frac {b^{2} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{c^{3} \sqrt {e}}+\frac {b d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 c^{2} e^{\frac {3}{2}}}+\frac {3 d^{2} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{8 c \,e^{\frac {5}{2}}}-\frac {\sqrt {e \,x^{2}+d}\, b x}{2 c^{2} e}-\frac {3 \sqrt {e \,x^{2}+d}\, d x}{8 c \,e^{2}}-\frac {\sqrt {e}\, \left (2 a b c \,d^{2}-b^{3} d^{2}+\left (2 a c -b^{2}\right ) \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} b +2 \left (2 a^{2} c e -2 a \,b^{2} e -2 a b c d +b^{3} d \right ) \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )+\left (-\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )^{2}\right )}{2 c^{3} \left (\RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{3} c +3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} b e -3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right )^{2} c d +8 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) a \,e^{2}-4 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) b d e +3 \RootOf \left (\textit {\_Z}^{4} c +c \,d^{4}+\left (4 b e -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 b \,d^{2} e -4 c \,d^{3}\right ) \textit {\_Z} \right ) c \,d^{2}+b \,d^{2} e -c \,d^{3}\right )} \]
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 \[ \int \frac {x^{8}}{{\left (c x^{4} + b x^{2} + a\right )} \sqrt {e x^{2} + d}}\,{d x} \]
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 \[ \int \frac {x^8}{\sqrt {e\,x^2+d}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]
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 \[ \int \frac {x^{8}}{\sqrt {d + e x^{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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