Optimal. Leaf size=154 \[ -\frac {b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac {2 a-b x^2}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac {\left (16 a c+15 b^2\right ) \sqrt {-a+b x^2+c x^4}}{48 a^3 x^2}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\sqrt {-a+b x^2+c x^4}}{6 a x^6} \]
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Rubi [A] time = 0.17, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1114, 744, 834, 806, 724, 204} \begin {gather*} \frac {\left (16 a c+15 b^2\right ) \sqrt {-a+b x^2+c x^4}}{48 a^3 x^2}-\frac {b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac {2 a-b x^2}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\sqrt {-a+b x^2+c x^4}}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 724
Rule 744
Rule 806
Rule 834
Rule 1114
Rubi steps
\begin {align*} \int \frac {1}{x^7 \sqrt {-a+b x^2+c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {-a+b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-a+b x^2+c x^4}}{6 a x^6}+\frac {\operatorname {Subst}\left (\int \frac {\frac {5 b}{2}+2 c x}{x^3 \sqrt {-a+b x+c x^2}} \, dx,x,x^2\right )}{6 a}\\ &=\frac {\sqrt {-a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} \left (15 b^2+16 a c\right )+\frac {5 b c x}{2}}{x^2 \sqrt {-a+b x+c x^2}} \, dx,x,x^2\right )}{12 a^2}\\ &=\frac {\sqrt {-a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\left (15 b^2+16 a c\right ) \sqrt {-a+b x^2+c x^4}}{48 a^3 x^2}+\frac {\left (b \left (5 b^2+12 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-a+b x+c x^2}} \, dx,x,x^2\right )}{32 a^3}\\ &=\frac {\sqrt {-a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\left (15 b^2+16 a c\right ) \sqrt {-a+b x^2+c x^4}}{48 a^3 x^2}-\frac {\left (b \left (5 b^2+12 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 a-x^2} \, dx,x,\frac {-2 a+b x^2}{\sqrt {-a+b x^2+c x^4}}\right )}{16 a^3}\\ &=\frac {\sqrt {-a+b x^2+c x^4}}{6 a x^6}+\frac {5 b \sqrt {-a+b x^2+c x^4}}{24 a^2 x^4}+\frac {\left (15 b^2+16 a c\right ) \sqrt {-a+b x^2+c x^4}}{48 a^3 x^2}-\frac {b \left (5 b^2+12 a c\right ) \tan ^{-1}\left (\frac {2 a-b x^2}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{32 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 116, normalized size = 0.75 \begin {gather*} \frac {b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac {b x^2-2 a}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac {\sqrt {-a+b x^2+c x^4} \left (8 a^2+2 a \left (5 b x^2+8 c x^4\right )+15 b^2 x^4\right )}{48 a^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 114, normalized size = 0.74 \begin {gather*} \frac {\left (-12 a b c-5 b^3\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {-a+b x^2+c x^4}}{\sqrt {a}}\right )}{16 a^{7/2}}+\frac {\sqrt {-a+b x^2+c x^4} \left (8 a^2+10 a b x^2+16 a c x^4+15 b^2 x^4\right )}{48 a^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 4.01, size = 272, normalized size = 1.77 \begin {gather*} \left [-\frac {3 \, {\left (5 \, b^{3} + 12 \, a b c\right )} \sqrt {-a} x^{6} \log \left (\frac {{\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} - a} {\left (b x^{2} - 2 \, a\right )} \sqrt {-a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left (10 \, a^{2} b x^{2} + {\left (15 \, a b^{2} + 16 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} - a}}{192 \, a^{4} x^{6}}, \frac {3 \, {\left (5 \, b^{3} + 12 \, a b c\right )} \sqrt {a} x^{6} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} - a} {\left (b x^{2} - 2 \, a\right )} \sqrt {a}}{2 \, {\left (a c x^{4} + a b x^{2} - a^{2}\right )}}\right ) + 2 \, {\left (10 \, a^{2} b x^{2} + {\left (15 \, a b^{2} + 16 \, a^{2} c\right )} x^{4} + 8 \, a^{3}\right )} \sqrt {c x^{4} + b x^{2} - a}}{96 \, a^{4} x^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 344, normalized size = 2.23 \begin {gather*} \frac {{\left (5 \, b^{3} + 12 \, a b c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}}{\sqrt {a}}\right )}{16 \, a^{\frac {7}{2}}} - \frac {15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{5} b^{3} + 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{5} a b c + 40 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{3} a b^{3} + 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{3} a^{2} b c - 96 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{2} a^{3} c^{\frac {3}{2}} + 33 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )} a^{2} b^{3} - 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )} a^{3} b c - 48 \, a^{3} b^{2} \sqrt {c} - 32 \, a^{4} c^{\frac {3}{2}}}{48 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{2} + a\right )}^{3} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 202, normalized size = 1.31 \begin {gather*} -\frac {3 b c \ln \left (\frac {b \,x^{2}-2 a +2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{8 \sqrt {-a}\, a^{2}}-\frac {5 b^{3} \ln \left (\frac {b \,x^{2}-2 a +2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{32 \sqrt {-a}\, a^{3}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}-a}\, c}{3 a^{2} x^{2}}+\frac {5 \sqrt {c \,x^{4}+b \,x^{2}-a}\, b^{2}}{16 a^{3} x^{2}}+\frac {5 \sqrt {c \,x^{4}+b \,x^{2}-a}\, b}{24 a^{2} x^{4}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}-a}}{6 a \,x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.26, size = 179, normalized size = 1.16 \begin {gather*} -\frac {5 \, b^{3} \arcsin \left (-\frac {b}{\sqrt {b^{2} + 4 \, a c}} + \frac {2 \, a}{\sqrt {b^{2} + 4 \, a c} x^{2}}\right )}{32 \, a^{\frac {7}{2}}} - \frac {3 \, b c \arcsin \left (-\frac {b}{\sqrt {b^{2} + 4 \, a c}} + \frac {2 \, a}{\sqrt {b^{2} + 4 \, a c} x^{2}}\right )}{8 \, a^{\frac {5}{2}}} + \frac {5 \, \sqrt {c x^{4} + b x^{2} - a} b^{2}}{16 \, a^{3} x^{2}} + \frac {\sqrt {c x^{4} + b x^{2} - a} c}{3 \, a^{2} x^{2}} + \frac {5 \, \sqrt {c x^{4} + b x^{2} - a} b}{24 \, a^{2} x^{4}} + \frac {\sqrt {c x^{4} + b x^{2} - a}}{6 \, a x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^7\,\sqrt {c\,x^4+b\,x^2-a}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{7} \sqrt {- a + b x^{2} + c x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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