Optimal. Leaf size=89 \[ \frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 a^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1114, 740, 12, 724, 206} \[ \frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 740
Rule 1114
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {b^2}{2}+2 a c}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\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 )}{a}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 89, normalized size = 1.00 \[ \frac {-2 a c+b^2+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 389, normalized size = 4.37 \[ \left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, {\left (a b c x^{2} + a b^{2} - 2 \, a^{2} c\right )} \sqrt {c x^{4} + b x^{2} + a}}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}}, \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left (a b c x^{2} + a b^{2} - 2 \, a^{2} c\right )} \sqrt {c x^{4} + b x^{2} + a}}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{4} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 110, normalized size = 1.24 \[ \frac {\frac {a b c x^{2}}{a^{2} b^{2} - 4 \, a^{3} c} + \frac {a b^{2} - 2 \, a^{2} c}{a^{2} b^{2} - 4 \, a^{3} c}}{\sqrt {c x^{4} + b x^{2} + a}} + \frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 99, normalized size = 1.11 \[ -\frac {\left (2 c \,x^{2}+b \right ) b}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, a}-\frac {\ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}+\frac {1}{2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {1}{x\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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