Optimal. Leaf size=139 \[ \frac {3 b \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{4 a^{5/2}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
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Rubi [A] time = 0.13, antiderivative size = 139, 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, 806, 724, 206} \[ -\frac {\left (3 b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac {3 b \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{4 a^{5/2}}+\frac {-2 a c+b^2+b c x^2}{a x^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 740
Rule 806
Rule 1114
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \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 ) x^2 \sqrt {a+b x^2+c x^4}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (-3 b^2+8 a c\right )-b c x}{x^2 \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 ) x^2 \sqrt {a+b x^2+c x^4}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^2 \sqrt {a+b x^2+c x^4}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac {(3 b) \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 )}{2 a^2}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^2 \sqrt {a+b x^2+c x^4}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^2}+\frac {3 b \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{4 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 137, normalized size = 0.99 \[ \frac {\frac {2 \sqrt {a} \left (-4 a^2 c+a \left (b^2-10 b c x^2-8 c^2 x^4\right )+3 b^2 x^2 \left (b+c x^2\right )\right )}{x^2 \sqrt {a+b x^2+c x^4}}-3 b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{4 a^{5/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 485, normalized size = 3.49 \[ \left [\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{4} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left ({\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + {\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{8 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{6} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{4} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2}\right )}}, -\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{4} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + {\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{4 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{6} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{4} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 200, normalized size = 1.44 \[ -\frac {\frac {{\left (a^{2} b^{2} c - 2 \, a^{3} c^{2}\right )} x^{2}}{a^{4} b^{2} - 4 \, a^{5} c} + \frac {a^{2} b^{3} - 3 \, a^{3} b c}{a^{4} b^{2} - 4 \, a^{5} c}}{\sqrt {c x^{4} + b x^{2} + a}} - \frac {3 \, b \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{2}} + \frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} b + 2 \, a \sqrt {c}}{2 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 195, normalized size = 1.40 \[ \frac {3 b^{2} c \,x^{2}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}}+\frac {3 b^{3}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}}-\frac {2 \left (2 c \,x^{2}+b \right ) c}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, a}+\frac {3 b \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{4 a^{\frac {5}{2}}}-\frac {3 b}{4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}}-\frac {1}{2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,x^{2}} \]
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 \[ \int \frac {1}{x^3\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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