Optimal. Leaf size=77 \[ \frac {b \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}-\frac {\sqrt {a x^2+b x^3+c x^4}}{a x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1927, 1904, 206} \[ \frac {b \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}-\frac {\sqrt {a x^2+b x^3+c x^4}}{a x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1904
Rule 1927
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a x^2+b x^3+c x^4}} \, dx &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{a x^2}-\frac {b \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2 a}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{a x^2}+\frac {b \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )}{a}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{a x^2}+\frac {b \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 1.16 \[ \frac {b x \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {a} (a+x (b+c x))}{2 a^{3/2} \sqrt {x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 194, normalized size = 2.52 \[ \left [\frac {\sqrt {a} b x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a^{2} x^{2}}, -\frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a^{2} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 88, normalized size = 1.14 \[ -\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-a b x \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+2 \sqrt {c \,x^{2}+b x +a}\, a^{\frac {3}{2}}\right )}{2 \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, a^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{3} + a x^{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,\sqrt {c\,x^4+b\,x^3+a\,x^2}} \,d x \]
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 \[ \int \frac {1}{x \sqrt {x^{2} \left (a + b x + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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