Optimal. Leaf size=114 \[ \frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3} \]
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Rubi [A] time = 0.15, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1920, 1951, 12, 1904, 206} \begin {gather*} \frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 1904
Rule 1920
Rule 1951
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}+\frac {1}{4} \int \frac {b+2 c x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}-\frac {\int \frac {b^2-4 a c}{2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{4 a}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 a}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}+\frac {\left (b^2-4 a c\right ) \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 )}{4 a}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 112, normalized size = 0.98 \begin {gather*} \frac {\sqrt {x^2 (a+x (b+c x))} \left (x^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}\right )}{8 a^{3/2} x^3 \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.53, size = 100, normalized size = 0.88 \begin {gather*} \frac {\left (4 a c-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {c} x^2-\sqrt {a x^2+b x^3+c x^4}}\right )}{4 a^{3/2}}+\frac {(-2 a-b x) \sqrt {a x^2+b x^3+c x^4}}{4 a x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 226, normalized size = 1.98 \begin {gather*} \left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {a} x^{3} \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}} {\left (a b x + 2 \, a^{2}\right )}}{16 \, a^{2} x^{3}}, -\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-a} x^{3} \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}} {\left (a b x + 2 \, a^{2}\right )}}{8 \, a^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 207, normalized size = 1.82 \begin {gather*} -\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (4 a^{\frac {3}{2}} c \,x^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-\sqrt {a}\, b^{2} x^{2} \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}\, b c \,x^{3}-4 \sqrt {c \,x^{2}+b x +a}\, a c \,x^{2}+2 \sqrt {c \,x^{2}+b x +a}\, b^{2} x^{2}-2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b x +4 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \right )}{8 \sqrt {c \,x^{2}+b x +a}\, a^{2} x^{3}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x^{4}}\,{d x} \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 {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x^4} \,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 {\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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