Optimal. Leaf size=143 \[ \frac {x \left (3 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c} \]
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Rubi [A] time = 0.17, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1928, 1949, 12, 1914, 621, 206} \begin {gather*} \frac {x \left (3 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 621
Rule 1914
Rule 1928
Rule 1949
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx &=\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\int \frac {x \left (a+\frac {3 b x}{2}\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2 c}\\ &=\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\int \frac {\left (3 b^2-4 a c\right ) x}{4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{2 c^2}\\ &=\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (3 b^2-4 a c\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 c^2}\\ &=\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (\left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^2 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (\left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{4 c^2 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 105, normalized size = 0.73 \begin {gather*} \frac {x \left (\left (3 b^2-4 a c\right ) \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} (2 c x-3 b) (a+x (b+c x))\right )}{8 c^{5/2} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 122, normalized size = 0.85 \begin {gather*} \frac {\log (x) \left (3 b^2-4 a c\right )}{8 c^{5/2}}+\frac {\left (4 a c-3 b^2\right ) \log \left (-2 c^{5/2} \sqrt {a x^2+b x^3+c x^4}+b c^2 x+2 c^3 x^2\right )}{8 c^{5/2}}+\frac {(2 c x-3 b) \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 226, normalized size = 1.58 \begin {gather*} \left [-\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x - 3 \, b c\right )}}{16 \, c^{3} x}, -\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x - 3 \, b c\right )}}{8 \, c^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\sqrt {c x^{4} + b x^{3} + a x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 144, normalized size = 1.01 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, \left (-4 a \,c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+3 b^{2} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+4 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {5}{2}} x -6 \sqrt {c \,x^{2}+b x +a}\, b \,c^{\frac {3}{2}}\right ) x}{8 \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, c^{\frac {7}{2}}} \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 {x^{3}}{\sqrt {c x^{4} + b x^{3} + a x^{2}}}\,{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 {x^3}{\sqrt {c\,x^4+b\,x^3+a\,x^2}} \,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 {x^{3}}{\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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