Optimal. Leaf size=173 \[ -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {c} x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.12, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1920, 1933, 843, 621, 206, 724} \[ -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {c} x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 843
Rule 1920
Rule 1933
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}+\frac {1}{2} \int \frac {b+2 c x}{\sqrt {a x^2+b x^3+c x^4}} \, dx\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}+\frac {\left (x \sqrt {a+b x+c x^2}\right ) \int \frac {b+2 c x}{x \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}+\frac {\left (b x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (c x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{\sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {\left (b x \sqrt {a+b x+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}}+\frac {\left (2 c 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 )}{\sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {c} 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 )}{\sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 131, normalized size = 0.76 \[ -\frac {\sqrt {a+x (b+c x)} \left (b x \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {a} \left (\sqrt {a+x (b+c x)}-\sqrt {c} x \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )}{2 \sqrt {a} \sqrt {x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 653, normalized size = 3.77 \[ \left [\frac {2 \, a \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + \sqrt {a} b x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a x^{2}}, -\frac {4 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - \sqrt {a} b x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + a \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 173, normalized size = 1.00 \[ -\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (-2 a \,c^{2} x \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+\sqrt {a}\, b \,c^{\frac {3}{2}} x \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-2 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {5}{2}} x^{2}-2 \sqrt {c \,x^{2}+b x +a}\, b \,c^{\frac {3}{2}} x +2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {3}{2}}\right )}{2 \sqrt {c \,x^{2}+b x +a}\, a \,c^{\frac {3}{2}} x^{2}} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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