Optimal. Leaf size=119 \[ -\frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{5/2}}+\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^2}-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 a x^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 119, 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 = {1929, 1951, 12, 1904, 206} \begin {gather*} -\frac {\left (3 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^{5/2}}+\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^2}-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 a x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 206
Rule 1904
Rule 1929
Rule 1951
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 a x^3}+\frac {\int \frac {-\frac {3 b}{2}-c x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{2 a}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 a x^3}+\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^2}-\frac {\int \frac {-\frac {3 b^2}{4}+a c}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2 a^2}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 a x^3}+\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^2}+\frac {\left (3 b^2-4 a c\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 a^2}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 a x^3}+\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^2}-\frac {\left (3 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^2}\\ &=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 a x^3}+\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^2}-\frac {\left (3 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^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 112, normalized size = 0.94 \begin {gather*} \frac {-\left (x^2 \left (3 b^2-4 a c\right ) \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )\right )-2 \sqrt {a} (2 a-3 b x) (a+x (b+c x))}{8 a^{5/2} x \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.44, size = 100, normalized size = 0.84 \begin {gather*} \frac {\left (3 b^2-4 a c\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^{5/2}}+\frac {(3 b x-2 a) \sqrt {a x^2+b x^3+c x^4}}{4 a^2 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.29, size = 232, normalized size = 1.95 \begin {gather*} \left [-\frac {{\left (3 \, 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 (3 \, a b x - 2 \, a^{2}\right )}}{16 \, a^{3} x^{3}}, \frac {{\left (3 \, 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 (3 \, a b x - 2 \, a^{2}\right )}}{8 \, a^{3} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 152, normalized size = 1.28 \begin {gather*} -\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-4 a^{2} c \,x^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+3 a \,b^{2} x^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )-6 \sqrt {c \,x^{2}+b x +a}\, a^{\frac {3}{2}} b x +4 \sqrt {c \,x^{2}+b x +a}\, a^{\frac {5}{2}}\right )}{8 \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, a^{\frac {7}{2}} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c x^{4} + b x^{3} + a x^{2}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,\sqrt {c\,x^4+b\,x^3+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt {x^{2} \left (a + b x + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________