Optimal. Leaf size=114 \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{4 a x^2}-\frac{\sqrt{a x^2+b x^3+c x^4}}{2 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.147701, antiderivative size = 114, normalized size of antiderivative = 1., 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} \[ \frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{4 a x^2}-\frac{\sqrt{a x^2+b x^3+c x^4}}{2 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1920
Rule 1951
Rule 12
Rule 1904
Rule 206
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*}
Mathematica [A] time = 0.100333, size = 112, normalized size = 0.98 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (x^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{a} (2 a+b x) \sqrt{a+x (b+c x)}\right )}{8 a^{3/2} x^3 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.006, size = 207, normalized size = 1.8 \begin{align*} -{\frac{1}{8\,{a}^{2}{x}^{3}}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 4\,c{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{2}+2\,c\sqrt{c{x}^{2}+bx+a}{x}^{3}b-4\,c\sqrt{c{x}^{2}+bx+a}{x}^{2}a-\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){x}^{2}{b}^{2}-2\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}xb+2\,\sqrt{c{x}^{2}+bx+a}{x}^{2}{b}^{2}+4\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}a \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69192, size = 517, normalized size = 4.54 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (a + b x + c x^{2}\right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]