Optimal. Leaf size=155 \[ \frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{16 a^{5/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.255838, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac{b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{16 a^{5/2}}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4} \]
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^5} \, dx &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4}+\frac{1}{6} \int \frac{b+2 c x}{x^2 \sqrt{a x^2+b x^3+c x^4}} \, dx\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}-\frac{\int \frac{\frac{1}{2} \left (3 b^2-8 a c\right )+b c x}{x \sqrt{a x^2+b x^3+c x^4}} \, dx}{12 a}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}+\frac{\int \frac{3 b \left (b^2-4 a c\right )}{4 \sqrt{a x^2+b x^3+c x^4}} \, dx}{12 a^2}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}+\frac{\left (b \left (b^2-4 a c\right )\right ) \int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{16 a^2}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac{\left (b \left (b^2-4 a c\right )\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 )}{8 a^2}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{3 x^4}-\frac{b \sqrt{a x^2+b x^3+c x^4}}{12 a x^3}+\frac{\left (3 b^2-8 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac{b \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 )}{16 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.143892, size = 131, normalized size = 0.85 \[ \frac{\sqrt{x^2 (a+x (b+c x))} \left (-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (8 a^2+2 a x (b+4 c x)-3 b^2 x^2\right )-3 b x^3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )\right )}{48 a^{5/2} x^4 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 234, normalized size = 1.5 \begin{align*}{\frac{1}{48\,{x}^{4}{a}^{3}}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 12\,c{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{3}b+6\,c\sqrt{c{x}^{2}+bx+a}{x}^{4}{b}^{2}-12\,c\sqrt{c{x}^{2}+bx+a}{x}^{3}ab-3\,\sqrt{a}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){x}^{3}{b}^{3}-6\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{x}^{2}{b}^{2}+6\,\sqrt{c{x}^{2}+bx+a}{x}^{3}{b}^{3}+12\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}xab-16\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{a}^{2} \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.84133, size = 609, normalized size = 3.93 \begin{align*} \left [-\frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{a} x^{4} \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 (2 \, a^{2} b x + 8 \, a^{3} -{\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{2}\right )}}{96 \, a^{3} x^{4}}, \frac{3 \,{\left (b^{3} - 4 \, a b c\right )} \sqrt{-a} x^{4} \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 (2 \, a^{2} b x + 8 \, a^{3} -{\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{2}\right )}}{48 \, a^{3} x^{4}}\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^{5}}\, 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]