Optimal. Leaf size=112 \[ -\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{6 \sqrt{a}}+\frac{1}{3} \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.114291, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1357, 732, 843, 621, 206, 724} \[ -\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{6 \sqrt{a}}+\frac{1}{3} \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1357
Rule 732
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^3+c x^6}}{x^4} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{b+2 c x}{x \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}+\frac{1}{6} b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^3\right )+\frac{1}{3} c \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{1}{3} b \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^3}{\sqrt{a+b x^3+c x^6}}\right )+\frac{1}{3} (2 c) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^3}{\sqrt{a+b x^3+c x^6}}\right )\\ &=-\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{6 \sqrt{a}}+\frac{1}{3} \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )\\ \end{align*}
Mathematica [A] time = 0.0493902, size = 112, normalized size = 1. \[ -\frac{\sqrt{a+b x^3+c x^6}}{3 x^3}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{6 \sqrt{a}}+\frac{1}{3} \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}\sqrt{c{x}^{6}+b{x}^{3}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.8756, size = 1411, normalized size = 12.6 \begin{align*} \left [\frac{2 \, a \sqrt{c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{c} - 4 \, a c\right ) + \sqrt{a} b x^{3} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, \sqrt{c x^{6} + b x^{3} + a} a}{12 \, a x^{3}}, -\frac{4 \, a \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - \sqrt{a} b x^{3} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, \sqrt{c x^{6} + b x^{3} + a} a}{12 \, a x^{3}}, \frac{\sqrt{-a} b x^{3} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + a \sqrt{c} x^{3} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{c} - 4 \, a c\right ) - 2 \, \sqrt{c x^{6} + b x^{3} + a} a}{6 \, a x^{3}}, \frac{\sqrt{-a} b x^{3} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) - 2 \, a \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) - 2 \, \sqrt{c x^{6} + b x^{3} + a} a}{6 \, a 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{a + b x^{3} + c x^{6}}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{6} + b x^{3} + a}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]