Optimal. Leaf size=558 \[ \frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.573605, antiderivative size = 558, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {1374, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1374
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{a+b x^3+c x^6} \, dx &=-\left (\frac{1}{2} \left (-1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx\right )+\frac{1}{2} \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx\\ &=-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \int \frac{1}{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \int \frac{2^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}-\sqrt [3]{c} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \int \frac{1}{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt [3]{2} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \int \frac{2^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}-\sqrt [3]{c} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} \sqrt{b^2-4 a c}}\\ &=-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \int \frac{1}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2\ 2^{2/3} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3} \int \frac{1}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2\ 2^{2/3} \sqrt{b^2-4 a c}}\\ &=-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}\\ &=\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}} \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}\\ \end{align*}
Mathematica [C] time = 0.0160134, size = 42, normalized size = 0.08 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^3 b+\text{$\#$1}^6 c+a\& ,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\& \right ] \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.003, size = 43, normalized size = 0.1 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{{{\it \_R}}^{3}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \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{x^{3}}{c x^{6} + b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.99071, size = 5501, normalized size = 9.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.8966, size = 122, normalized size = 0.22 \begin{align*} \operatorname{RootSum}{\left (t^{6} \left (46656 a^{3} c^{4} - 34992 a^{2} b^{2} c^{3} + 8748 a b^{4} c^{2} - 729 b^{6} c\right ) + t^{3} \left (432 a^{2} c^{2} - 216 a b^{2} c + 27 b^{4}\right ) + a, \left ( t \mapsto t \log{\left (x + \frac{2592 t^{4} a^{2} c^{3} - 1296 t^{4} a b^{2} c^{2} + 162 t^{4} b^{4} c + 12 t a c - 3 t b^{2}}{b} \right )} \right )\right )} \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{x^{3}}{c x^{6} + b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]