Optimal. Leaf size=204 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{x^3}{4 c \left (a+c x^4\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.125864, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {288, 297, 1162, 617, 204, 1165, 628} \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{x^3}{4 c \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 288
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a+c x^4\right )^2} \, dx &=-\frac{x^3}{4 c \left (a+c x^4\right )}+\frac{3 \int \frac{x^2}{a+c x^4} \, dx}{4 c}\\ &=-\frac{x^3}{4 c \left (a+c x^4\right )}-\frac{3 \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{8 c^{3/2}}+\frac{3 \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{8 c^{3/2}}\\ &=-\frac{x^3}{4 c \left (a+c x^4\right )}+\frac{3 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 c^2}+\frac{3 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 c^2}+\frac{3 \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}\\ &=-\frac{x^3}{4 c \left (a+c x^4\right )}+\frac{3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}\\ &=-\frac{x^3}{4 c \left (a+c x^4\right )}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}\\ \end{align*}
Mathematica [A] time = 0.110255, size = 185, normalized size = 0.91 \[ \frac{-\frac{8 c^{3/4} x^3}{a+c x^4}+\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}}{32 c^{7/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 145, normalized size = 0.7 \begin{align*} -{\frac{{x}^{3}}{4\,c \left ( c{x}^{4}+a \right ) }}+{\frac{3\,\sqrt{2}}{32\,{c}^{2}}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,\sqrt{2}}{16\,{c}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,\sqrt{2}}{16\,{c}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \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 = 1.89238, size = 423, normalized size = 2.07 \begin{align*} -\frac{4 \, x^{3} + 12 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \arctan \left (-c^{2} x \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} + \sqrt{-a c^{3} \sqrt{-\frac{1}{a c^{7}}} + x^{2}} c^{2} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}}\right ) - 3 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \log \left (a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x\right ) + 3 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \log \left (-a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x\right )}{16 \,{\left (c^{2} x^{4} + a c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.615309, size = 44, normalized size = 0.22 \begin{align*} - \frac{x^{3}}{4 a c + 4 c^{2} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a c^{7} + 81, \left ( t \mapsto t \log{\left (\frac{4096 t^{3} a c^{5}}{27} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15847, size = 265, normalized size = 1.3 \begin{align*} -\frac{x^{3}}{4 \,{\left (c x^{4} + a\right )} c} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a c^{4}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a c^{4}} - \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a c^{4}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]