Optimal. Leaf size=768 \[ -\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{8 \sqrt [4]{c} \sqrt{c+d x^8} (a d+b c)}-\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}+\sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{8 \sqrt [4]{c} \sqrt{c+d x^8} (a d+b c)}+\frac{\tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{8 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{b c-a d}}-\frac{\tan ^{-1}\left (\frac{x^2 \sqrt{a d-b c}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{8 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{a d-b c}}-\frac{\left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8} (a d+b c)}+\frac{\left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )^2 \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8} (a d+b c)} \]
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Rubi [A] time = 0.935099, antiderivative size = 768, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {465, 490, 1217, 220, 1707} \[ \frac{\tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{8 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{b c-a d}}-\frac{\tan ^{-1}\left (\frac{x^2 \sqrt{a d-b c}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{8 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{a d-b c}}-\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} \sqrt{c+d x^8} (a d+b c)}-\frac{\sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}+\sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} \sqrt{c+d x^8} (a d+b c)}-\frac{\left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8} (a d+b c)}+\frac{\left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \left (\sqrt{-a} \sqrt{d}+\sqrt{b} \sqrt{c}\right )^2 \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8} (a d+b c)} \]
Antiderivative was successfully verified.
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Rule 465
Rule 490
Rule 1217
Rule 220
Rule 1707
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-a}-\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 \sqrt{b}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-a}+\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 \sqrt{b}}\\ &=-\frac{\left (\sqrt{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (\sqrt{-a}-\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 (b c+a d)}+\frac{\left (\sqrt{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (\sqrt{-a}+\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 (b c+a d)}-\frac{\left (\left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 (b c+a d)}-\frac{\left (\left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 (b c+a d)}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{8 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{b c-a d}}-\frac{\tan ^{-1}\left (\frac{\sqrt{-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{8 \sqrt [4]{-a} \sqrt [4]{b} \sqrt{-b c+a d}}-\frac{\left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} (b c+a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 \sqrt [4]{c} (b c+a d) \sqrt{c+d x^8}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 \left (\sqrt{c}+\sqrt{d} x^4\right ) \sqrt{\frac{c+d x^8}{\left (\sqrt{c}+\sqrt{d} x^4\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 \sqrt{-a} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^8}}\\ \end{align*}
Mathematica [C] time = 0.0365615, size = 65, normalized size = 0.08 \[ \frac{x^6 \sqrt{\frac{c+d x^8}{c}} F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{6 a \sqrt{c+d x^8}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (b x^{8} + a\right )} \sqrt{d x^{8} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\left (a + b x^{8}\right ) \sqrt{c + d x^{8}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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