Optimal. Leaf size=1162 \[ \text{result too large to display} \]
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Rubi [A] time = 1.90973, antiderivative size = 1162, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {465, 472, 584, 305, 220, 1196, 490, 1217, 1707} \[ \frac{b \sqrt{d x^8+c} x^6}{8 a (b c-a d) \left (b x^8+a\right )}-\frac{\sqrt{d} \sqrt{d x^8+c} x^2}{8 a (b c-a d) \left (\sqrt{d} x^4+\sqrt{c}\right )}-\frac{(b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^8+c}}\right )}{32 (-a)^{5/4} \sqrt [4]{b} (b c-a d)^{3/2}}-\frac{(b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{a d-b c} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^8+c}}\right )}{32 (-a)^{5/4} \sqrt [4]{b} (a d-b c)^{3/2}}+\frac{\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a (b c-a d) \sqrt{d x^8+c}}-\frac{\sqrt [4]{c} \sqrt [4]{d} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 a (b c-a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt [4]{d} (b c-3 a d) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{32 a \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt [4]{d} (b c-3 a d) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{32 a \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (b c-3 a d) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\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 )}{64 (-a)^{3/2} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 (b c-3 a d) \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\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 )}{64 (-a)^{3/2} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}} \]
Antiderivative was successfully verified.
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Rule 465
Rule 472
Rule 584
Rule 305
Rule 220
Rule 1196
Rule 490
Rule 1217
Rule 1707
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx,x,x^2\right )\\ &=\frac{b x^6 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-b c+4 a d+b d x^4\right )}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{8 a (b c-a d)}\\ &=\frac{b x^6 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{d x^2}{\sqrt{c+d x^4}}+\frac{(-b c+3 a d) x^2}{\left (a+b x^4\right ) \sqrt{c+d x^4}}\right ) \, dx,x,x^2\right )}{8 a (b c-a d)}\\ &=\frac{b x^6 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}-\frac{d \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{8 a (b c-a d)}+\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{8 a (b c-a d)}\\ &=\frac{b x^6 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}-\frac{\left (\sqrt{c} \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{8 a (b c-a d)}+\frac{\left (\sqrt{c} \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{d} x^2}{\sqrt{c}}}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{8 a (b c-a d)}-\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-a}-\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{16 a \sqrt{b} (b c-a d)}+\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-a}+\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{16 a \sqrt{b} (b c-a d)}\\ &=-\frac{\sqrt{d} x^2 \sqrt{c+d x^8}}{8 a (b c-a d) \left (\sqrt{c}+\sqrt{d} x^4\right )}+\frac{b x^6 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}+\frac{\sqrt [4]{c} \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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a (b c-a d) \sqrt{c+d x^8}}-\frac{\sqrt [4]{c} \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 )}{16 a (b c-a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (b c-3 a d)\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 )}{16 a (b c-a d) (b c+a d)}+\frac{\left (\sqrt{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (b c-3 a d)\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 )}{16 a (b c-a d) (b c+a d)}-\frac{\left (\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d} (b c-3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{16 a \sqrt{b} (b c-a d) (b c+a d)}-\frac{\left (\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt{d} (b c-3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{16 a \sqrt{b} (b c-a d) (b c+a d)}\\ &=-\frac{\sqrt{d} x^2 \sqrt{c+d x^8}}{8 a (b c-a d) \left (\sqrt{c}+\sqrt{d} x^4\right )}+\frac{b x^6 \sqrt{c+d x^8}}{8 a (b c-a d) \left (a+b x^8\right )}-\frac{(b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{32 (-a)^{5/4} \sqrt [4]{b} (b c-a d)^{3/2}}-\frac{(b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{32 (-a)^{5/4} \sqrt [4]{b} (-b c+a d)^{3/2}}+\frac{\sqrt [4]{c} \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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 a (b c-a d) \sqrt{c+d x^8}}-\frac{\sqrt [4]{c} \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 )}{16 a (b c-a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (b c-3 a 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 )}{32 a \sqrt{b} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (b c-3 a 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 )}{32 a \sqrt{b} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (b c-3 a d) \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 )}{64 (-a)^{3/2} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{c+d x^8}}+\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 (b c-3 a d) \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 )}{64 (-a)^{3/2} \sqrt{b} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{c+d x^8}}\\ \end{align*}
Mathematica [C] time = 0.161663, size = 169, normalized size = 0.15 \[ \frac{x^6 \left (-3 b d x^8 \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+7 \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} (b c-4 a d) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+21 a b \left (c+d x^8\right )\right )}{168 a^2 \left (a+b x^8\right ) \sqrt{c+d x^8} (b c-a d)} \]
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}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\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 )}^{2} \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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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