Optimal. Leaf size=851 \[ -\frac{\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 ) \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{16 b \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{d x^8+c}}-\frac{\sqrt [4]{-a} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^8+c}}\right )}{8 b^{3/4} \sqrt{b c-a d}}-\frac{\sqrt [4]{-a} \tan ^{-1}\left (\frac{\sqrt{a d-b c} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^8+c}}\right )}{8 b^{3/4} \sqrt{a d-b c}}+\frac{\left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{4 b \sqrt [4]{c} \sqrt [4]{d} \sqrt{d x^8+c}}-\frac{a \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \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}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{8 b \sqrt [4]{c} (b c+a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \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}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{8 b \sqrt [4]{c} (b c+a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 \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 )}{16 b \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{d x^8+c}} \]
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Rubi [A] time = 1.10131, antiderivative size = 851, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {465, 483, 220, 409, 1217, 1707} \[ -\frac{\left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right ) \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{16 b \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{d x^8+c}}-\frac{\sqrt [4]{-a} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^8+c}}\right )}{8 b^{3/4} \sqrt{b c-a d}}-\frac{\sqrt [4]{-a} \tan ^{-1}\left (\frac{\sqrt{a d-b c} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^8+c}}\right )}{8 b^{3/4} \sqrt{a d-b c}}+\frac{\left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b \sqrt [4]{c} \sqrt [4]{d} \sqrt{d x^8+c}}-\frac{a \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt [4]{d} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b \sqrt [4]{c} (b c+a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b \sqrt [4]{c} (b c+a d) \sqrt{d x^8+c}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 \left (\sqrt{d} x^4+\sqrt{c}\right ) \sqrt{\frac{d x^8+c}{\left (\sqrt{d} x^4+\sqrt{c}\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x^2}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 b \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{d x^8+c}} \]
Antiderivative was successfully verified.
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Rule 465
Rule 483
Rule 220
Rule 409
Rule 1217
Rule 1707
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{2 b}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{2 b}\\ &=\frac{\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 )}{4 b \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 b}\\ &=\frac{\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 )}{4 b \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8}}-\frac{\left (\sqrt{c} \left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right )\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 (b c+a d)}-\frac{\left (\sqrt{c} \left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right )\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{-a}}\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 (b c+a d)}-\frac{\left (a \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 b (b c+a d)}-\frac{\left (\left (\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\right ) \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{4 b (b c+a d)}\\ &=-\frac{\sqrt [4]{-a} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{8 b^{3/4} \sqrt{b c-a d}}-\frac{\sqrt [4]{-a} \tan ^{-1}\left (\frac{\sqrt{-b c+a d} x^2}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^8}}\right )}{8 b^{3/4} \sqrt{-b c+a d}}+\frac{\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 )}{4 b \sqrt [4]{c} \sqrt [4]{d} \sqrt{c+d x^8}}-\frac{a \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\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 b \sqrt [4]{c} (b c+a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{-a} \sqrt{b} \sqrt{c}+a \sqrt{d}\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 b \sqrt [4]{c} (b c+a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\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 [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^8}}-\frac{\left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\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 [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^8}}\\ \end{align*}
Mathematica [C] time = 0.0399451, size = 65, normalized size = 0.08 \[ \frac{x^{10} \sqrt{\frac{c+d x^8}{c}} F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{10 a \sqrt{c+d x^8}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{9}}{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^{9}}{{\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^{9}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{9}}{{\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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