Optimal. Leaf size=1060 \[ \text{result too large to display} \]
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Rubi [A] time = 1.98252, antiderivative size = 1060, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {465, 472, 583, 523, 220, 409, 1217, 1707} \[ -\frac{b (7 b c-9 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 ) \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{64 a^3 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{b \sqrt [4]{d} (7 b c-9 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 ) \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )}{32 (-a)^{5/2} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{b^{5/4} (7 b c-9 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)^{11/4} (b c-a d)^{3/2}}-\frac{b^{5/4} (7 b c-9 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)^{11/4} (a d-b c)^{3/2}}-\frac{d^{3/4} (7 b c-4 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 )}{48 a^2 c^{5/4} (b c-a d) \sqrt{d x^8+c}}-\frac{b \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (7 b c-9 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)^{5/2} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{d x^8+c}}-\frac{b \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (7 b c-9 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 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{d x^8+c}}+\frac{b \sqrt{d x^8+c}}{8 a (b c-a d) x^6 \left (b x^8+a\right )}-\frac{(7 b c-4 a d) \sqrt{d x^8+c}}{24 a^2 c (b c-a d) x^6} \]
Antiderivative was successfully verified.
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Rule 465
Rule 472
Rule 583
Rule 523
Rule 220
Rule 409
Rule 1217
Rule 1707
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx,x,x^2\right )\\ &=\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^6 \left (a+b x^8\right )}-\frac{\operatorname{Subst}\left (\int \frac{-7 b c+4 a d-5 b d x^4}{x^4 \left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{8 a (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^6}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^6 \left (a+b x^8\right )}+\frac{\operatorname{Subst}\left (\int \frac{-21 b^2 c^2+20 a b c d+4 a^2 d^2-b d (7 b c-4 a d) x^4}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{24 a^2 c (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^6}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^6 \left (a+b x^8\right )}-\frac{(b (7 b c-9 a d)) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx,x,x^2\right )}{8 a^2 (b c-a d)}-\frac{(d (7 b c-4 a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{24 a^2 c (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^6}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^6 \left (a+b x^8\right )}-\frac{d^{3/4} (7 b c-4 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 )}{48 a^2 c^{5/4} (b c-a d) \sqrt{c+d x^8}}-\frac{(b (7 b c-9 a d)) \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 )}{16 a^3 (b c-a d)}-\frac{(b (7 b c-9 a d)) \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 )}{16 a^3 (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^6}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^6 \left (a+b x^8\right )}-\frac{d^{3/4} (7 b c-4 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 )}{48 a^2 c^{5/4} (b c-a d) \sqrt{c+d x^8}}-\frac{\left (b^{3/2} \sqrt{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (7 b c-9 a d)\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 )}{16 a^3 (b c-a d) (b c+a d)}-\frac{\left (b^{3/2} \sqrt{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (7 b c-9 a d)\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 )}{16 a^3 (b c-a d) (b c+a d)}-\frac{\left (b \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt{d} (7 b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{16 a^2 (b c-a d) (b c+a d)}+\frac{\left (b \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d} (7 b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x^4}} \, dx,x,x^2\right )}{16 (-a)^{5/2} (b c-a d) (b c+a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^6}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^6 \left (a+b x^8\right )}+\frac{b^{5/4} (7 b c-9 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)^{11/4} (b c-a d)^{3/2}}-\frac{b^{5/4} (7 b c-9 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)^{11/4} (-b c+a d)^{3/2}}-\frac{d^{3/4} (7 b c-4 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 )}{48 a^2 c^{5/4} (b c-a d) \sqrt{c+d x^8}}-\frac{b \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{-a}}+\sqrt{d}\right ) \sqrt [4]{d} (7 b c-9 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^2 \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{c+d x^8}}+\frac{b \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (7 b c-9 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)^{5/2} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{c+d x^8}}-\frac{b \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (7 b c-9 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 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{c+d x^8}}-\frac{b \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 (7 b c-9 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 \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{c+d x^8}}\\ \end{align*}
Mathematica [C] time = 0.272972, size = 225, normalized size = 0.21 \[ \frac{5 x^8 \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} \left (4 a^2 d^2+20 a b c d-21 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+5 a \left (c+d x^8\right ) \left (4 a^2 d-4 a b \left (c-d x^8\right )-7 b^2 c x^8\right )+b d x^{16} \left (a+b x^8\right ) \sqrt{\frac{d x^8}{c}+1} (4 a d-7 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )}{120 a^3 c x^6 \left (a+b x^8\right ) \sqrt{c+d x^8} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7} \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{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{7}}\,{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{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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