Optimal. Leaf size=308 \[ \frac{\sqrt{x}}{4 a \left (a+c x^4\right )}-\frac{7 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}} \]
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Rubi [A] time = 0.266109, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {290, 329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt{x}}{4 a \left (a+c x^4\right )}-\frac{7 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 329
Rule 214
Rule 212
Rule 208
Rule 205
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (a+c x^4\right )^2} \, dx &=\frac{\sqrt{x}}{4 a \left (a+c x^4\right )}+\frac{7 \int \frac{1}{\sqrt{x} \left (a+c x^4\right )} \, dx}{8 a}\\ &=\frac{\sqrt{x}}{4 a \left (a+c x^4\right )}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{a+c x^8} \, dx,x,\sqrt{x}\right )}{4 a}\\ &=\frac{\sqrt{x}}{4 a \left (a+c x^4\right )}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}-\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{8 (-a)^{3/2}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{8 (-a)^{3/2}}\\ &=\frac{\sqrt{x}}{4 a \left (a+c x^4\right )}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{16 (-a)^{7/4}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt{x}\right )}{16 (-a)^{7/4}}+\frac{7 \operatorname{Subst}\left (\int \frac{\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{16 (-a)^{7/4}}+\frac{7 \operatorname{Subst}\left (\int \frac{\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt{-a}+\sqrt{c} x^4} \, dx,x,\sqrt{x}\right )}{16 (-a)^{7/4}}\\ &=\frac{\sqrt{x}}{4 a \left (a+c x^4\right )}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{32 (-a)^{7/4} \sqrt [4]{c}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{32 (-a)^{7/4} \sqrt [4]{c}}-\frac{7 \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac{\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}-\frac{7 \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac{\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}\\ &=\frac{\sqrt{x}}{4 a \left (a+c x^4\right )}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}-\frac{7 \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}\\ &=\frac{\sqrt{x}}{4 a \left (a+c x^4\right )}-\frac{7 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{16 (-a)^{15/8} \sqrt [8]{c}}-\frac{7 \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}+\frac{7 \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{c} x\right )}{32 \sqrt{2} (-a)^{15/8} \sqrt [8]{c}}\\ \end{align*}
Mathematica [C] time = 0.0113292, size = 51, normalized size = 0.17 \[ \frac{7 \sqrt{x} \, _2F_1\left (\frac{1}{8},1;\frac{9}{8};-\frac{c x^4}{a}\right )}{4 a^2}+\frac{\sqrt{x}}{4 a \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 50, normalized size = 0.2 \begin{align*}{\frac{1}{4\,a \left ( c{x}^{4}+a \right ) }\sqrt{x}}+{\frac{7}{32\,ac}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{7}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \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*} -7 \, c \int \frac{x^{\frac{7}{2}}}{8 \,{\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} + \frac{7 \, c x^{\frac{9}{2}} + 8 \, a \sqrt{x}}{4 \,{\left (a^{2} c x^{4} + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68283, size = 1381, normalized size = 4.48 \begin{align*} \frac{28 \, \sqrt{2}{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \arctan \left (\sqrt{2} \sqrt{a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} + \sqrt{2} a^{2} \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + x} a^{13} c \left (-\frac{1}{a^{15} c}\right )^{\frac{7}{8}} - \sqrt{2} a^{13} c \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{7}{8}} + 1\right ) + 28 \, \sqrt{2}{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \arctan \left (\sqrt{2} \sqrt{a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + x} a^{13} c \left (-\frac{1}{a^{15} c}\right )^{\frac{7}{8}} - \sqrt{2} a^{13} c \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{7}{8}} - 1\right ) + 7 \, \sqrt{2}{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \log \left (a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} + \sqrt{2} a^{2} \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + x\right ) - 7 \, \sqrt{2}{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \log \left (a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} - \sqrt{2} a^{2} \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + x\right ) + 56 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \arctan \left (\sqrt{a^{4} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{4}} + x} a^{13} c \left (-\frac{1}{a^{15} c}\right )^{\frac{7}{8}} - a^{13} c \sqrt{x} \left (-\frac{1}{a^{15} c}\right )^{\frac{7}{8}}\right ) + 14 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \log \left (a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + \sqrt{x}\right ) - 14 \,{\left (a c x^{4} + a^{2}\right )} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} \log \left (-a^{2} \left (-\frac{1}{a^{15} c}\right )^{\frac{1}{8}} + \sqrt{x}\right ) + 16 \, \sqrt{x}}{64 \,{\left (a c x^{4} + a^{2}\right )}} \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 [B] time = 1.36933, size = 613, normalized size = 1.99 \begin{align*} \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{32 \, a^{2}} + \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \log \left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} - \frac{7 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \log \left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} + \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \log \left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} - \frac{7 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} \log \left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{64 \, a^{2}} + \frac{\sqrt{x}}{4 \,{\left (c x^{4} + a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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