Optimal. Leaf size=223 \[ -\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}+\frac{21 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}+\frac{21 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{x^7}{8 c \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.142328, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {288, 297, 1162, 617, 204, 1165, 628} \[ -\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}+\frac{21 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}+\frac{21 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{x^7}{8 c \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 288
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{10}}{\left (a+c x^4\right )^3} \, dx &=-\frac{x^7}{8 c \left (a+c x^4\right )^2}+\frac{7 \int \frac{x^6}{\left (a+c x^4\right )^2} \, dx}{8 c}\\ &=-\frac{x^7}{8 c \left (a+c x^4\right )^2}-\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}+\frac{21 \int \frac{x^2}{a+c x^4} \, dx}{32 c^2}\\ &=-\frac{x^7}{8 c \left (a+c x^4\right )^2}-\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}-\frac{21 \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{64 c^{5/2}}+\frac{21 \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{64 c^{5/2}}\\ &=-\frac{x^7}{8 c \left (a+c x^4\right )^2}-\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}+\frac{21 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 c^3}+\frac{21 \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 c^3}+\frac{21 \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}+\frac{21 \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}\\ &=-\frac{x^7}{8 c \left (a+c x^4\right )^2}-\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}+\frac{21 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}+\frac{21 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}\\ &=-\frac{x^7}{8 c \left (a+c x^4\right )^2}-\frac{7 x^3}{32 c^2 \left (a+c x^4\right )}-\frac{21 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}+\frac{21 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} \sqrt [4]{a} c^{11/4}}+\frac{21 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}-\frac{21 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{128 \sqrt{2} \sqrt [4]{a} c^{11/4}}\\ \end{align*}
Mathematica [A] time = 0.0895378, size = 205, normalized size = 0.92 \[ \frac{-\frac{88 c^{3/4} x^3}{a+c x^4}+\frac{32 a c^{3/4} x^3}{\left (a+c x^4\right )^2}+\frac{21 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{21 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{42 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{42 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}}{256 c^{11/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 156, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ( -{\frac{11\,{x}^{7}}{32\,c}}-{\frac{7\,a{x}^{3}}{32\,{c}^{2}}} \right ) }+{\frac{21\,\sqrt{2}}{256\,{c}^{3}}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{21\,\sqrt{2}}{128\,{c}^{3}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{21\,\sqrt{2}}{128\,{c}^{3}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84284, size = 554, normalized size = 2.48 \begin{align*} -\frac{44 \, c x^{7} + 28 \, a x^{3} + 84 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a c^{11}}\right )^{\frac{1}{4}} \arctan \left (-c^{3} x \left (-\frac{1}{a c^{11}}\right )^{\frac{1}{4}} + \sqrt{-a c^{5} \sqrt{-\frac{1}{a c^{11}}} + x^{2}} c^{3} \left (-\frac{1}{a c^{11}}\right )^{\frac{1}{4}}\right ) - 21 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a c^{11}}\right )^{\frac{1}{4}} \log \left (a c^{8} \left (-\frac{1}{a c^{11}}\right )^{\frac{3}{4}} + x\right ) + 21 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a c^{11}}\right )^{\frac{1}{4}} \log \left (-a c^{8} \left (-\frac{1}{a c^{11}}\right )^{\frac{3}{4}} + x\right )}{128 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.79546, size = 68, normalized size = 0.3 \begin{align*} - \frac{7 a x^{3} + 11 c x^{7}}{32 a^{2} c^{2} + 64 a c^{3} x^{4} + 32 c^{4} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a c^{11} + 194481, \left ( t \mapsto t \log{\left (\frac{2097152 t^{3} a c^{8}}{9261} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15113, size = 278, normalized size = 1.25 \begin{align*} -\frac{11 \, c x^{7} + 7 \, a x^{3}}{32 \,{\left (c x^{4} + a\right )}^{2} c^{2}} + \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a c^{5}} + \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a c^{5}} - \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a c^{5}} + \frac{21 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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