Optimal. Leaf size=266 \[ \frac{21 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{11/4}}-\frac{21 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{11/4}}-\frac{21 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{11/4}}+\frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{11/4}}-\frac{7 a x^3 \sqrt [4]{a-b x^4}}{32 b^2}-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b} \]
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Rubi [A] time = 0.140351, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {321, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{21 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{11/4}}-\frac{21 a^2 \log \left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt{2} b^{11/4}}-\frac{21 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{11/4}}+\frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{11/4}}-\frac{7 a x^3 \sqrt [4]{a-b x^4}}{32 b^2}-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{10}}{\left (a-b x^4\right )^{3/4}} \, dx &=-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b}+\frac{(7 a) \int \frac{x^6}{\left (a-b x^4\right )^{3/4}} \, dx}{8 b}\\ &=-\frac{7 a x^3 \sqrt [4]{a-b x^4}}{32 b^2}-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b}+\frac{\left (21 a^2\right ) \int \frac{x^2}{\left (a-b x^4\right )^{3/4}} \, dx}{32 b^2}\\ &=-\frac{7 a x^3 \sqrt [4]{a-b x^4}}{32 b^2}-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b}+\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{32 b^2}\\ &=-\frac{7 a x^3 \sqrt [4]{a-b x^4}}{32 b^2}-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b}-\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt{b} x^2}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{64 b^{5/2}}+\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{b} x^2}{1+b x^4} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{64 b^{5/2}}\\ &=-\frac{7 a x^3 \sqrt [4]{a-b x^4}}{32 b^2}-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b}+\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 b^3}+\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 b^3}+\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}+2 x}{-\frac{1}{\sqrt{b}}-\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{11/4}}+\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2}}{\sqrt [4]{b}}-2 x}{-\frac{1}{\sqrt{b}}+\frac{\sqrt{2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{11/4}}\\ &=-\frac{7 a x^3 \sqrt [4]{a-b x^4}}{32 b^2}-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b}+\frac{21 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{11/4}}-\frac{21 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{11/4}}+\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{11/4}}-\frac{\left (21 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{11/4}}\\ &=-\frac{7 a x^3 \sqrt [4]{a-b x^4}}{32 b^2}-\frac{x^7 \sqrt [4]{a-b x^4}}{8 b}-\frac{21 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{11/4}}+\frac{21 a^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{11/4}}+\frac{21 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{11/4}}-\frac{21 a^2 \log \left (1+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt{2} b^{11/4}}\\ \end{align*}
Mathematica [A] time = 0.0461487, size = 104, normalized size = 0.39 \[ \frac{21 a^2 \sqrt [4]{-b} \tan ^{-1}\left (\frac{\sqrt [4]{-b} x}{\sqrt [4]{a-b x^4}}\right )-21 a^2 \sqrt [4]{-b} \tanh ^{-1}\left (\frac{\sqrt [4]{-b} x}{\sqrt [4]{a-b x^4}}\right )-2 b x^3 \sqrt [4]{a-b x^4} \left (7 a+4 b x^4\right )}{64 b^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{{x}^{10} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \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 = 1.89209, size = 551, normalized size = 2.07 \begin{align*} -\frac{84 \, b^{2} \left (-\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2} b^{8} \left (-\frac{a^{8}}{b^{11}}\right )^{\frac{3}{4}} - b^{8} x \sqrt{\frac{b^{6} x^{2} \sqrt{-\frac{a^{8}}{b^{11}}} + \sqrt{-b x^{4} + a} a^{4}}{x^{2}}} \left (-\frac{a^{8}}{b^{11}}\right )^{\frac{3}{4}}}{a^{8} x}\right ) + 21 \, b^{2} \left (-\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} \log \left (\frac{21 \,{\left (b^{3} x \left (-\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) - 21 \, b^{2} \left (-\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} \log \left (-\frac{21 \,{\left (b^{3} x \left (-\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) + 4 \,{\left (4 \, b x^{7} + 7 \, a x^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.70951, size = 39, normalized size = 0.15 \begin{align*} \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{15}{4}\right )} \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^{10}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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