Optimal. Leaf size=146 \[ \frac{4 a^{9/2} \left (\frac{b x^4}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac{a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4} \]
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Rubi [A] time = 0.103049, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {275, 279, 321, 233, 231} \[ -\frac{2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac{4 a^{9/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4} \]
Antiderivative was successfully verified.
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Rule 275
Rule 279
Rule 321
Rule 233
Rule 231
Rubi steps
\begin{align*} \int x^9 \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^4 \left (a+b x^2\right )^{5/4} \, dx,x,x^2\right )\\ &=\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{1}{6} a \operatorname{Subst}\left (\int x^4 \sqrt [4]{a+b x^2} \, dx,x,x^2\right )\\ &=\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{1}{66} a^2 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac{a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}-\frac{a^3 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b}\\ &=-\frac{2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac{a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{\left (2 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{231 b^2}\\ &=-\frac{2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac{a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{\left (2 a^4 \left (1+\frac{b x^4}{a}\right )^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{231 b^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac{a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{4 a^{9/2} \left (1+\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0802982, size = 81, normalized size = 0.55 \[ \frac{x^2 \sqrt [4]{a+b x^4} \left (\frac{6 a^3 \, _2F_1\left (-\frac{5}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )}{\sqrt [4]{\frac{b x^4}{a}+1}}-\left (6 a-11 b x^4\right ) \left (a+b x^4\right )^2\right )}{165 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{x}^{9} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, 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{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{9}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{13} + a x^{9}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.30584, size = 29, normalized size = 0.2 \begin{align*} \frac{a^{\frac{5}{4}} x^{10}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10} \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{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{9}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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