Optimal. Leaf size=59 \[ \frac{a^2 \left (a+c x^4\right )^{3/2}}{6 c^3}+\frac{\left (a+c x^4\right )^{7/2}}{14 c^3}-\frac{a \left (a+c x^4\right )^{5/2}}{5 c^3} \]
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Rubi [A] time = 0.0346376, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^2 \left (a+c x^4\right )^{3/2}}{6 c^3}+\frac{\left (a+c x^4\right )^{7/2}}{14 c^3}-\frac{a \left (a+c x^4\right )^{5/2}}{5 c^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{11} \sqrt{a+c x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^2 \sqrt{a+c x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt{a+c x}}{c^2}-\frac{2 a (a+c x)^{3/2}}{c^2}+\frac{(a+c x)^{5/2}}{c^2}\right ) \, dx,x,x^4\right )\\ &=\frac{a^2 \left (a+c x^4\right )^{3/2}}{6 c^3}-\frac{a \left (a+c x^4\right )^{5/2}}{5 c^3}+\frac{\left (a+c x^4\right )^{7/2}}{14 c^3}\\ \end{align*}
Mathematica [A] time = 0.019856, size = 39, normalized size = 0.66 \[ \frac{\left (a+c x^4\right )^{3/2} \left (8 a^2-12 a c x^4+15 c^2 x^8\right )}{210 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.6 \begin{align*}{\frac{15\,{x}^{8}{c}^{2}-12\,a{x}^{4}c+8\,{a}^{2}}{210\,{c}^{3}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971849, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{7}{2}}}{14 \, c^{3}} - \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}} a}{5 \, c^{3}} + \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}} a^{2}}{6 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48873, size = 104, normalized size = 1.76 \begin{align*} \frac{{\left (15 \, c^{3} x^{12} + 3 \, a c^{2} x^{8} - 4 \, a^{2} c x^{4} + 8 \, a^{3}\right )} \sqrt{c x^{4} + a}}{210 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.87388, size = 87, normalized size = 1.47 \begin{align*} \begin{cases} \frac{4 a^{3} \sqrt{a + c x^{4}}}{105 c^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + c x^{4}}}{105 c^{2}} + \frac{a x^{8} \sqrt{a + c x^{4}}}{70 c} + \frac{x^{12} \sqrt{a + c x^{4}}}{14} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1018, size = 58, normalized size = 0.98 \begin{align*} \frac{15 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a^{2}}{210 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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