Optimal. Leaf size=59 \[ \frac{a^2 \left (a+c x^4\right )^{5/2}}{10 c^3}+\frac{\left (a+c x^4\right )^{9/2}}{18 c^3}-\frac{a \left (a+c x^4\right )^{7/2}}{7 c^3} \]
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Rubi [A] time = 0.0342912, 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} \[ \frac{a^2 \left (a+c x^4\right )^{5/2}}{10 c^3}+\frac{\left (a+c x^4\right )^{9/2}}{18 c^3}-\frac{a \left (a+c x^4\right )^{7/2}}{7 c^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{11} \left (a+c x^4\right )^{3/2} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^2 (a+c x)^{3/2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+c x)^{3/2}}{c^2}-\frac{2 a (a+c x)^{5/2}}{c^2}+\frac{(a+c x)^{7/2}}{c^2}\right ) \, dx,x,x^4\right )\\ &=\frac{a^2 \left (a+c x^4\right )^{5/2}}{10 c^3}-\frac{a \left (a+c x^4\right )^{7/2}}{7 c^3}+\frac{\left (a+c x^4\right )^{9/2}}{18 c^3}\\ \end{align*}
Mathematica [A] time = 0.0208882, size = 39, normalized size = 0.66 \[ \frac{\left (a+c x^4\right )^{5/2} \left (8 a^2-20 a c x^4+35 c^2 x^8\right )}{630 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 36, normalized size = 0.6 \begin{align*}{\frac{35\,{x}^{8}{c}^{2}-20\,a{x}^{4}c+8\,{a}^{2}}{630\,{c}^{3}} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971273, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{9}{2}}}{18 \, c^{3}} - \frac{{\left (c x^{4} + a\right )}^{\frac{7}{2}} a}{7 \, c^{3}} + \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}} a^{2}}{10 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44932, size = 128, normalized size = 2.17 \begin{align*} \frac{{\left (35 \, c^{4} x^{16} + 50 \, a c^{3} x^{12} + 3 \, a^{2} c^{2} x^{8} - 4 \, a^{3} c x^{4} + 8 \, a^{4}\right )} \sqrt{c x^{4} + a}}{630 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.61937, size = 109, normalized size = 1.85 \begin{align*} \begin{cases} \frac{4 a^{4} \sqrt{a + c x^{4}}}{315 c^{3}} - \frac{2 a^{3} x^{4} \sqrt{a + c x^{4}}}{315 c^{2}} + \frac{a^{2} x^{8} \sqrt{a + c x^{4}}}{210 c} + \frac{5 a x^{12} \sqrt{a + c x^{4}}}{63} + \frac{c x^{16} \sqrt{a + c x^{4}}}{18} & \text{for}\: c \neq 0 \\\frac{a^{\frac{3}{2}} x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12925, size = 143, normalized size = 2.42 \begin{align*} \frac{\frac{3 \,{\left (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}\right )} a}{c^{2}} + \frac{35 \,{\left (c x^{4} + a\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a^{3}}{c^{2}}}{630 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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