Integrand size = 15, antiderivative size = 59 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {a^2 \left (a+c x^4\right )^{5/2}}{10 c^3}-\frac {a \left (a+c x^4\right )^{7/2}}{7 c^3}+\frac {\left (a+c x^4\right )^{9/2}}{18 c^3} \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {a^2 \left (a+c x^4\right )^{5/2}}{10 c^3}+\frac {\left (a+c x^4\right )^{9/2}}{18 c^3}-\frac {a \left (a+c x^4\right )^{7/2}}{7 c^3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int x^2 (a+c x)^{3/2} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {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*}
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.66 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {\left (a+c x^4\right )^{5/2} \left (8 a^2-20 a c x^4+35 c^2 x^8\right )}{630 c^3} \]
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Time = 4.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(\frac {\left (x^{4} c +a \right )^{\frac {5}{2}} \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right )}{630 c^{3}}\) | \(36\) |
pseudoelliptic | \(\frac {\left (x^{4} c +a \right )^{\frac {5}{2}} \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right )}{630 c^{3}}\) | \(36\) |
default | \(\frac {\sqrt {x^{4} c +a}\, \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right ) \left (c^{2} x^{8}+2 a \,x^{4} c +a^{2}\right )}{630 c^{3}}\) | \(54\) |
elliptic | \(\frac {\sqrt {x^{4} c +a}\, \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right ) \left (c^{2} x^{8}+2 a \,x^{4} c +a^{2}\right )}{630 c^{3}}\) | \(54\) |
trager | \(\frac {\left (35 c^{4} x^{16}+50 a \,c^{3} x^{12}+3 a^{2} c^{2} x^{8}-4 a^{3} c \,x^{4}+8 a^{4}\right ) \sqrt {x^{4} c +a}}{630 c^{3}}\) | \(58\) |
risch | \(\frac {\left (35 c^{4} x^{16}+50 a \,c^{3} x^{12}+3 a^{2} c^{2} x^{8}-4 a^{3} c \,x^{4}+8 a^{4}\right ) \sqrt {x^{4} c +a}}{630 c^{3}}\) | \(58\) |
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Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {{\left (35 \, c^{4} x^{16} + 50 \, a c^{3} x^{12} + 3 \, a^{2} c^{2} x^{8} - 4 \, a^{3} c x^{4} + 8 \, a^{4}\right )} \sqrt {c x^{4} + a}}{630 \, c^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (49) = 98\).
Time = 0.59 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.85 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\begin {cases} \frac {4 a^{4} \sqrt {a + c x^{4}}}{315 c^{3}} - \frac {2 a^{3} x^{4} \sqrt {a + c x^{4}}}{315 c^{2}} + \frac {a^{2} x^{8} \sqrt {a + c x^{4}}}{210 c} + \frac {5 a x^{12} \sqrt {a + c x^{4}}}{63} + \frac {c x^{16} \sqrt {a + c x^{4}}}{18} & \text {for}\: c \neq 0 \\\frac {a^{\frac {3}{2}} x^{12}}{12} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {{\left (c x^{4} + a\right )}^{\frac {9}{2}}}{18 \, c^{3}} - \frac {{\left (c x^{4} + a\right )}^{\frac {7}{2}} a}{7 \, c^{3}} + \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}} a^{2}}{10 \, c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {35 \, {\left (c x^{4} + a\right )}^{\frac {9}{2}} - 90 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}} a + 63 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} a^{2}}{630 \, c^{3}} \]
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Time = 6.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\sqrt {c\,x^4+a}\,\left (\frac {5\,a\,x^{12}}{63}+\frac {c\,x^{16}}{18}+\frac {4\,a^4}{315\,c^3}-\frac {2\,a^3\,x^4}{315\,c^2}+\frac {a^2\,x^8}{210\,c}\right ) \]
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