Integrand size = 16, antiderivative size = 82 \[ \int \frac {x^{15}}{\left (a-b x^4\right )^{3/4}} \, dx=-\frac {a^3 \sqrt [4]{a-b x^4}}{b^4}+\frac {3 a^2 \left (a-b x^4\right )^{5/4}}{5 b^4}-\frac {a \left (a-b x^4\right )^{9/4}}{3 b^4}+\frac {\left (a-b x^4\right )^{13/4}}{13 b^4} \]
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Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {272, 45} \[ \int \frac {x^{15}}{\left (a-b x^4\right )^{3/4}} \, dx=-\frac {a^3 \sqrt [4]{a-b x^4}}{b^4}+\frac {3 a^2 \left (a-b x^4\right )^{5/4}}{5 b^4}+\frac {\left (a-b x^4\right )^{13/4}}{13 b^4}-\frac {a \left (a-b x^4\right )^{9/4}}{3 b^4} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x^3}{(a-b x)^{3/4}} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (\frac {a^3}{b^3 (a-b x)^{3/4}}-\frac {3 a^2 \sqrt [4]{a-b x}}{b^3}+\frac {3 a (a-b x)^{5/4}}{b^3}-\frac {(a-b x)^{9/4}}{b^3}\right ) \, dx,x,x^4\right ) \\ & = -\frac {a^3 \sqrt [4]{a-b x^4}}{b^4}+\frac {3 a^2 \left (a-b x^4\right )^{5/4}}{5 b^4}-\frac {a \left (a-b x^4\right )^{9/4}}{3 b^4}+\frac {\left (a-b x^4\right )^{13/4}}{13 b^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.62 \[ \int \frac {x^{15}}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{a-b x^4} \left (-128 a^3-32 a^2 b x^4-20 a b^2 x^8-15 b^3 x^{12}\right )}{195 b^4} \]
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Time = 4.50 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (15 b^{3} x^{12}+20 a \,b^{2} x^{8}+32 a^{2} b \,x^{4}+128 a^{3}\right )}{195 b^{4}}\) | \(48\) |
trager | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (15 b^{3} x^{12}+20 a \,b^{2} x^{8}+32 a^{2} b \,x^{4}+128 a^{3}\right )}{195 b^{4}}\) | \(48\) |
pseudoelliptic | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (15 b^{3} x^{12}+20 a \,b^{2} x^{8}+32 a^{2} b \,x^{4}+128 a^{3}\right )}{195 b^{4}}\) | \(48\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} {\left (\left (-b \,x^{4}+a \right )^{3}\right )}^{\frac {1}{4}} \left (15 b^{3} x^{12}+20 a \,b^{2} x^{8}+32 a^{2} b \,x^{4}+128 a^{3}\right )}{195 b^{4} {\left (-\left (b \,x^{4}-a \right )^{3}\right )}^{\frac {1}{4}}}\) | \(75\) |
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Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.57 \[ \int \frac {x^{15}}{\left (a-b x^4\right )^{3/4}} \, dx=-\frac {{\left (15 \, b^{3} x^{12} + 20 \, a b^{2} x^{8} + 32 \, a^{2} b x^{4} + 128 \, a^{3}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{195 \, b^{4}} \]
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Time = 0.66 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15 \[ \int \frac {x^{15}}{\left (a-b x^4\right )^{3/4}} \, dx=\begin {cases} - \frac {128 a^{3} \sqrt [4]{a - b x^{4}}}{195 b^{4}} - \frac {32 a^{2} x^{4} \sqrt [4]{a - b x^{4}}}{195 b^{3}} - \frac {4 a x^{8} \sqrt [4]{a - b x^{4}}}{39 b^{2}} - \frac {x^{12} \sqrt [4]{a - b x^{4}}}{13 b} & \text {for}\: b \neq 0 \\\frac {x^{16}}{16 a^{\frac {3}{4}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {x^{15}}{\left (a-b x^4\right )^{3/4}} \, dx=\frac {{\left (-b x^{4} + a\right )}^{\frac {13}{4}}}{13 \, b^{4}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {9}{4}} a}{3 \, b^{4}} + \frac {3 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{5 \, b^{4}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06 \[ \int \frac {x^{15}}{\left (a-b x^4\right )^{3/4}} \, dx=-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{b^{4}} - \frac {15 \, {\left (b x^{4} - a\right )}^{3} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} + 65 \, {\left (b x^{4} - a\right )}^{2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a - 117 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{195 \, b^{4}} \]
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Time = 5.56 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.60 \[ \int \frac {x^{15}}{\left (a-b x^4\right )^{3/4}} \, dx=-{\left (a-b\,x^4\right )}^{1/4}\,\left (\frac {128\,a^3}{195\,b^4}+\frac {x^{12}}{13\,b}+\frac {4\,a\,x^8}{39\,b^2}+\frac {32\,a^2\,x^4}{195\,b^3}\right ) \]
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