Optimal. Leaf size=106 \[ -\frac {a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}+\frac {4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac {6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}+\frac {4 a \left (a-b x^4\right )^{17/4}}{17 b^5}-\frac {\left (a-b x^4\right )^{21/4}}{21 b^5} \]
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Rubi [A]
time = 0.05, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {272, 45}
\begin {gather*} -\frac {a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}+\frac {4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac {6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}-\frac {\left (a-b x^4\right )^{21/4}}{21 b^5}+\frac {4 a \left (a-b x^4\right )^{17/4}}{17 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{19} \sqrt [4]{a-b x^4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x^4 \sqrt [4]{a-b x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^4 \sqrt [4]{a-b x}}{b^4}-\frac {4 a^3 (a-b x)^{5/4}}{b^4}+\frac {6 a^2 (a-b x)^{9/4}}{b^4}-\frac {4 a (a-b x)^{13/4}}{b^4}+\frac {(a-b x)^{17/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=-\frac {a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}+\frac {4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac {6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}+\frac {4 a \left (a-b x^4\right )^{17/4}}{17 b^5}-\frac {\left (a-b x^4\right )^{21/4}}{21 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 73, normalized size = 0.69 \begin {gather*} \frac {\sqrt [4]{a-b x^4} \left (-2048 a^5-512 a^4 b x^4-320 a^3 b^2 x^8-240 a^2 b^3 x^{12}-195 a b^4 x^{16}+3315 b^5 x^{20}\right )}{69615 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 59, normalized size = 0.56
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (3315 x^{16} b^{4}+3120 a \,b^{3} x^{12}+2880 a^{2} b^{2} x^{8}+2560 a^{3} b \,x^{4}+2048 a^{4}\right )}{69615 b^{5}}\) | \(59\) |
trager | \(-\frac {\left (-3315 b^{5} x^{20}+195 a \,b^{4} x^{16}+240 a^{2} b^{3} x^{12}+320 a^{3} b^{2} x^{8}+512 a^{4} b \,x^{4}+2048 a^{5}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{69615 b^{5}}\) | \(70\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (-3315 b^{5} x^{20}+195 a \,b^{4} x^{16}+240 a^{2} b^{3} x^{12}+320 a^{3} b^{2} x^{8}+512 a^{4} b \,x^{4}+2048 a^{5}\right )}{69615 b^{5} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 86, normalized size = 0.81 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {21}{4}}}{21 \, b^{5}} + \frac {4 \, {\left (-b x^{4} + a\right )}^{\frac {17}{4}} a}{17 \, b^{5}} - \frac {6 \, {\left (-b x^{4} + a\right )}^{\frac {13}{4}} a^{2}}{13 \, b^{5}} + \frac {4 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}} a^{3}}{9 \, b^{5}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{4}}{5 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 69, normalized size = 0.65 \begin {gather*} \frac {{\left (3315 \, b^{5} x^{20} - 195 \, a b^{4} x^{16} - 240 \, a^{2} b^{3} x^{12} - 320 \, a^{3} b^{2} x^{8} - 512 \, a^{4} b x^{4} - 2048 \, a^{5}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{69615 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.06, size = 134, normalized size = 1.26 \begin {gather*} \begin {cases} - \frac {2048 a^{5} \sqrt [4]{a - b x^{4}}}{69615 b^{5}} - \frac {512 a^{4} x^{4} \sqrt [4]{a - b x^{4}}}{69615 b^{4}} - \frac {64 a^{3} x^{8} \sqrt [4]{a - b x^{4}}}{13923 b^{3}} - \frac {16 a^{2} x^{12} \sqrt [4]{a - b x^{4}}}{4641 b^{2}} - \frac {a x^{16} \sqrt [4]{a - b x^{4}}}{357 b} + \frac {x^{20} \sqrt [4]{a - b x^{4}}}{21} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{20}}{20} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 120, normalized size = 1.13 \begin {gather*} \frac {3315 \, {\left (b x^{4} - a\right )}^{5} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} + 16380 \, {\left (b x^{4} - a\right )}^{4} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a + 32130 \, {\left (b x^{4} - a\right )}^{3} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2} + 30940 \, {\left (b x^{4} - a\right )}^{2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{3} - 13923 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{4}}{69615 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 68, normalized size = 0.64 \begin {gather*} -{\left (a-b\,x^4\right )}^{1/4}\,\left (\frac {2048\,a^5}{69615\,b^5}-\frac {x^{20}}{21}+\frac {a\,x^{16}}{357\,b}+\frac {512\,a^4\,x^4}{69615\,b^4}+\frac {64\,a^3\,x^8}{13923\,b^3}+\frac {16\,a^2\,x^{12}}{4641\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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