Optimal. Leaf size=101 \[ \frac {a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac {\left (a+b x^4\right )^{19/4}}{19 b^5}-\frac {4 a \left (a+b x^4\right )^{15/4}}{15 b^5} \]
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Rubi [A] time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.00, 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} \[ \frac {6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac {a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac {\left (a+b x^4\right )^{19/4}}{19 b^5}-\frac {4 a \left (a+b x^4\right )^{15/4}}{15 b^5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{19}}{\sqrt [4]{a+b x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {a^4}{b^4 \sqrt [4]{a+b x}}-\frac {4 a^3 (a+b x)^{3/4}}{b^4}+\frac {6 a^2 (a+b x)^{7/4}}{b^4}-\frac {4 a (a+b x)^{11/4}}{b^4}+\frac {(a+b x)^{15/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac {a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}-\frac {4 a \left (a+b x^4\right )^{15/4}}{15 b^5}+\frac {\left (a+b x^4\right )^{19/4}}{19 b^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.60 \[ \frac {\left (a+b x^4\right )^{3/4} \left (2048 a^4-1536 a^3 b x^4+1344 a^2 b^2 x^8-1232 a b^3 x^{12}+1155 b^4 x^{16}\right )}{21945 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 57, normalized size = 0.56 \[ \frac {{\left (1155 \, b^{4} x^{16} - 1232 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} - 1536 \, a^{3} b x^{4} + 2048 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{21945 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 71, normalized size = 0.70 \[ \frac {1155 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}} - 5852 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} a + 11970 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{2} - 12540 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{3} + 7315 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{4}}{21945 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.57 \[ \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (1155 x^{16} b^{4}-1232 a \,x^{12} b^{3}+1344 a^{2} x^{8} b^{2}-1536 a^{3} x^{4} b +2048 a^{4}\right )}{21945 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 81, normalized size = 0.80 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {19}{4}}}{19 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} a}{15 \, b^{5}} + \frac {6 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{2}}{11 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{3}}{7 \, b^{5}} + \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{4}}{3 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 58, normalized size = 0.57 \[ {\left (b\,x^4+a\right )}^{3/4}\,\left (\frac {2048\,a^4}{21945\,b^5}+\frac {x^{16}}{19\,b}-\frac {16\,a\,x^{12}}{285\,b^2}-\frac {512\,a^3\,x^4}{7315\,b^4}+\frac {64\,a^2\,x^8}{1045\,b^3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.29, size = 116, normalized size = 1.15 \[ \begin {cases} \frac {2048 a^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{21945 b^{5}} - \frac {512 a^{3} x^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{7315 b^{4}} + \frac {64 a^{2} x^{8} \left (a + b x^{4}\right )^{\frac {3}{4}}}{1045 b^{3}} - \frac {16 a x^{12} \left (a + b x^{4}\right )^{\frac {3}{4}}}{285 b^{2}} + \frac {x^{16} \left (a + b x^{4}\right )^{\frac {3}{4}}}{19 b} & \text {for}\: b \neq 0 \\\frac {x^{20}}{20 \sqrt [4]{a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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