Optimal. Leaf size=99 \[ -\frac {a^4}{b^5 \sqrt [4]{a+b x^4}}-\frac {4 a^3 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac {4 a \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac {\left (a+b x^4\right )^{15/4}}{15 b^5} \]
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Rubi [A] time = 0.05, antiderivative size = 99, 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} \[ -\frac {a^4}{b^5 \sqrt [4]{a+b x^4}}-\frac {4 a^3 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac {4 a \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac {\left (a+b x^4\right )^{15/4}}{15 b^5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{19}}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x^4}{(a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {a^4}{b^4 (a+b x)^{5/4}}-\frac {4 a^3}{b^4 \sqrt [4]{a+b x}}+\frac {6 a^2 (a+b x)^{3/4}}{b^4}-\frac {4 a (a+b x)^{7/4}}{b^4}+\frac {(a+b x)^{11/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=-\frac {a^4}{b^5 \sqrt [4]{a+b x^4}}-\frac {4 a^3 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac {4 a \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac {\left (a+b x^4\right )^{15/4}}{15 b^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.62 \[ \frac {-2048 a^4-512 a^3 b x^4+192 a^2 b^2 x^8-112 a b^3 x^{12}+77 b^4 x^{16}}{1155 b^5 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 69, normalized size = 0.70 \[ \frac {{\left (77 \, b^{4} x^{16} - 112 \, a b^{3} x^{12} + 192 \, a^{2} b^{2} x^{8} - 512 \, a^{3} b x^{4} - 2048 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1155 \, {\left (b^{6} x^{4} + a b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 87, normalized size = 0.88 \[ -\frac {a^{4}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{5}} + \frac {77 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} b^{70} - 420 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a b^{70} + 990 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2} b^{70} - 1540 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{3} b^{70}}{1155 \, b^{75}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.59 \[ -\frac {-77 x^{16} b^{4}+112 a \,x^{12} b^{3}-192 a^{2} x^{8} b^{2}+512 a^{3} x^{4} b +2048 a^{4}}{1155 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 81, normalized size = 0.82 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {15}{4}}}{15 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a}{11 \, b^{5}} + \frac {6 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2}}{7 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{3}}{3 \, b^{5}} - \frac {a^{4}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 69, normalized size = 0.70 \[ -\frac {420\,a\,{\left (b\,x^4+a\right )}^3+1540\,a^3\,\left (b\,x^4+a\right )-77\,{\left (b\,x^4+a\right )}^4+1155\,a^4-990\,a^2\,{\left (b\,x^4+a\right )}^2}{1155\,b^5\,{\left (b\,x^4+a\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.12, size = 116, normalized size = 1.17 \[ \begin {cases} - \frac {2048 a^{4}}{1155 b^{5} \sqrt [4]{a + b x^{4}}} - \frac {512 a^{3} x^{4}}{1155 b^{4} \sqrt [4]{a + b x^{4}}} + \frac {64 a^{2} x^{8}}{385 b^{3} \sqrt [4]{a + b x^{4}}} - \frac {16 a x^{12}}{165 b^{2} \sqrt [4]{a + b x^{4}}} + \frac {x^{16}}{15 b \sqrt [4]{a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{20}}{20 a^{\frac {5}{4}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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