Optimal. Leaf size=92 \[ -\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac {128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7} \]
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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270}
\begin {gather*} \frac {128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{3/4}}{x^{20}} \, dx &=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}-\frac {(12 b) \int \frac {\left (a+b x^4\right )^{3/4}}{x^{16}} \, dx}{19 a}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}+\frac {\left (32 b^2\right ) \int \frac {\left (a+b x^4\right )^{3/4}}{x^{12}} \, dx}{95 a^2}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}-\frac {\left (128 b^3\right ) \int \frac {\left (a+b x^4\right )^{3/4}}{x^8} \, dx}{1045 a^3}\\ &=-\frac {\left (a+b x^4\right )^{7/4}}{19 a x^{19}}+\frac {4 b \left (a+b x^4\right )^{7/4}}{95 a^2 x^{15}}-\frac {32 b^2 \left (a+b x^4\right )^{7/4}}{1045 a^3 x^{11}}+\frac {128 b^3 \left (a+b x^4\right )^{7/4}}{7315 a^4 x^7}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 53, normalized size = 0.58 \begin {gather*} \frac {\left (a+b x^4\right )^{7/4} \left (-385 a^3+308 a^2 b x^4-224 a b^2 x^8+128 b^3 x^{12}\right )}{7315 a^4 x^{19}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 50, normalized size = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {7}{4}} \left (-128 b^{3} x^{12}+224 a \,b^{2} x^{8}-308 a^{2} b \,x^{4}+385 a^{3}\right )}{7315 x^{19} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (-128 x^{16} b^{4}+96 a \,b^{3} x^{12}-84 a^{2} b^{2} x^{8}+77 a^{3} b \,x^{4}+385 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{7315 x^{19} a^{4}}\) | \(61\) |
risch | \(-\frac {\left (-128 x^{16} b^{4}+96 a \,b^{3} x^{12}-84 a^{2} b^{2} x^{8}+77 a^{3} b \,x^{4}+385 a^{4}\right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{7315 x^{19} a^{4}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 69, normalized size = 0.75 \begin {gather*} \frac {\frac {1045 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{3}}{x^{7}} - \frac {1995 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} b^{2}}{x^{11}} + \frac {1463 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} b}{x^{15}} - \frac {385 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}}}{x^{19}}}{7315 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 60, normalized size = 0.65 \begin {gather*} \frac {{\left (128 \, b^{4} x^{16} - 96 \, a b^{3} x^{12} + 84 \, a^{2} b^{2} x^{8} - 77 \, a^{3} b x^{4} - 385 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{7315 \, a^{4} x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 847 vs.
\(2 (85) = 170\).
time = 2.23, size = 847, normalized size = 9.21 \begin {gather*} - \frac {1155 a^{7} b^{\frac {39}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {3696 a^{6} b^{\frac {43}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {3906 a^{5} b^{\frac {47}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} - \frac {1380 a^{4} b^{\frac {51}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {45 a^{3} b^{\frac {55}{4}} x^{16} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {540 a^{2} b^{\frac {59}{4}} x^{20} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {864 a b^{\frac {63}{4}} x^{24} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} + \frac {384 b^{\frac {67}{4}} x^{28} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {19}{4}\right )}{256 a^{7} b^{9} x^{16} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{6} b^{10} x^{20} \Gamma \left (- \frac {3}{4}\right ) + 768 a^{5} b^{11} x^{24} \Gamma \left (- \frac {3}{4}\right ) + 256 a^{4} b^{12} x^{28} \Gamma \left (- \frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.81, size = 93, normalized size = 1.01 \begin {gather*} \frac {128\,b^4\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^4\,x^3}-\frac {b\,{\left (b\,x^4+a\right )}^{3/4}}{95\,a\,x^{15}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{19\,x^{19}}-\frac {96\,b^3\,{\left (b\,x^4+a\right )}^{3/4}}{7315\,a^3\,x^7}+\frac {12\,b^2\,{\left (b\,x^4+a\right )}^{3/4}}{1045\,a^2\,x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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