20.44 Problem number 185

\[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx \]

Optimal antiderivative \[ \frac {8388608 b^{12} \sqrt {b \,x^{\frac {2}{3}}+a x}}{11700675 a^{13}}-\frac {16777216 b^{13} \sqrt {b \,x^{\frac {2}{3}}+a x}}{11700675 a^{14} x^{\frac {1}{3}}}-\frac {2097152 b^{11} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2340135 a^{11}}-\frac {131072 b^{9} x \sqrt {b \,x^{\frac {2}{3}}+a x}}{334305 a^{10}}+\frac {65536 b^{8} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{185725 a^{9}}-\frac {180224 b^{7} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{557175 a^{8}}+\frac {1171456 b^{6} x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}}{3900225 a^{7}}-\frac {73216 b^{5} x^{\frac {7}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{260015 a^{6}}+\frac {36608 b^{4} x^{\frac {8}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{137655 a^{5}}-\frac {9152 b^{3} x^{3} \sqrt {b \,x^{\frac {2}{3}}+a x}}{36225 a^{4}}+\frac {416 b^{2} x^{\frac {10}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{1725 a^{3}}-\frac {52 b \,x^{\frac {11}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{225 a^{2}}+\frac {2 x^{4} \sqrt {b \,x^{\frac {2}{3}}+a x}}{9 a} \]

command

integrate(x^4/(b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ \frac {{\left (211106232532992 \, b^{19} + 43980465111040 \, b^{18} + 206158430208 \, {\left (64 \, a^{3} - 3\right )} b^{16} - 4123168604160 \, b^{17} - 1073741824 \, {\left (11264 \, a^{3} - 53\right )} b^{15} + 15143273600 \, a^{15} - 402653184 \, {\left (5504 \, a^{3} + 1\right )} b^{14} + 12582912 \, {\left (3194880 \, a^{6} - 114688 \, a^{3} - 3\right )} b^{13} + 469762048 \, {\left (18816 \, a^{6} + 103 \, a^{3}\right )} b^{12} - 50331648 \, {\left (48816 \, a^{6} + 23 \, a^{3}\right )} b^{11} - 786432 \, {\left (45731840 \, a^{9} - 495872 \, a^{6} - 15 \, a^{3}\right )} b^{10} - 7340032 \, {\left (1349120 \, a^{9} + 3439 \, a^{6}\right )} b^{9} + 250478592 \, {\left (5600 \, a^{9} + 3 \, a^{6}\right )} b^{8} + 12288 \, {\left (2616979456 \, a^{12} - 21542400 \, a^{9} - 693 \, a^{6}\right )} b^{7} + 212992 \, {\left (43743616 \, a^{12} + 89111 \, a^{9}\right )} b^{6} - 638976 \, {\left (1652476 \, a^{12} + 935 \, a^{9}\right )} b^{5} + 3264 \, {\left (3608543232 \, a^{15} + 64599808 \, a^{12} + 2145 \, a^{9}\right )} b^{4} + 578816 \, {\left (13049856 \, a^{15} - 27313 \, a^{12}\right )} b^{3} + 217056 \, {\left (6211584 \, a^{15} + 2353 \, a^{12}\right )} b^{2} - 156009 \, {\left (2547712 \, a^{15} + 39 \, a^{12}\right )} b\right )} x + 2 \, {\left (1300075 \, {\left (16777216 \, a^{13} b^{6} + 6291456 \, a^{13} b^{5} + 196608 \, a^{13} b^{4} - 262144 \, a^{16} - 114688 \, a^{13} b^{3} - 2304 \, a^{13} b^{2} + 864 \, a^{13} b - 27 \, a^{13}\right )} x^{5} - 1478048 \, {\left (16777216 \, a^{10} b^{9} + 6291456 \, a^{10} b^{8} + 196608 \, a^{10} b^{7} - 114688 \, a^{10} b^{6} - 2304 \, a^{10} b^{5} + 864 \, a^{10} b^{4} - {\left (262144 \, a^{13} + 27 \, a^{10}\right )} b^{3}\right )} x^{4} + 1757184 \, {\left (16777216 \, a^{7} b^{12} + 6291456 \, a^{7} b^{11} + 196608 \, a^{7} b^{10} - 114688 \, a^{7} b^{9} - 2304 \, a^{7} b^{8} + 864 \, a^{7} b^{7} - {\left (262144 \, a^{10} + 27 \, a^{7}\right )} b^{6}\right )} x^{3} - 2293760 \, {\left (16777216 \, a^{4} b^{15} + 6291456 \, a^{4} b^{14} + 196608 \, a^{4} b^{13} - 114688 \, a^{4} b^{12} - 2304 \, a^{4} b^{11} + 864 \, a^{4} b^{10} - {\left (262144 \, a^{7} + 27 \, a^{4}\right )} b^{9}\right )} x^{2} + 4194304 \, {\left (16777216 \, a b^{18} + 6291456 \, a b^{17} + 196608 \, a b^{16} - 114688 \, a b^{15} - 2304 \, a b^{14} + 864 \, a b^{13} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{12}\right )} x - 2 \, {\left (70368744177664 \, b^{19} + 26388279066624 \, b^{18} + 824633720832 \, b^{17} - 481036337152 \, b^{16} - 9663676416 \, b^{15} - 4194304 \, {\left (262144 \, a^{3} + 27\right )} b^{13} + 3623878656 \, b^{14} + 676039 \, {\left (16777216 \, a^{12} b^{7} + 6291456 \, a^{12} b^{6} + 196608 \, a^{12} b^{5} - 114688 \, a^{12} b^{4} - 2304 \, a^{12} b^{3} + 864 \, a^{12} b^{2} - {\left (262144 \, a^{15} + 27 \, a^{12}\right )} b\right )} x^{4} - 777920 \, {\left (16777216 \, a^{9} b^{10} + 6291456 \, a^{9} b^{9} + 196608 \, a^{9} b^{8} - 114688 \, a^{9} b^{7} - 2304 \, a^{9} b^{6} + 864 \, a^{9} b^{5} - {\left (262144 \, a^{12} + 27 \, a^{9}\right )} b^{4}\right )} x^{3} + 946176 \, {\left (16777216 \, a^{6} b^{13} + 6291456 \, a^{6} b^{12} + 196608 \, a^{6} b^{11} - 114688 \, a^{6} b^{10} - 2304 \, a^{6} b^{9} + 864 \, a^{6} b^{8} - {\left (262144 \, a^{9} + 27 \, a^{6}\right )} b^{7}\right )} x^{2} - 1310720 \, {\left (16777216 \, a^{3} b^{16} + 6291456 \, a^{3} b^{15} + 196608 \, a^{3} b^{14} - 114688 \, a^{3} b^{13} - 2304 \, a^{3} b^{12} + 864 \, a^{3} b^{11} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b^{10}\right )} x\right )} x^{\frac {2}{3}} + 48 \, {\left (29393 \, {\left (16777216 \, a^{11} b^{8} + 6291456 \, a^{11} b^{7} + 196608 \, a^{11} b^{6} - 114688 \, a^{11} b^{5} - 2304 \, a^{11} b^{4} + 864 \, a^{11} b^{3} - {\left (262144 \, a^{14} + 27 \, a^{11}\right )} b^{2}\right )} x^{4} - 34320 \, {\left (16777216 \, a^{8} b^{11} + 6291456 \, a^{8} b^{10} + 196608 \, a^{8} b^{9} - 114688 \, a^{8} b^{8} - 2304 \, a^{8} b^{7} + 864 \, a^{8} b^{6} - {\left (262144 \, a^{11} + 27 \, a^{8}\right )} b^{5}\right )} x^{3} + 43008 \, {\left (16777216 \, a^{5} b^{14} + 6291456 \, a^{5} b^{13} + 196608 \, a^{5} b^{12} - 114688 \, a^{5} b^{11} - 2304 \, a^{5} b^{10} + 864 \, a^{5} b^{9} - {\left (262144 \, a^{8} + 27 \, a^{5}\right )} b^{8}\right )} x^{2} - 65536 \, {\left (16777216 \, a^{2} b^{17} + 6291456 \, a^{2} b^{16} + 196608 \, a^{2} b^{15} - 114688 \, a^{2} b^{14} - 2304 \, a^{2} b^{13} + 864 \, a^{2} b^{12} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{11}\right )} x\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}}{11700675 \, {\left (16777216 \, a^{14} b^{6} + 6291456 \, a^{14} b^{5} + 196608 \, a^{14} b^{4} - 262144 \, a^{17} - 114688 \, a^{14} b^{3} - 2304 \, a^{14} b^{2} + 864 \, a^{14} b - 27 \, a^{14}\right )} x} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ \text {Timed out} \]