Optimal. Leaf size=276 \[ \frac {\left (19712 a^4 b^2-9843 b^5\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{4096 a^{23/4}}-\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{a^{23/4}}+\frac {\left (9843 b^5-19712 a^4 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{4096 a^{23/4}}+\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{a^{23/4}}+\frac {\sqrt [4]{a x^4+b x^3} \left (-15360 a^5 x-65280 a^4 b+6144 a^4 x^4+8064 a^3 b x^3+10400 a^2 b^2 x^2+16420 a b^3 x+32705 b^4\right )}{30720 a^5} \]
________________________________________________________________________________________
Rubi [C] time = 1.76, antiderivative size = 707, normalized size of antiderivative = 2.56, number of steps used = 41, number of rules used = 11, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {2056, 6733, 6725, 279, 331, 298, 203, 206, 321, 511, 510} \begin {gather*} -\frac {371 b^5 \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {371 b^5 \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {371 b^4 \sqrt [4]{a x^4+b x^3}}{6144 a^5}+\frac {53 b^3 x \sqrt [4]{a x^4+b x^3}}{1536 a^4}-\frac {1}{2} x \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{a x^4+b x^3}+\frac {65 b^2 x^2 \sqrt [4]{a x^4+b x^3}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{a x^4+b x^3}}{80 a^2}-\frac {3 b^2 \left (a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {3 b^2 \left (a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{\frac {a x}{b}+1}}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{a x^4+b x^3}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3}}{a^5}+\frac {x^4 \sqrt [4]{a x^4+b x^3}}{5 a} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 203
Rule 206
Rule 279
Rule 298
Rule 321
Rule 331
Rule 510
Rule 511
Rule 2056
Rule 6725
Rule 6733
Rubi steps
\begin {align*} \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x} \left (-b-a x+x^4\right )}{-b+a x} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{b+a x^4} \left (-b-a x^4+x^{16}\right )}{-b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {b \left (2 a^4-b^3\right ) x^2 \sqrt [4]{b+a x^4}}{a^5}-\left (1-\frac {b^3}{a^4}\right ) x^6 \sqrt [4]{b+a x^4}+\frac {b^2 x^{10} \sqrt [4]{b+a x^4}}{a^3}+\frac {b x^{14} \sqrt [4]{b+a x^4}}{a^2}+\frac {x^{18} \sqrt [4]{b+a x^4}}{a}-\frac {\left (2 a^4 b^2-b^5\right ) x^2 \sqrt [4]{b+a x^4}}{a^5 \left (-b+a x^4\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^{18} \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^{14} \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 b^2 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^{10} \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^3 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^2 \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{-b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^6 \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {b^2 x^2 \sqrt [4]{b x^3+a x^4}}{3 a^3}+\frac {b x^3 \sqrt [4]{b x^3+a x^4}}{4 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {\left (b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{18}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{5 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b^2 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{4 a^2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b^3 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{3 a^3 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{-b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^5 x^{3/4} \sqrt [4]{1+\frac {a x}{b}}}\\ &=-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {b^3 x \sqrt [4]{b x^3+a x^4}}{24 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {17 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{48 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}-\frac {\left (3 b^2 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{16 a^2 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (11 b^3 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{48 a^3 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (7 b^4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{24 a^4 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (3 b^2 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{8 a x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {7 b^4 \sqrt [4]{b x^3+a x^4}}{96 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {5 b^3 x \sqrt [4]{b x^3+a x^4}}{384 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}+\frac {\left (11 b^3 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{64 a^3 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (77 b^4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{384 a^4 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (7 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (3 b^2 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {35 b^4 \sqrt [4]{b x^3+a x^4}}{1536 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (77 b^4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 a^4 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (77 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 a^5 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (7 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{32 a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (3 b^2 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{3/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (3 b^2 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{3/2} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {371 b^4 \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (7 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (7 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (231 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (77 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{512 a^5 x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {371 b^4 \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}-\frac {7 b^5 \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {7 b^5 \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (77 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (77 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (231 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {371 b^4 \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}-\frac {35 b^5 \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {35 b^5 \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (231 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (231 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {371 b^4 \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}-\frac {371 b^5 \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {371 b^5 \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.75, size = 232, normalized size = 0.84 \begin {gather*} \frac {x^3 \left (35 b^2 \left (13056 a^4-6541 b^3\right ) \left (\frac {a x}{b}+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {2 a x}{b-a x}\right )+15 a b x \left (19712 a^4-9843 b^3\right ) \left (1-\frac {a^2 x^2}{b^2}\right )^{3/4} F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};-\frac {a x}{b},\frac {a x}{b}\right )+7 \left (1-\frac {a x}{b}\right )^{3/4} \left (-15360 a^6 x^2+768 a^5 \left (8 x^5-105 b x\right )-384 a^4 b \left (170 b-37 x^4\right )+18464 a^3 b^2 x^3+26820 a^2 b^3 x^2+49125 a b^4 x+32705 b^5\right )\right )}{215040 a^5 \left (x^3 (a x+b)\right )^{3/4} \left (1-\frac {a x}{b}\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.38, size = 276, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{b x^3+a x^4} \left (-65280 a^4 b+32705 b^4-15360 a^5 x+16420 a b^3 x+10400 a^2 b^2 x^2+8064 a^3 b x^3+6144 a^4 x^4\right )}{30720 a^5}+\frac {\left (19712 a^4 b^2-9843 b^5\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}-\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{23/4}}+\frac {\left (-19712 a^4 b^2+9843 b^5\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}+\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{23/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.71, size = 1239, normalized size = 4.49
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.33, size = 704, normalized size = 2.55
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-a x -b \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a x -b}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - a x - b\right )}}{a x - b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}\,\left (-x^4+a\,x+b\right )}{b-a\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (- a x - b + x^{4}\right )}{a x - b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________