Optimal. Leaf size=323 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12} a+6 \text {$\#$1}^8 a^2-4 \text {$\#$1}^4 a^3+a^4-a b^3\& ,\frac {-\text {$\#$1}^{12} \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^{12} \log (x)+3 \text {$\#$1}^8 a \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-3 \text {$\#$1}^8 a \log (x)-3 \text {$\#$1}^4 a^2 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+3 \text {$\#$1}^4 a^2 \log (x)+a^3 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-b^3 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-a^3 \log (x)+b^3 \log (x)}{-\text {$\#$1}^{15}+3 \text {$\#$1}^{11} a-3 \text {$\#$1}^7 a^2+\text {$\#$1}^3 a^3}\& \right ]-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{a^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{a^{3/4}} \]
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Rubi [B] time = 3.50, antiderivative size = 875, normalized size of antiderivative = 2.71, number of steps used = 53, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2056, 6725, 904, 50, 63, 331, 298, 203, 206, 21, 906, 93} \begin {gather*} -\frac {2 \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 63
Rule 93
Rule 203
Rule 206
Rule 298
Rule 331
Rule 904
Rule 906
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{11/4} \sqrt [4]{b+a x}}{-b+a x^4} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \left (-\frac {x^{11/4} \sqrt [4]{b+a x}}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}-\frac {x^{11/4} \sqrt [4]{b+a x}}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{11/4} \sqrt [4]{b+a x}}{\sqrt {b}-\sqrt {a} x^2} \, dx}{2 \sqrt {b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{11/4} \sqrt [4]{b+a x}}{\sqrt {b}+\sqrt {a} x^2} \, dx}{2 \sqrt {b} x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \left (-b^{3/2}-a \sqrt {b} x\right )}{(b+a x)^{3/4} \left (\sqrt {b}-\sqrt {a} x^2\right )} \, dx}{2 \sqrt {a} \sqrt {b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \left (-b^{3/2}-a \sqrt {b} x\right )}{(b+a x)^{3/4} \left (\sqrt {b}+\sqrt {a} x^2\right )} \, dx}{2 \sqrt {a} \sqrt {b} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x}}{\sqrt {b}-\sqrt {a} x^2} \, dx}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x}}{\sqrt {b}+\sqrt {a} x^2} \, dx}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \frac {\sqrt [4]{b x^3+a x^4} \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {-a \sqrt {b}-\sqrt {a} b x}{\sqrt [4]{x} (b+a x)^{3/4} \left (\sqrt {b}-\sqrt {a} x^2\right )} \, dx}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {-a \sqrt {b}+\sqrt {a} b x}{\sqrt [4]{x} (b+a x)^{3/4} \left (\sqrt {b}+\sqrt {a} x^2\right )} \, dx}{2 a x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \frac {\left (2 \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 )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {-a b^{3/4}+\frac {\sqrt {a} b^{3/2}}{\sqrt {-\sqrt {a}}}}{2 \sqrt {b} \sqrt [4]{x} \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right ) (b+a x)^{3/4}}+\frac {-a b^{3/4}-\frac {\sqrt {a} b^{3/2}}{\sqrt {-\sqrt {a}}}}{2 \sqrt {b} \sqrt [4]{x} \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right ) (b+a x)^{3/4}}\right ) \, dx}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {-a b^{3/4}-\sqrt [4]{a} b^{3/2}}{2 \sqrt {b} \sqrt [4]{x} \left (\sqrt [4]{b}-\sqrt [4]{a} x\right ) (b+a x)^{3/4}}+\frac {-a b^{3/4}+\sqrt [4]{a} b^{3/2}}{2 \sqrt {b} \sqrt [4]{x} \left (\sqrt [4]{b}+\sqrt [4]{a} x\right ) (b+a x)^{3/4}}\right ) \, dx}{2 a x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \frac {\left (2 \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 )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^{3/4}-b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{b}+\sqrt [4]{a} x\right ) (b+a x)^{3/4}} \, dx}{4 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^{3/4}+b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{b}-\sqrt [4]{a} x\right ) (b+a x)^{3/4}} \, dx}{4 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\sqrt {-\sqrt {a}} b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right ) (b+a x)^{3/4}} \, dx}{4 a x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\sqrt {-\sqrt {a}} b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right ) (b+a x)^{3/4}} \, dx}{4 a x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \left (\frac {\sqrt [4]{b x^3+a x^4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{b x^3+a x^4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a} x^{3/4} \sqrt [4]{b+a x}}\right )-\frac {\left (\left (a^{3/4}-b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b}-\left (a \sqrt [4]{b}-\sqrt [4]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^{3/4}+b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b}-\left (a \sqrt [4]{b}+\sqrt [4]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\sqrt {-\sqrt {a}} b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b}-\left (a \sqrt [4]{b}-\sqrt {-\sqrt {a}} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\sqrt {-\sqrt {a}} b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b}-\left (a \sqrt [4]{b}+\sqrt {-\sqrt {a}} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \left (-\frac {\sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}\right )-\frac {\left (\sqrt {a^{3/4}-b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [8]{a} \sqrt {a^{3/4}-b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{7/8} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\sqrt {a^{3/4}-b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [8]{a} \sqrt {a^{3/4}-b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{7/8} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a^{3/4}+b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [8]{a} \sqrt {a^{3/4}+b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{7/8} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\sqrt {a^{3/4}+b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [8]{a} \sqrt {a^{3/4}+b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{7/8} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-\sqrt {a}} b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\sqrt {a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-\sqrt {a}} b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-\sqrt {a}} b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\sqrt {a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-\sqrt {a}} b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+2 \left (-\frac {\sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}\right )-\frac {\sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
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Mathematica [C] time = 0.54, size = 359, normalized size = 1.11 \begin {gather*} \frac {x^3 \left (\left (b^{3/4}-a^{3/4}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x-\sqrt [4]{a} b^{3/4} x}{b+a x}\right )-a^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x-i \sqrt [4]{a} b^{3/4} x}{b+a x}\right )-a^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i \sqrt [4]{a} b^{3/4} x}{b+a x}\right )-a^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+\sqrt [4]{a} b^{3/4} x}{b+a x}\right )+4 a^{3/4} \left (\frac {a x}{b}+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {a x}{b}\right )+i b^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x-i \sqrt [4]{a} b^{3/4} x}{b+a x}\right )-i b^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i \sqrt [4]{a} b^{3/4} x}{b+a x}\right )-b^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+\sqrt [4]{a} b^{3/4} x}{b+a x}\right )\right )}{3 a^{3/4} \left (x^3 (a x+b)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.59, size = 322, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{3/4}}+\frac {1}{4} \text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {a^3 \log (x)-b^3 \log (x)-a^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+b^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-3 a^2 \log (x) \text {$\#$1}^4+3 a^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+3 a \log (x) \text {$\#$1}^8-3 a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^8-\log (x) \text {$\#$1}^{12}+\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^{12}}{-a^3 \text {$\#$1}^3+3 a^2 \text {$\#$1}^7-3 a \text {$\#$1}^{11}+\text {$\#$1}^{15}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a \,x^{4}-b}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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}} x^{2}}{a x^{4} - b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^2\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{b-a\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sqrt [4]{x^{3} \left (a x + b\right )}}{a x^{4} - b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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