Optimal. Leaf size=82 \[ \frac {1}{3} \text {RootSum}\left [-\text {$\#$1}^{12}+3 \text {$\#$1}^8 a-3 \text {$\#$1}^4 a^2+a^3+a b^2\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^4-a}\& \right ] \]
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Rubi [B] time = 1.66, antiderivative size = 1101, normalized size of antiderivative = 13.43, number of steps used = 33, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2056, 6725, 105, 63, 331, 298, 203, 206, 93} \begin {gather*} -\frac {2 (-1)^{2/3} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 (-1)^{2/3} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 105
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx &=\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (b+a x^3\right )} \, dx}{x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\sqrt [4]{-b x^3+a x^4} \int \left (-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )}-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )}-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{-b+a x}}\\ &=-\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (a^{2/3} \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\sqrt [3]{-1} a^{2/3} \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left ((-1)^{2/3} a^{2/3} \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) (-b+a x)^{3/4}} \, dx}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\left (a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) (-b+a x)^{3/4}} \, dx}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) (-b+a x)^{3/4}} \, dx}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (4 a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (4 \sqrt [3]{-1} a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 (-1)^{2/3} a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (4 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-\sqrt [3]{b}-\left (-a \sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 \left (a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-\sqrt [3]{b}-\left (-a \sqrt [3]{b}-\sqrt [3]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-\sqrt [3]{b}-\left (-a \sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (4 a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (4 \sqrt [3]{-1} a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 (-1)^{2/3} a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \sqrt {a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [6]{a} \sqrt {a^{2/3}+b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \sqrt {a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [6]{a} \sqrt {a^{2/3}+b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [6]{a} \sqrt {a^{2/3}-\sqrt [3]{-1} b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} \sqrt {a^{2/3}-\sqrt [3]{-1} b^{2/3}} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [6]{a} \sqrt {a^{2/3}-\sqrt [3]{-1} b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} \sqrt {a^{2/3}-\sqrt [3]{-1} b^{2/3}} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [6]{a} \sqrt {a^{2/3}+(-1)^{2/3} b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} \sqrt {a^{2/3}+(-1)^{2/3} b^{2/3}} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [6]{a} \sqrt {a^{2/3}+(-1)^{2/3} b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} \sqrt {a^{2/3}+(-1)^{2/3} b^{2/3}} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \sqrt [3]{-1} \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \sqrt [3]{-1} \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 (-1)^{2/3} \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 (-1)^{2/3} \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}\\ &=-\frac {2 \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 (-1)^{2/3} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 (-1)^{2/3} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}\\ \end {align*}
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Mathematica [B] time = 0.37, size = 194, normalized size = 2.37 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x-b)} \left (\left (a^{2/3}+b^{2/3}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+\sqrt [3]{a} b^{2/3} x}{a x-b}\right )+\left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\sqrt [3]{-1} \sqrt [3]{a} b^{2/3} x-a x}{b-a x}\right )+\left (b^{2/3}-\sqrt [3]{-1} a^{2/3}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+(-1)^{2/3} \sqrt [3]{a} b^{2/3} x}{a x-b}\right )\right )}{9 b^{2/3} (b-a x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.73, size = 82, normalized size = 1.00 \begin {gather*} \frac {1}{3} \text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^4+3 a \text {$\#$1}^8-\text {$\#$1}^{12}\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\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] 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 (a x^{3} + b\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}}}{x \left (a \,x^{3}+b \right )}\, dx\]
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}}}{{\left (a x^{3} + b\right )} x}\,{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 = -0.01 \begin {gather*} \int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x\,\left (a\,x^3+b\right )} \,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 {\sqrt [4]{x^{3} \left (a x - b\right )}}{x \left (a x^{3} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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