Optimal. Leaf size=215 \[ a \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+a \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-b \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-a \log (x)+b \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]+\frac {4 \sqrt [4]{a x^4+b x^3}}{x}-2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )+2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right ) \]
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Rubi [B] time = 2.20, antiderivative size = 491, normalized size of antiderivative = 2.28, number of steps used = 23, number of rules used = 13, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2056, 6725, 47, 63, 331, 298, 203, 206, 908, 37, 6688, 12, 93} \begin {gather*} \frac {4 \sqrt [4]{a x^4+b x^3}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 47
Rule 63
Rule 93
Rule 203
Rule 206
Rule 298
Rule 331
Rule 908
Rule 2056
Rule 6688
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {\sqrt [4]{b+a x} \left (b+a x^2\right )}{x^{5/4} \left (-b+a x^2\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}}{x^{5/4}}+\frac {2 b \sqrt [4]{b+a x}}{x^{5/4} \left (-b+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 {\sqrt [4]{b+a x}}{x^{5/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\sqrt [4]{b+a x}}{x^{5/4} \left (-b+a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {-a b-a b x}{\sqrt [4]{x} (b+a x)^{3/4} \left (-b+a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{x^{5/4} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {a b (1+x)}{\sqrt [4]{x} (b+a x)^{3/4} \left (b-a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 a \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 {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (4 a \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 (2 a b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1+x}{\sqrt [4]{x} (b+a x)^{3/4} \left (b-a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (2 \sqrt {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 )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \sqrt {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 )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a b \sqrt [4]{b x^3+a x^4}\right ) \int \left (\frac {\sqrt {b}+\frac {b}{\sqrt {a}}}{2 b \sqrt [4]{x} \left (\sqrt {b}-\sqrt {a} x\right ) (b+a x)^{3/4}}+\frac {\sqrt {b}-\frac {b}{\sqrt {a}}}{2 b \sqrt [4]{x} \left (\sqrt {b}+\sqrt {a} x\right ) (b+a x)^{3/4}}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt {b}+\sqrt {a} x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt {b}-\sqrt {a} x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 a \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 a \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a^{3/4} \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}-\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a^{3/4} \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}-\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a^{3/4} \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}+\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a^{3/4} \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}+\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 169, normalized size = 0.79 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x+b)} \left (\frac {\left (\sqrt {a} \sqrt {b} x-a x\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x-\sqrt {a} \sqrt {b} x}{b+a x}\right )-\left (\sqrt {a} \sqrt {b} x+a x\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+\sqrt {a} \sqrt {b} x}{b+a x}\right )+6 (a x+b)}{a x+b}-\frac {3 \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-\frac {a x}{b}\right )}{\sqrt [4]{\frac {a x}{b}+1}}\right )}{3 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.81, size = 214, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+a \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-b \log (x)-a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 567, normalized size = 2.64 \begin {gather*} \frac {4 \, {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \arctan \left (\frac {{\left (a x - \sqrt {a b} x\right )} {\left (a + \sqrt {a b}\right )}^{\frac {3}{4}} \sqrt {\frac {\sqrt {a + \sqrt {a b}} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a + \sqrt {a b}\right )}^{\frac {3}{4}} {\left (a - \sqrt {a b}\right )}}{{\left (a^{2} - a b\right )} x}\right ) - 4 \, {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {{\left (a x + \sqrt {a b} x\right )} {\left (a - \sqrt {a b}\right )}^{\frac {3}{4}} \sqrt {\frac {\sqrt {a - \sqrt {a b}} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a + \sqrt {a b}\right )} {\left (a - \sqrt {a b}\right )}^{\frac {3}{4}}}{{\left (a^{2} - a b\right )} x}\right ) - 4 \, a^{\frac {1}{4}} x \arctan \left (\frac {a^{\frac {3}{4}} x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{\frac {3}{4}}}{a x}\right ) - {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + a^{\frac {1}{4}} x \log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - a^{\frac {1}{4}} x \log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x} \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^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x^{2} \left (a \,x^{2}-b \right )}\, dx\]
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^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}}\,{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 {\left (a\,x^2+b\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^2\,\left (b-a\,x^2\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 )} \left (a x^{2} + b\right )}{x^{2} \left (a x^{2} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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