Optimal. Leaf size=63 \[ -\text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4 a+b\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{2 \text {$\#$1}^4-a}\& \right ] \]
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Rubi [B] time = 1.17, antiderivative size = 573, normalized size of antiderivative = 9.10, number of steps used = 23, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2056, 911, 105, 63, 331, 298, 203, 206, 93, 205, 208} \begin {gather*} \frac {2 \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt {a^2-4 b} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt {a^2-4 b} \sqrt [4]{a x+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 105
Rule 203
Rule 205
Rule 206
Rule 208
Rule 298
Rule 331
Rule 911
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{b x^3+a x^4}}{x \left (b+a x+x^2\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+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {2 \sqrt [4]{b+a x}}{\sqrt {a^2-4 b} \sqrt [4]{x} \left (a-\sqrt {a^2-4 b}+2 x\right )}-\frac {2 \sqrt [4]{b+a x}}{\sqrt {a^2-4 b} \sqrt [4]{x} \left (a+\sqrt {a^2-4 b}+2 x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\sqrt [4]{b+a x}}{\sqrt [4]{x} \left (a-\sqrt {a^2-4 b}+2 x\right )} \, dx}{\sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\sqrt [4]{b+a x}}{\sqrt [4]{x} \left (a+\sqrt {a^2-4 b}+2 x\right )} \, dx}{\sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a+\sqrt {a^2-4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{\sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^2-a \sqrt {a^2-4 b}-2 b\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a-\sqrt {a^2-4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{\sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (4 \left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2-4 b}-\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 \left (a^2-a \sqrt {a^2-4 b}-2 b\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a-\sqrt {a^2-4 b}-\left (a \left (a-\sqrt {a^2-4 b}\right )-2 b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {\left (2 \sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \sqrt {a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b} \sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {2 \sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
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Mathematica [B] time = 11.53, size = 392, normalized size = 6.22 \begin {gather*} -\frac {\sqrt [4]{x^3 (a x+b)} \left (-\sqrt [4]{a-\sqrt {a^2-4 b}} \log \left (2^{3/4} \sqrt [4]{a-\sqrt {a^2-4 b}}-2 \sqrt [4]{a+\frac {b}{x}}\right )+\sqrt [4]{\sqrt {a^2-4 b}+a} \log \left (2^{3/4} \sqrt [4]{\sqrt {a^2-4 b}+a}-2 \sqrt [4]{a+\frac {b}{x}}\right )+\sqrt [4]{a-\sqrt {a^2-4 b}} \log \left (2^{3/4} \sqrt [4]{a-\sqrt {a^2-4 b}}+2 \sqrt [4]{a+\frac {b}{x}}\right )-\sqrt [4]{\sqrt {a^2-4 b}+a} \log \left (2^{3/4} \sqrt [4]{\sqrt {a^2-4 b}+a}+2 \sqrt [4]{a+\frac {b}{x}}\right )+2 \sqrt [4]{a-\sqrt {a^2-4 b}} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a+\frac {b}{x}}}{\sqrt [4]{a-\sqrt {a^2-4 b}}}\right )-2 \sqrt [4]{\sqrt {a^2-4 b}+a} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a+\frac {b}{x}}}{\sqrt [4]{\sqrt {a^2-4 b}+a}}\right )\right )}{\sqrt [4]{2} x \sqrt {a^2-4 b} \sqrt [4]{a+\frac {b}{x}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 63, normalized size = 1.00 \begin {gather*} -\text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 1918, normalized size = 30.44
result too large to display
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 + x^{2} + b\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x \left (a x +x^{2}+b \right )}\, dx\]
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 + x^{2} + 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 = -0.02 \begin {gather*} \int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x\,\left (x^2+a\,x+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 + b + x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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