Optimal. Leaf size=221 \[ -\frac {2 \sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt [4]{b}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt [4]{b}}\right )}{\sqrt [4]{d}} \]
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Rubi [F]
time = 2.48, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx &=\frac {\left (\sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}} \sqrt [4]{-c+c x+d x^4}\right ) \int \frac {(-4+3 x) \sqrt [4]{-a+a x+b x^4}}{(-1+x) x \sqrt [4]{-c+c x+d x^4}} \, dx}{\sqrt [4]{-a+a x+b x^4}}\\ &=\frac {\left (\sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}} \sqrt [4]{-c+c x+d x^4}\right ) \int \left (\frac {\sqrt [4]{-a+a x+b x^4}}{(1-x) \sqrt [4]{-c+c x+d x^4}}+\frac {4 \sqrt [4]{-a+a x+b x^4}}{x \sqrt [4]{-c+c x+d x^4}}\right ) \, dx}{\sqrt [4]{-a+a x+b x^4}}\\ &=\frac {\left (\sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}} \sqrt [4]{-c+c x+d x^4}\right ) \int \frac {\sqrt [4]{-a+a x+b x^4}}{(1-x) \sqrt [4]{-c+c x+d x^4}} \, dx}{\sqrt [4]{-a+a x+b x^4}}+\frac {\left (4 \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}} \sqrt [4]{-c+c x+d x^4}\right ) \int \frac {\sqrt [4]{-a+a x+b x^4}}{x \sqrt [4]{-c+c x+d x^4}} \, dx}{\sqrt [4]{-a+a x+b x^4}}\\ \end {align*}
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Mathematica [F]
time = 3.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (-4+3 x \right ) \left (\frac {b \,x^{4}+a x -a}{d \,x^{4}+c x -c}\right )^{\frac {1}{4}}}{\left (-1+x \right ) x}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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 (3\,x-4\right )\,{\left (\frac {b\,x^4+a\,x-a}{d\,x^4+c\,x-c}\right )}^{1/4}}{x\,\left (x-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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