Optimal. Leaf size=67 \[ -\frac {2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{\sqrt [4]{c}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{\sqrt [4]{c}} \]
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Rubi [F]
time = 1.00, 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 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {4 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx &=\int \left (\frac {1}{\sqrt [4]{-b+c x^4+a x^5}}+\frac {5 b}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}}\right ) \, dx\\ &=(5 b) \int \frac {1}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx+\int \frac {1}{\sqrt [4]{-b+c x^4+a x^5}} \, dx\\ &=(5 b) \int \left (-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}+\sqrt [5]{-1} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}-(-1)^{2/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}+(-1)^{3/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 b^{4/5} \left (\sqrt [5]{b}-(-1)^{4/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}\right ) \, dx+\int \frac {1}{\sqrt [4]{-b+c x^4+a x^5}} \, dx\\ &=-\left (\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx\right )-\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}+\sqrt [5]{-1} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx-\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}-(-1)^{2/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx-\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}+(-1)^{3/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx-\sqrt [5]{b} \int \frac {1}{\left (\sqrt [5]{b}-(-1)^{4/5} \sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx+\int \frac {1}{\sqrt [4]{-b+c x^4+a x^5}} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.62, size = 58, normalized size = 0.87 \begin {gather*} -\frac {2 \left (\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )\right )}{\sqrt [4]{c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{5}+4 b}{\left (a \,x^{5}-b \right ) \left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{5} + 4 b}{\left (a x^{5} - b\right ) \sqrt [4]{a x^{5} - b + c x^{4}}}\, dx \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 = -0.01 \begin {gather*} \int -\frac {a\,x^5+4\,b}{\left (b-a\,x^5\right )\,{\left (a\,x^5+c\,x^4-b\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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