Optimal. Leaf size=104 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^5-b+c x^4}}\right )}{2 c^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^5-b+c x^4}}\right )}{2 c^{5/4}}+\frac {x \left (a x^5-b+c x^4\right )^{3/4}}{c \left (b-a x^5\right )} \]
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Rubi [F] time = 2.76, 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 {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \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 {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx &=\int \left (\frac {5 b x^4}{\left (b-a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}}+\frac {x^4}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}}\right ) \, dx\\ &=(5 b) \int \frac {x^4}{\left (b-a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx+\int \frac {x^4}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx\\ &=(5 b) \int \frac {x^4}{\left (b-a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx+\int \left (-\frac {1}{5 a^{4/5} \left (\sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{-1} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 a^{4/5} \left ((-1)^{2/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 a^{4/5} \left (-(-1)^{3/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 a^{4/5} \left ((-1)^{4/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}\right ) \, dx\\ &=-\frac {\int \frac {1}{\left (\sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{\left (-\sqrt [5]{-1} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{\left ((-1)^{2/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{\left (-(-1)^{3/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{\left ((-1)^{4/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}+(5 b) \int \frac {x^4}{\left (b-a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx\\ \end {align*}
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Mathematica [F] time = 2.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.10, size = 104, normalized size = 1.00 \begin {gather*} \frac {x \left (-b+c x^4+a x^5\right )^{3/4}}{c \left (b-a x^5\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{2 c^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{2 c^{5/4}} \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^{5} + 4 \, b\right )} x^{4}}{{\left (a x^{5} + c x^{4} - b\right )}^{\frac {1}{4}} {\left (a x^{5} - b\right )}^{2}}\,{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 \[\int \frac {x^{4} \left (a \,x^{5}+4 b \right )}{\left (a \,x^{5}-b \right )^{2} \left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}\, dx\]
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*} \int \frac {{\left (a x^{5} + 4 \, b\right )} x^{4}}{{\left (a x^{5} + c x^{4} - b\right )}^{\frac {1}{4}} {\left (a x^{5} - b\right )}^{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 = -0.01 \begin {gather*} \int \frac {x^4\,\left (a\,x^5+4\,b\right )}{{\left (b-a\,x^5\right )}^2\,{\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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sympy [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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