Optimal. Leaf size=152 \[ \sqrt {2} c^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^5-b}}{\sqrt {a x^5-b}-\sqrt {c} x^2}\right )+\sqrt {2} c^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5-b}}{\sqrt {2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^5-b}}\right )+\frac {4 \left (a x^5-b\right )^{3/4}}{3 x^3} \]
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Rubi [F] time = 1.23, 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 {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx &=\int \left (-\frac {4 \left (-b+a x^5\right )^{3/4}}{x^4}+\frac {(4 c+5 a x) \left (-b+a x^5\right )^{3/4}}{-b+c x^4+a x^5}\right ) \, dx\\ &=-\left (4 \int \frac {\left (-b+a x^5\right )^{3/4}}{x^4} \, dx\right )+\int \frac {(4 c+5 a x) \left (-b+a x^5\right )^{3/4}}{-b+c x^4+a x^5} \, dx\\ &=-\frac {\left (4 \left (-b+a x^5\right )^{3/4}\right ) \int \frac {\left (1-\frac {a x^5}{b}\right )^{3/4}}{x^4} \, dx}{\left (1-\frac {a x^5}{b}\right )^{3/4}}+\int \left (\frac {4 c \left (-b+a x^5\right )^{3/4}}{-b+c x^4+a x^5}+\frac {5 a x \left (-b+a x^5\right )^{3/4}}{-b+c x^4+a x^5}\right ) \, dx\\ &=\frac {4 \left (-b+a x^5\right )^{3/4} \, _2F_1\left (-\frac {3}{4},-\frac {3}{5};\frac {2}{5};\frac {a x^5}{b}\right )}{3 x^3 \left (1-\frac {a x^5}{b}\right )^{3/4}}+(5 a) \int \frac {x \left (-b+a x^5\right )^{3/4}}{-b+c x^4+a x^5} \, dx+(4 c) \int \frac {\left (-b+a x^5\right )^{3/4}}{-b+c x^4+a x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.73, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.18, size = 152, normalized size = 1.00 \begin {gather*} \frac {4 \left (-b+a x^5\right )^{3/4}}{3 x^3}+\sqrt {2} c^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{-\sqrt {c} x^2+\sqrt {-b+a x^5}}\right )+\sqrt {2} c^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^5}}\right ) \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 )} {\left (a x^{5} - b\right )}^{\frac {3}{4}}}{{\left (a x^{5} + c x^{4} - b\right )} x^{4}}\,{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 \[\int \frac {\left (a \,x^{5}-b \right )^{\frac {3}{4}} \left (a \,x^{5}+4 b \right )}{x^{4} \left (a \,x^{5}+c \,x^{4}-b \right )}\, dx\]
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^{5} + 4 \, b\right )} {\left (a x^{5} - b\right )}^{\frac {3}{4}}}{{\left (a x^{5} + c x^{4} - b\right )} x^{4}}\,{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 {{\left (a\,x^5-b\right )}^{3/4}\,\left (a\,x^5+4\,b\right )}{x^4\,\left (a\,x^5+c\,x^4-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 {\left (a x^{5} - b\right )^{\frac {3}{4}} \left (a x^{5} + 4 b\right )}{x^{4} \left (a x^{5} - b + c x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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