Optimal. Leaf size=154 \[ \frac {c^{3/4} \tan ^{-1}\left (\frac {2^{3/4} \sqrt [4]{c} x \sqrt [4]{a x^5+b}}{\sqrt {2} \sqrt {a x^5+b}-\sqrt {c} x^2}\right )}{2 \sqrt [4]{2}}+\frac {c^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b}}{\sqrt [4]{2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{2^{3/4}}}{x \sqrt [4]{a x^5+b}}\right )}{2 \sqrt [4]{2}}+\frac {2 \left (a x^5+b\right )^{3/4}}{3 x^3} \]
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Rubi [F] time = 1.19, 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 (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 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 (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx &=\int \left (-\frac {2 \left (b+a x^5\right )^{3/4}}{x^4}+\frac {(2 c+5 a x) \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}\right ) \, dx\\ &=-\left (2 \int \frac {\left (b+a x^5\right )^{3/4}}{x^4} \, dx\right )+\int \frac {(2 c+5 a x) \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5} \, dx\\ &=-\frac {\left (2 \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 {2 c \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}+\frac {5 a x \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}\right ) \, dx\\ &=\frac {2 \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}}{2 b+c x^4+2 a x^5} \, dx+(2 c) \int \frac {\left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.37, size = 154, normalized size = 1.00 \begin {gather*} -\frac {c^{3/4} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^5+b}}{\sqrt [4]{2} \sqrt [4]{c}}-\frac {\sqrt [4]{c} x^2}{2^{3/4}}}{x \sqrt [4]{a x^5+b}}\right )}{2 \sqrt [4]{2}}+\frac {c^{3/4} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5+b}}{\sqrt [4]{2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{2^{3/4}}}{x \sqrt [4]{a x^5+b}}\right )}{2 \sqrt [4]{2}}+\frac {2 \left (a x^5+b\right )^{3/4}}{3 x^3} \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} + b\right )}^{\frac {3}{4}} {\left (a x^{5} - 4 \, b\right )}}{{\left (2 \, a x^{5} + c x^{4} + 2 \, 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 = 0.41, 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}}}{x^{4} \left (2 a \,x^{5}+c \,x^{4}+2 b \right )}\, dx \end {gather*}
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} + b\right )}^{\frac {3}{4}} {\left (a x^{5} - 4 \, b\right )}}{{\left (2 \, a x^{5} + c x^{4} + 2 \, 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 (4\,b-a\,x^5\right )}{x^4\,\left (2\,a\,x^5+c\,x^4+2\,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} - 4 b\right ) \left (a x^{5} + b\right )^{\frac {3}{4}}}{x^{4} \left (2 a x^{5} + 2 b + c x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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