Integrand size = 41, antiderivative size = 154 \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\frac {2 \left (b+a x^5\right )^{3/4}}{3 x^3}+\frac {c^{3/4} \arctan \left (\frac {2^{3/4} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{-\sqrt {c} x^2+\sqrt {2} \sqrt {b+a x^5}}\right )}{2 \sqrt [4]{2}}+\frac {c^{3/4} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{2^{3/4}}+\frac {\sqrt {b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right )}{2 \sqrt [4]{2}} \]
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\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \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} \operatorname {Hypergeometric2F1}\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*}
Time = 1.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\frac {1}{12} \left (\frac {8 \left (b+a x^5\right )^{3/4}}{x^3}+3\ 2^{3/4} c^{3/4} \arctan \left (\frac {\sqrt [4]{c} x}{2^{3/4} \sqrt [4]{b+a x^5}}-\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c} x}\right )+3\ 2^{3/4} c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{c} x}{2^{3/4} \sqrt [4]{b+a x^5}}+\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c} x}\right )\right ) \]
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Time = 1.54 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.21
method | result | size |
pseudoelliptic | \(\frac {\left (8 \left (a \,x^{5}+b \right )^{\frac {3}{4}} 2^{\frac {1}{4}} c^{\frac {1}{4}}-3 \ln \left (\frac {\sqrt {2}\, \sqrt {c}\, x^{2}-2 \,2^{\frac {1}{4}} c^{\frac {1}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} x +2 \sqrt {a \,x^{5}+b}}{\sqrt {2}\, \sqrt {c}\, x^{2}+2 \,2^{\frac {1}{4}} c^{\frac {1}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} x +2 \sqrt {a \,x^{5}+b}}\right ) c \,x^{3}-6 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c \,x^{3}-6 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c \,x^{3}\right ) 2^{\frac {3}{4}}}{24 x^{3} c^{\frac {1}{4}}}\) | \(187\) |
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Timed out. \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int \frac {\left (a x^{5} - 4 b\right ) \left (a x^{5} + b\right )^{\frac {3}{4}}}{x^{4} \cdot \left (2 a x^{5} + 2 b + c x^{4}\right )}\, dx \]
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\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int { \frac {{\left (a x^{5} + b\right )}^{\frac {3}{4}} {\left (a x^{5} - 4 \, b\right )}}{{\left (2 \, a x^{5} + c x^{4} + 2 \, b\right )} x^{4}} \,d x } \]
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\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int { \frac {{\left (a x^{5} + b\right )}^{\frac {3}{4}} {\left (a x^{5} - 4 \, b\right )}}{{\left (2 \, a x^{5} + c x^{4} + 2 \, b\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int -\frac {{\left (a\,x^5+b\right )}^{3/4}\,\left (4\,b-a\,x^5\right )}{x^4\,\left (2\,a\,x^5+c\,x^4+2\,b\right )} \,d x \]
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