Integrand size = 21, antiderivative size = 545 \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{b c-a d} x}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}+\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sqrt [5]{b c-a d} x}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\log \left (\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (\frac {2 (b c-a d)^{2/5} x^2+\sqrt [5]{c} \sqrt [5]{b c-a d} x \sqrt [5]{a+b x^5}-\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x \sqrt [5]{a+b x^5}+2 c^{2/5} \left (a+b x^5\right )^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (\frac {2 (b c-a d)^{2/5} x^2+\sqrt [5]{c} \sqrt [5]{b c-a d} x \sqrt [5]{a+b x^5}+\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x \sqrt [5]{a+b x^5}+2 c^{2/5} \left (a+b x^5\right )^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \]
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Time = 0.86 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {385, 208, 648, 632, 210, 642, 31} \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x \sqrt [5]{b c-a d}}{\sqrt [5]{c} \sqrt [5]{a+b x^5}}+\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\log \left (\sqrt [5]{c}-\frac {x \sqrt [5]{b c-a d}}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (\frac {2 c^{2/5} \left (a+b x^5\right )^{2/5}-\sqrt {5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (\frac {2 c^{2/5} \left (a+b x^5\right )^{2/5}+\sqrt {5} \sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+\sqrt [5]{c} x \sqrt [5]{a+b x^5} \sqrt [5]{b c-a d}+2 x^2 (b c-a d)^{2/5}}{\left (a+b x^5\right )^{2/5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \]
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Rule 31
Rule 208
Rule 210
Rule 385
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^5} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right ) \\ & = \frac {2 \text {Subst}\left (\int \frac {\sqrt [5]{c}+\frac {1}{4} \left (1-\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{c^{2/5}+\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}}+\frac {2 \text {Subst}\left (\int \frac {\sqrt [5]{c}+\frac {1}{4} \left (1+\sqrt {5}\right ) \sqrt [5]{b c-a d} x}{c^{2/5}+\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [5]{c}-\sqrt [5]{b c-a d} x} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5}} \\ & = -\frac {\log \left (\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (5-\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{c^{2/5}+\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{3/5}}+\frac {\left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{c^{2/5}+\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{3/5}}+\frac {\left (1-\sqrt {5}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+2 (b c-a d)^{2/5} x}{c^{2/5}+\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+2 (b c-a d)^{2/5} x}{c^{2/5}+\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d} x+(b c-a d)^{2/5} x^2} \, dx,x,\frac {x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \\ & = -\frac {\log \left (\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 c^{2/5}+\frac {2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac {\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}-\frac {\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 c^{2/5}+\frac {2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac {\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}+\frac {\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\left (5-\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5-\sqrt {5}\right ) c^{2/5} (b c-a d)^{2/5}-x^2} \, dx,x,\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+\frac {2 (b c-a d)^{2/5} x}{\sqrt [5]{a+b x^5}}\right )}{10 c^{3/5}}-\frac {\left (5+\sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \left (5+\sqrt {5}\right ) c^{2/5} (b c-a d)^{2/5}-x^2} \, dx,x,\frac {1}{2} \left (1-\sqrt {5}\right ) \sqrt [5]{c} \sqrt [5]{b c-a d}+\frac {2 (b c-a d)^{2/5} x}{\sqrt [5]{a+b x^5}}\right )}{10 c^{3/5}} \\ & = \frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\left (1-\sqrt {5}\right ) \sqrt [5]{c}+\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}}{\sqrt {2 \left (5+\sqrt {5}\right )} \sqrt [5]{c}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {5+\sqrt {5}} \left (\left (1+\sqrt {5}\right ) \sqrt [5]{c}+\frac {4 \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{2 \sqrt {10} \sqrt [5]{c}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}-\frac {\log \left (\sqrt [5]{c}-\frac {\sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{5 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1-\sqrt {5}\right ) \log \left (2 c^{2/5}+\frac {2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac {\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}-\frac {\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}}+\frac {\left (1+\sqrt {5}\right ) \log \left (2 c^{2/5}+\frac {2 (b c-a d)^{2/5} x^2}{\left (a+b x^5\right )^{2/5}}+\frac {\sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}+\frac {\sqrt {5} \sqrt [5]{c} \sqrt [5]{b c-a d} x}{\sqrt [5]{a+b x^5}}\right )}{20 c^{4/5} \sqrt [5]{b c-a d}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\frac {x \operatorname {Hypergeometric2F1}\left (\frac {1}{5},1,\frac {6}{5},\frac {(b c-a d) x^5}{c \left (a+b x^5\right )}\right )}{c \sqrt [5]{a+b x^5}} \]
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Time = 10.58 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(-\frac {\left (\frac {\sqrt {5+\sqrt {5}}\, \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}+1\right ) \ln \left (\frac {2 \left (\frac {a d -b c}{c}\right )^{\frac {2}{5}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \left (b \,x^{5}+a \right )^{\frac {1}{5}} \left (\sqrt {5}+1\right ) x +2 \left (b \,x^{5}+a \right )^{\frac {2}{5}}}{x^{2}}\right )}{2}-\frac {\sqrt {5+\sqrt {5}}\, \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}-1\right ) \ln \left (\frac {2 \left (\frac {a d -b c}{c}\right )^{\frac {2}{5}} x^{2}+\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \left (b \,x^{5}+a \right )^{\frac {1}{5}} \left (\sqrt {5}-1\right ) x +2 \left (b \,x^{5}+a \right )^{\frac {2}{5}}}{x^{2}}\right )}{2}+\sqrt {2}\, \sqrt {5-\sqrt {5}}\, \left (5+\sqrt {5}\right ) \arctan \left (\frac {\left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \left (\sqrt {5}-1\right ) x +4 \left (b \,x^{5}+a \right )^{\frac {1}{5}}\right ) \sqrt {2}}{2 \left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \sqrt {5+\sqrt {5}}\, x}\right )+\sqrt {5+\sqrt {5}}\, \left (-2 \sqrt {5-\sqrt {5}}\, \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} x +\left (b \,x^{5}+a \right )^{\frac {1}{5}}}{x}\right )+\arctan \left (\frac {\sqrt {2}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \left (\sqrt {5}+1\right ) x -4 \left (b \,x^{5}+a \right )^{\frac {1}{5}}\right )}{2 \left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} \sqrt {5-\sqrt {5}}\, x}\right ) \sqrt {2}\, \left (\sqrt {5}-5\right )\right )\right ) \sqrt {5}}{100 \left (\frac {a d -b c}{c}\right )^{\frac {1}{5}} c}\) | \(409\) |
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Exception generated. \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\int \frac {1}{\sqrt [5]{a + b x^{5}} \left (c + d x^{5}\right )}\, dx \]
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\[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\int { \frac {1}{{\left (b x^{5} + a\right )}^{\frac {1}{5}} {\left (d x^{5} + c\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\int { \frac {1}{{\left (b x^{5} + a\right )}^{\frac {1}{5}} {\left (d x^{5} + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [5]{a+b x^5} \left (c+d x^5\right )} \, dx=\int \frac {1}{{\left (b\,x^5+a\right )}^{1/5}\,\left (d\,x^5+c\right )} \,d x \]
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