Integrand size = 15, antiderivative size = 261 \[ \int \frac {x}{a+b (c+d x)^4} \, dx=\frac {\arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^2}+\frac {c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}-\frac {c \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}+\frac {c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}-\frac {c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2} \]
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Time = 0.19 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {378, 1890, 217, 1179, 642, 1176, 631, 210, 281, 211} \[ \int \frac {x}{a+b (c+d x)^4} \, dx=\frac {c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}-\frac {c \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}+\frac {c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}-\frac {c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}+\frac {\arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^2} \]
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Rule 210
Rule 211
Rule 217
Rule 281
Rule 378
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-c+x}{a+b x^4} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {c}{a+b x^4}+\frac {x}{a+b x^4}\right ) \, dx,x,c+d x\right )}{d^2} \\ & = \frac {\text {Subst}\left (\int \frac {x}{a+b x^4} \, dx,x,c+d x\right )}{d^2}-\frac {c \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 d^2}-\frac {c \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \sqrt {a} d^2}-\frac {c \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \sqrt {a} d^2} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^2}-\frac {c \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,c+d x\right )}{4 \sqrt {a} \sqrt {b} d^2}-\frac {c \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,c+d x\right )}{4 \sqrt {a} \sqrt {b} d^2}+\frac {c \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,c+d x\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}+\frac {c \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,c+d x\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^2}+\frac {c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}-\frac {c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}-\frac {c \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}+\frac {c \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} d^2}+\frac {c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}-\frac {c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}+\frac {c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2}-\frac {c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} d^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.40 \[ \int \frac {x}{a+b (c+d x)^4} \, dx=\frac {\text {RootSum}\left [a+b c^4+4 b c^3 d \text {$\#$1}+6 b c^2 d^2 \text {$\#$1}^2+4 b c d^3 \text {$\#$1}^3+b d^4 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{c^3+3 c^2 d \text {$\#$1}+3 c d^2 \text {$\#$1}^2+d^3 \text {$\#$1}^3}\&\right ]}{4 b d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.36
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}}{4 b d}\) | \(95\) |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}}{4 b d}\) | \(95\) |
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Result contains complex when optimal does not.
Time = 11.05 (sec) , antiderivative size = 40785, normalized size of antiderivative = 156.26 \[ \int \frac {x}{a+b (c+d x)^4} \, dx=\text {Too large to display} \]
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Time = 0.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.50 \[ \int \frac {x}{a+b (c+d x)^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} b^{2} d^{8} + 32 t^{2} a^{2} b d^{4} - 16 t a b c^{2} d^{2} + a + b c^{4}, \left ( t \mapsto t \log {\left (x + \frac {128 t^{3} a^{3} b d^{6} + 16 t^{2} a^{2} b c^{2} d^{4} + 8 t a^{2} d^{2} + 4 t a b c^{4} d^{2} - a c^{2} - b c^{6}}{4 a c d - b c^{5} d} \right )} \right )\right )} \]
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\[ \int \frac {x}{a+b (c+d x)^4} \, dx=\int { \frac {x}{{\left (d x + c\right )}^{4} b + a} \,d x } \]
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\[ \int \frac {x}{a+b (c+d x)^4} \, dx=\int { \frac {x}{{\left (d x + c\right )}^{4} b + a} \,d x } \]
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Time = 9.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.79 \[ \int \frac {x}{a+b (c+d x)^4} \, dx=\sum _{k=1}^4\ln \left (-\mathrm {root}\left (256\,a^3\,b^2\,d^8\,z^4+32\,a^2\,b\,d^4\,z^2-16\,a\,b\,c^2\,d^2\,z+b\,c^4+a,z,k\right )\,\left (-\mathrm {root}\left (256\,a^3\,b^2\,d^8\,z^4+32\,a^2\,b\,d^4\,z^2-16\,a\,b\,c^2\,d^2\,z+b\,c^4+a,z,k\right )\,\left (16\,a\,x\,b^3\,d^{12}+32\,a\,c\,b^3\,d^{11}\right )+4\,b^3\,c^3\,d^9+4\,b^3\,c^2\,d^{10}\,x\right )+b^2\,d^8\,x\right )\,\mathrm {root}\left (256\,a^3\,b^2\,d^8\,z^4+32\,a^2\,b\,d^4\,z^2-16\,a\,b\,c^2\,d^2\,z+b\,c^4+a,z,k\right ) \]
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