Integrand size = 17, antiderivative size = 496 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=-\frac {1}{\left (a+b c^4\right ) x}-\frac {\sqrt {b} c \left (a-b c^4\right ) d \arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) d \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) d \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) d \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2} \]
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Time = 0.66 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {378, 6857, 1890, 1262, 649, 211, 266, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt {b} c d \left (a-b c^4\right ) \arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}-\frac {1}{x \left (a+b c^4\right )}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2} \]
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Rule 210
Rule 211
Rule 266
Rule 378
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1262
Rule 1890
Rule 6857
Rubi steps \begin{align*} \text {integral}& = d \text {Subst}\left (\int \frac {1}{(-c+x)^2 \left (a+b x^4\right )} \, dx,x,c+d x\right ) \\ & = d \text {Subst}\left (\int \left (\frac {1}{\left (a+b c^4\right ) (c-x)^2}+\frac {4 b c^3}{\left (a+b c^4\right )^2 (c-x)}+\frac {b \left (-c^2 \left (3 a-b c^4\right )-2 c \left (a-b c^4\right ) x-\left (a-3 b c^4\right ) x^2+4 b c^3 x^3\right )}{\left (a+b c^4\right )^2 \left (a+b x^4\right )}\right ) \, dx,x,c+d x\right ) \\ & = -\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {(b d) \text {Subst}\left (\int \frac {-c^2 \left (3 a-b c^4\right )-2 c \left (a-b c^4\right ) x-\left (a-3 b c^4\right ) x^2+4 b c^3 x^3}{a+b x^4} \, dx,x,c+d x\right )}{\left (a+b c^4\right )^2} \\ & = -\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {(b d) \text {Subst}\left (\int \left (\frac {x \left (-2 c \left (a-b c^4\right )+4 b c^3 x^2\right )}{a+b x^4}+\frac {-c^2 \left (3 a-b c^4\right )+\left (-a+3 b c^4\right ) x^2}{a+b x^4}\right ) \, dx,x,c+d x\right )}{\left (a+b c^4\right )^2} \\ & = -\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {(b d) \text {Subst}\left (\int \frac {x \left (-2 c \left (a-b c^4\right )+4 b c^3 x^2\right )}{a+b x^4} \, dx,x,c+d x\right )}{\left (a+b c^4\right )^2}+\frac {(b d) \text {Subst}\left (\int \frac {-c^2 \left (3 a-b c^4\right )+\left (-a+3 b c^4\right ) x^2}{a+b x^4} \, dx,x,c+d x\right )}{\left (a+b c^4\right )^2} \\ & = -\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {(b d) \text {Subst}\left (\int \frac {-2 c \left (a-b c^4\right )+4 b c^3 x}{a+b x^2} \, dx,x,(c+d x)^2\right )}{2 \left (a+b c^4\right )^2}+\frac {\left (\left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \left (a+b c^4\right )^2}-\frac {\left (\left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \text {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx,x,c+d x\right )}{2 \left (a+b c^4\right )^2} \\ & = -\frac {1}{\left (a+b c^4\right ) x}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {\left (2 b^2 c^3 d\right ) \text {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,(c+d x)^2\right )}{\left (a+b c^4\right )^2}-\frac {\left (b c \left (a-b c^4\right ) d\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,(c+d x)^2\right )}{\left (a+b c^4\right )^2}-\frac {\left (\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \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} \sqrt [4]{a} \left (a+b c^4\right )^2}-\frac {\left (\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \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} \sqrt [4]{a} \left (a+b c^4\right )^2}-\frac {\left (\left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \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 \left (a+b c^4\right )^2}-\frac {\left (\left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \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 \left (a+b c^4\right )^2} \\ & = -\frac {1}{\left (a+b c^4\right ) x}-\frac {\sqrt {b} c \left (a-b c^4\right ) d \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2}-\frac {\left (\sqrt [4]{b} \left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {\left (\sqrt [4]{b} \left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d\right ) \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} \sqrt [4]{a} \left (a+b c^4\right )^2} \\ & = -\frac {1}{\left (a+b c^4\right ) x}-\frac {\sqrt {b} c \left (a-b c^4\right ) d \tan ^{-1}\left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (a-3 b c^4+\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (a+b c^4\right )^2}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (a-3 b c^4-\frac {\sqrt {b} c^2 \left (3 a-b c^4\right )}{\sqrt {a}}\right ) d \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} \sqrt [4]{a} \left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\frac {-4 \left (a+b c^4+4 b c^3 d x \log (x)\right )+d x \text {RootSum}\left [a+b c^4+4 b c^3 d \text {$\#$1}+6 b c^2 d^2 \text {$\#$1}^2+4 b c d^3 \text {$\#$1}^3+b d^4 \text {$\#$1}^4\&,\frac {-6 a c^2 \log (x-\text {$\#$1})+10 b c^6 \log (x-\text {$\#$1})-4 a c d \log (x-\text {$\#$1}) \text {$\#$1}+20 b c^5 d \log (x-\text {$\#$1}) \text {$\#$1}-a d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2+15 b c^4 d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2+4 b c^3 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^3}{c^3+3 c^2 d \text {$\#$1}+3 c d^2 \text {$\#$1}^2+d^3 \text {$\#$1}^3}\&\right ]}{4 \left (a+b c^4\right )^2 x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.37
method | result | size |
default | \(-\frac {1}{\left (b \,c^{4}+a \right ) x}-\frac {4 b \,c^{3} d \ln \left (x \right )}{\left (b \,c^{4}+a \right )^{2}}-\frac {d \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\left (-4 b \,d^{3} c^{3} \textit {\_R}^{3}+d^{2} \left (-15 b \,c^{4}+a \right ) \textit {\_R}^{2}+4 c d \left (-5 b \,c^{4}+a \right ) \textit {\_R} -10 b \,c^{6}+6 a \,c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}\right )}{4 \left (b \,c^{4}+a \right )^{2}}\) | \(184\) |
risch | \(-\frac {1}{\left (b \,c^{4}+a \right ) x}-\frac {4 b \,c^{3} d \ln \left (x \right )}{b^{2} c^{8}+2 a b \,c^{4}+a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b^{2} c^{8}+2 a^{4} b \,c^{4}+a^{5}\right ) \textit {\_Z}^{4}-16 a^{3} b \,c^{3} d \,\textit {\_Z}^{3}+20 a^{2} b \,c^{2} d^{2} \textit {\_Z}^{2}-8 a b c \,d^{3} \textit {\_Z} +b \,d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-3 a^{2} b^{3} c^{12} d -a^{3} b^{2} c^{8} d +7 a^{4} b \,c^{4} d +5 d \,a^{5}\right ) \textit {\_R}^{4}+\left (4 a \,b^{3} c^{11} d^{2}-40 a^{2} b^{2} c^{7} d^{2}-44 a^{3} b \,c^{3} d^{2}\right ) \textit {\_R}^{3}+\left (-b^{3} c^{10} d^{3}+62 a \,b^{2} c^{6} d^{3}+79 a^{2} b \,c^{2} d^{3}\right ) \textit {\_R}^{2}+\left (-16 b^{2} c^{5} d^{4}-32 b c \,d^{4} a \right ) \textit {\_R} +4 b \,d^{5}\right ) x +\left (-a^{2} b^{3} c^{13}-3 a^{3} b^{2} c^{9}-3 a^{4} b \,c^{5}-c \,a^{5}\right ) \textit {\_R}^{4}+\left (3 a \,b^{3} c^{12} d -9 a^{2} b^{2} c^{8} d -11 a^{3} b \,c^{4} d +a^{4} d \right ) \textit {\_R}^{3}+\left (-b^{3} c^{11} d^{2}+46 a \,b^{2} c^{7} d^{2}+15 a^{2} b \,c^{3} d^{2}\right ) \textit {\_R}^{2}-16 b^{2} c^{6} d^{3} \textit {\_R} \right )\right )}{4}\) | \(420\) |
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Result contains complex when optimal does not.
Time = 78.19 (sec) , antiderivative size = 1128605, normalized size of antiderivative = 2275.41 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{4} b + a\right )} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{4} b + a\right )} x^{2}} \,d x } \]
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Time = 9.53 (sec) , antiderivative size = 2440, normalized size of antiderivative = 4.92 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\text {Too large to display} \]
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