Integrand size = 31, antiderivative size = 221 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}} \]
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Time = 0.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1872, 1868, 1890, 281, 214, 1181, 211} \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac {3 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}+\frac {x \left (-a g+7 b c+6 b d x+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac {x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]
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Rule 211
Rule 214
Rule 281
Rule 1181
Rule 1868
Rule 1872
Rule 1890
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {\int \frac {7 b c-a g+6 b d x+5 b e x^2+4 b f x^3}{\left (a-b x^4\right )^2} \, dx}{8 a b} \\ & = \frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac {\int \frac {-3 (7 b c-a g)-12 b d x-5 b e x^2}{a-b x^4} \, dx}{32 a^2 b} \\ & = \frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac {\int \left (-\frac {12 b d x}{a-b x^4}+\frac {-3 (7 b c-a g)-5 b e x^2}{a-b x^4}\right ) \, dx}{32 a^2 b} \\ & = \frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac {\int \frac {-3 (7 b c-a g)-5 b e x^2}{a-b x^4} \, dx}{32 a^2 b}+\frac {(3 d) \int \frac {x}{a-b x^4} \, dx}{8 a^2} \\ & = \frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {(3 d) \text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^{5/2} \sqrt {b}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^{5/2} \sqrt {b}} \\ & = \frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {3 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.19 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (a^2 (4 f+3 g x)-b^2 x^5 (7 c+x (6 d+5 e x))+a b x \left (11 c+x \left (10 d+9 e x+g x^3\right )\right )\right )}{\left (a-b x^4\right )^2}+2 \left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (21 b c+12 \sqrt [4]{a} b^{3/4} d+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\left (21 b c-12 \sqrt [4]{a} b^{3/4} d+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+12 \sqrt [4]{a} b^{3/4} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{128 a^{11/4} b^{5/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.51 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {-\frac {5 b e \,x^{7}}{32 a^{2}}-\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {\left (3 a g +11 b c \right ) x}{32 a b}+\frac {f}{8 b}}{\left (-b \,x^{4}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (5 \textit {\_R}^{2} e +12 \textit {\_R} d -\frac {3 \left (a g -7 b c \right )}{b}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b}\) | \(147\) |
default | \(\frac {-\frac {5 b e \,x^{7}}{32 a^{2}}-\frac {3 b d \,x^{6}}{16 a^{2}}+\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}+\frac {9 e \,x^{3}}{32 a}+\frac {5 d \,x^{2}}{16 a}+\frac {\left (3 a g +11 b c \right ) x}{32 a b}+\frac {f}{8 b}}{\left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {\left (-3 a g +21 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {3 b d \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{\sqrt {a b}}-\frac {5 e \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 a^{2} b}\) | \(244\) |
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Result contains complex when optimal does not.
Time = 42.83 (sec) , antiderivative size = 343626, normalized size of antiderivative = 1554.87 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\text {Timed out} \]
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Time = 0.37 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.29 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=-\frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} - 9 \, a b e x^{3} + {\left (7 \, b^{2} c - a b g\right )} x^{5} - 10 \, a b d x^{2} - 4 \, a^{2} f - {\left (11 \, a b c + 3 \, a^{2} g\right )} x}{32 \, {\left (a^{2} b^{3} x^{8} - 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {12 \, \sqrt {b} d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a}} - \frac {12 \, \sqrt {b} d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a}} + \frac {2 \, {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, b^{\frac {3}{2}} c + 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (181) = 362\).
Time = 0.43 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.75 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=-\frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} + \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {5 \, b^{2} e x^{7} + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} - a b g x^{5} - 9 \, a b e x^{3} - 10 \, a b d x^{2} - 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2} b} \]
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Time = 9.66 (sec) , antiderivative size = 1002, normalized size of antiderivative = 4.53 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^3} \, dx=\frac {\frac {f}{8\,b}+\frac {5\,d\,x^2}{16\,a}+\frac {9\,e\,x^3}{32\,a}-\frac {x^5\,\left (7\,b\,c-a\,g\right )}{32\,a^2}+\frac {x\,\left (11\,b\,c+3\,a\,g\right )}{32\,a\,b}-\frac {3\,b\,d\,x^6}{16\,a^2}-\frac {5\,b\,e\,x^7}{32\,a^2}}{a^2-2\,a\,b\,x^4+b^2\,x^8}+\left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2-6881280\,a^6\,b^4\,c\,e\,z^2-4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z+55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z+2709504\,a^3\,b^4\,c^2\,d\,z+8640\,a^2\,b^2\,d^2\,e\,g-6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3-23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2-625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4-81\,a^4\,g^4-194481\,b^4\,c^4,z,k\right )\,\left (\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2-6881280\,a^6\,b^4\,c\,e\,z^2-4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z+55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z+2709504\,a^3\,b^4\,c^2\,d\,z+8640\,a^2\,b^2\,d^2\,e\,g-6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3-23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2-625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4-81\,a^4\,g^4-194481\,b^4\,c^4,z,k\right )\,\left (\frac {344064\,a^5\,b^3\,c-49152\,a^6\,b^2\,g}{32768\,a^6}-\frac {6\,b^3\,d\,x}{a}\right )+\frac {x\,\left (144\,a^4\,b\,g^2-2016\,a^3\,b^2\,c\,g+400\,a^3\,b^2\,e^2+7056\,a^2\,b^3\,c^2\right )}{4096\,a^6}-\frac {15\,b^2\,d\,e}{32\,a^3}\right )-\frac {-45\,a^2\,e\,g^2+630\,a\,b\,c\,e\,g-432\,a\,b\,d^2\,g+125\,a\,b\,e^3-2205\,b^2\,c^2\,e+3024\,b^2\,c\,d^2}{32768\,a^6}-\frac {x\,\left (216\,b^2\,d^3-315\,c\,e\,b^2\,d+45\,a\,e\,g\,b\,d\right )}{4096\,a^6}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^5\,z^4+983040\,a^7\,b^3\,e\,g\,z^2-6881280\,a^6\,b^4\,c\,e\,z^2-4718592\,a^6\,b^4\,d^2\,z^2-774144\,a^4\,b^3\,c\,d\,g\,z+55296\,a^5\,b^2\,d\,g^2\,z+153600\,a^4\,b^3\,d\,e^2\,z+2709504\,a^3\,b^4\,c^2\,d\,z+8640\,a^2\,b^2\,d^2\,e\,g-6300\,a^2\,b^2\,c\,e^2\,g-60480\,a\,b^3\,c\,d^2\,e+111132\,a\,b^3\,c^3\,g+2268\,a^3\,b\,c\,g^3-23814\,a^2\,b^2\,c^2\,g^2+450\,a^3\,b\,e^2\,g^2+22050\,a\,b^3\,c^2\,e^2-625\,a^2\,b^2\,e^4+20736\,a\,b^3\,d^4-81\,a^4\,g^4-194481\,b^4\,c^4,z,k\right )\right ) \]
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