Integrand size = 46, antiderivative size = 285 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x+b (5 b e-3 a i) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}-\frac {\left (5 b e-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac {\left (5 b e+\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac {(3 b d-a h) \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}} \]
[Out]
Time = 0.25 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1872, 1868, 1890, 281, 214, 1181, 211} \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (3 b d-a h)}{16 a^{5/2} b^{3/2}}+\frac {x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{32 a^2 b^2 \left (a-b x^4\right )}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2} \]
[In]
[Out]
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+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}-\frac {\int \frac {-b (7 b c-a g)-2 b (3 b d-a h) x-b (5 b e-3 a i) x^2-4 b (b f-a j) x^3}{\left (a-b x^4\right )^2} \, dx}{8 a b^2} \\ & = \frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x+b (5 b e-3 a i) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac {\int \frac {3 b (7 b c-a g)+4 b (3 b d-a h) x+b (5 b e-3 a i) x^2}{a-b x^4} \, dx}{32 a^2 b^2} \\ & = \frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x+b (5 b e-3 a i) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac {\int \left (\frac {4 b (3 b d-a h) x}{a-b x^4}+\frac {3 b (7 b c-a g)+b (5 b e-3 a i) x^2}{a-b x^4}\right ) \, dx}{32 a^2 b^2} \\ & = \frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x+b (5 b e-3 a i) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac {\int \frac {3 b (7 b c-a g)+b (5 b e-3 a i) x^2}{a-b x^4} \, dx}{32 a^2 b^2}+\frac {(3 b d-a h) \int \frac {x}{a-b x^4} \, dx}{8 a^2 b} \\ & = \frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x+b (5 b e-3 a i) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac {(3 b d-a h) \text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2 b}+\frac {\left (5 b e-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^2 b}+\frac {\left (5 b e+\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^2 b} \\ & = \frac {x \left (b c+a g+(b d+a h) x+(b e+a i) x^2+(b f+a j) x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a (b f-a j)+x \left (b (7 b c-a g)+2 b (3 b d-a h) x+b (5 b e-3 a i) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}-\frac {\left (5 b e-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac {\left (5 b e+\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.33 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\frac {-\frac {4 a^{3/4} \left (8 a^2 j-b^2 x (7 c+x (6 d+5 e x))+a b x (g+x (2 h+3 i x))\right )}{a-b x^4}+\frac {16 a^{7/4} \left (a^2 j+b^2 x (c+x (d+e x))+a b (f+x (g+x (h+i x)))\right )}{\left (a-b x^4\right )^2}+2 \sqrt [4]{b} \left (21 b^{3/2} c-5 \sqrt {a} b e-3 a \sqrt {b} g+3 a^{3/2} i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt [4]{b} \left (-21 b^{3/2} c-12 \sqrt [4]{a} b^{5/4} d-5 \sqrt {a} b e+3 a \sqrt {b} g+4 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\sqrt [4]{b} \left (21 b^{3/2} c-12 \sqrt [4]{a} b^{5/4} d+5 \sqrt {a} b e-3 a \sqrt {b} g+4 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )-4 \sqrt [4]{a} \sqrt {b} (-3 b d+a h) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{128 a^{11/4} b^2} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.62 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {\frac {\left (3 a i -5 b e \right ) x^{7}}{32 a^{2}}+\frac {\left (a h -3 b d \right ) x^{6}}{16 a^{2}}+\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}+\frac {j \,x^{4}}{4 b}+\frac {\left (a i +9 b e \right ) x^{3}}{32 a b}+\frac {\left (a h +5 b d \right ) x^{2}}{16 a b}+\frac {\left (3 a g +11 b c \right ) x}{32 a b}-\frac {a j -b f}{8 b^{2}}}{\left (-b \,x^{4}+a \right )^{2}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (-\left (3 a i -5 b e \right ) \textit {\_R}^{2}-4 \left (a h -3 b d \right ) \textit {\_R} -3 a g +21 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} b^{2}}\) | \(206\) |
default | \(\frac {\frac {\left (3 a i -5 b e \right ) x^{7}}{32 a^{2}}+\frac {\left (a h -3 b d \right ) x^{6}}{16 a^{2}}+\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}+\frac {j \,x^{4}}{4 b}+\frac {\left (a i +9 b e \right ) x^{3}}{32 a b}+\frac {\left (a h +5 b d \right ) x^{2}}{16 a b}+\frac {\left (3 a g +11 b c \right ) x}{32 a b}-\frac {a j -b f}{8 b^{2}}}{\left (-b \,x^{4}+a \right )^{2}}+\frac {\frac {\left (-3 a g +21 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {\left (-4 a h +12 b d \right ) \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {\left (-3 a i +5 b e \right ) \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 a^{2} b}\) | \(311\) |
[In]
[Out]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\frac {8 \, a^{2} b j x^{4} - {\left (5 \, b^{3} e - 3 \, a b^{2} i\right )} x^{7} - 2 \, {\left (3 \, b^{3} d - a b^{2} h\right )} x^{6} - {\left (7 \, b^{3} c - a b^{2} g\right )} x^{5} + 4 \, a^{2} b f - 4 \, a^{3} j + {\left (9 \, a b^{2} e + a^{2} b i\right )} x^{3} + 2 \, {\left (5 \, a b^{2} d + a^{2} b h\right )} x^{2} + {\left (11 \, a b^{2} c + 3 \, a^{2} b g\right )} x}{32 \, {\left (a^{2} b^{4} x^{8} - 2 \, a^{3} b^{3} x^{4} + a^{4} b^{2}\right )}} + \frac {\frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g + 3 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, b^{\frac {3}{2}} c + 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2} b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (245) = 490\).
Time = 0.29 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.97 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=-\frac {\sqrt {2} {\left (21 \, b^{3} c - 3 \, a b^{2} g - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d + 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - 5 \, \sqrt {-a b} b^{2} e + 3 \, \sqrt {-a b} a b i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2} b} - \frac {\sqrt {2} {\left (21 \, b^{3} c - 3 \, a b^{2} g + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - 5 \, \sqrt {-a b} b^{2} e - 3 \, \sqrt {-a b} a b i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2} b} - \frac {\sqrt {2} {\left (21 \, b^{3} c - 3 \, a b^{2} g - 5 \, \sqrt {-a b} b^{2} e + 3 \, \sqrt {-a b} a b i\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2} b} + \frac {\sqrt {2} {\left (21 \, b^{3} c - 3 \, a b^{2} g - 5 \, \sqrt {-a b} b^{2} e + 3 \, \sqrt {-a b} a b i\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2} b} - \frac {5 \, b^{3} e x^{7} - 3 \, a b^{2} i x^{7} + 6 \, b^{3} d x^{6} - 2 \, a b^{2} h x^{6} + 7 \, b^{3} c x^{5} - a b^{2} g x^{5} - 8 \, a^{2} b j x^{4} - 9 \, a b^{2} e x^{3} - a^{2} b i x^{3} - 10 \, a b^{2} d x^{2} - 2 \, a^{2} b h x^{2} - 11 \, a b^{2} c x - 3 \, a^{2} b g x - 4 \, a^{2} b f + 4 \, a^{3} j}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2} b^{2}} \]
[In]
[Out]
Time = 10.02 (sec) , antiderivative size = 2696, normalized size of antiderivative = 9.46 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{\left (a-b x^4\right )^3} \, dx=\text {Too large to display} \]
[In]
[Out]