Optimal. Leaf size=285 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}+\frac {x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{32 a^2 b^2 \left (a-b x^4\right )}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2} \]
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Rubi [A] time = 0.39, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1858, 1854, 1876, 275, 208, 1167, 205} \[ \frac {x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{32 a^2 b^2 \left (a-b x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 275
Rule 1167
Rule 1854
Rule 1858
Rule 1876
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+200 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+(200 a+b e) 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+5 b (120 a-b e) 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+(200 a+b e) 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-5 b (120 a-b e) 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-5 b (120 a-b e) x^2}{a-b x^4} \, dx}{32 a^2 b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(200 a+b e) 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-5 b (120 a-b e) 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)-5 b (120 a-b e) 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+(200 a+b e) 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-5 b (120 a-b e) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac {\int \frac {3 b (7 b c-a g)-5 b (120 a-b e) 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+(200 a+b e) 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-5 b (120 a-b e) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}-\frac {\left (600 a-5 b e-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^2 b}-\frac {\left (600 a-5 b e+\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^2 b}+\frac {(3 b d-a h) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2 b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(200 a+b e) 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-5 b (120 a-b e) x^2\right )}{32 a^2 b^2 \left (a-b x^4\right )}+\frac {\left (600 a-5 b e+\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}-\frac {\left (600 a-5 b e-\frac {3 \sqrt {b} (7 b c-a g)}{\sqrt {a}}\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*}
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Mathematica [A] time = 0.34, size = 380, normalized size = 1.33 \[ \frac {\sqrt [4]{b} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d-5 \sqrt {a} b e+3 a \sqrt {b} g-21 b^{3/2} c\right )+\sqrt [4]{b} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d+5 \sqrt {a} b e-3 a \sqrt {b} g+21 b^{3/2} c\right )+2 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-5 \sqrt {a} b e-3 a \sqrt {b} g+21 b^{3/2} c\right )+\frac {16 a^{7/4} \left (a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{\left (a-b x^4\right )^2}-\frac {4 a^{3/4} \left (8 a^2 j+a b x (g+x (2 h+3 i x))-b^2 x (7 c+x (6 d+5 e x))\right )}{a-b x^4}-4 \sqrt [4]{a} \sqrt {b} (a h-3 b d) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{128 a^{11/4} b^2} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 684, normalized size = 2.40 \[ -\frac {3}{256} \, i {\left (\frac {2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \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 )}{a^{2} b^{4}} - \frac {\sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{a^{2} b^{4}}\right )} - \frac {3}{256} \, i {\left (\frac {2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \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 )}{a^{2} b^{4}} + \frac {\sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{a^{2} b^{4}}\right )} - \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 + 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h + 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 - 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h - 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 {3 \, a b^{2} i x^{7} - 5 \, b^{3} x^{7} e - 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} + a^{2} b i x^{3} + 9 \, a b^{2} x^{3} e + 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}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 488, normalized size = 1.71 \[ \frac {h \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{32 \sqrt {a b}\, a b}-\frac {3 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{32 \sqrt {a b}\, a^{2}}+\frac {3 i \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}-\frac {3 i \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}-\frac {5 e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 a^{2} b}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 a^{2} b}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 a^{3}}-\frac {-\frac {\left (3 a i -5 b e \right ) x^{7}}{32 a^{2}}-\frac {\left (a h -3 b d \right ) x^{6}}{16 a^{2}}-\frac {j \,x^{4}}{4 b}-\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}-\frac {\left (a i +9 b e \right ) x^{3}}{32 a b}-\frac {\left (a h +5 b d \right ) x^{2}}{16 a b}-\frac {\left (3 a g +11 b c \right ) x}{32 a b}+\frac {a j -b f}{8 b^{2}}}{\left (b \,x^{4}-a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.13, size = 377, normalized size = 1.32 \[ \frac {8 \, a^{2} b j x^{4} - {\left (5 \, b^{3} e - 3 \, a b^{2} i\right )} x^{7} - 2 \, {\left (3 \, b^{3} d - a b^{2} h\right )} x^{6} - {\left (7 \, b^{3} c - a b^{2} g\right )} x^{5} + 4 \, a^{2} b f - 4 \, a^{3} j + {\left (9 \, a b^{2} e + a^{2} b i\right )} x^{3} + 2 \, {\left (5 \, a b^{2} d + a^{2} b h\right )} x^{2} + {\left (11 \, a b^{2} c + 3 \, a^{2} b g\right )} x}{32 \, {\left (a^{2} b^{4} x^{8} - 2 \, a^{3} b^{3} x^{4} + a^{4} b^{2}\right )}} + \frac {\frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g + 3 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, b^{\frac {3}{2}} c + 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.91, size = 2696, normalized size = 9.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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