Optimal. Leaf size=463 \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (3 \sqrt {b} (a g+7 b c)-\sqrt {a} (3 a i+5 b e)\right )}{128 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (3 \sqrt {b} (a g+7 b c)-\sqrt {a} (3 a i+5 b e)\right )}{128 \sqrt {2} a^{11/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt {b} (a g+7 b c)+\sqrt {a} (3 a i+5 b e)\right )}{64 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (3 \sqrt {b} (a g+7 b c)+\sqrt {a} (3 a i+5 b e)\right )}{64 \sqrt {2} a^{11/4} b^{7/4}}+\frac {(a h+3 b d) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}-\frac {4 a f-x \left (2 x (a h+3 b d)+x^2 (3 a i+5 b e)+a g+7 b c\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]
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Rubi [A] time = 0.69, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {1858, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (3 \sqrt {b} (a g+7 b c)-\sqrt {a} (3 a i+5 b e)\right )}{128 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (3 \sqrt {b} (a g+7 b c)-\sqrt {a} (3 a i+5 b e)\right )}{128 \sqrt {2} a^{11/4} b^{7/4}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt {b} (a g+7 b c)+\sqrt {a} (3 a i+5 b e)\right )}{64 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (3 \sqrt {b} (a g+7 b c)+\sqrt {a} (3 a i+5 b e)\right )}{64 \sqrt {2} a^{11/4} b^{7/4}}+\frac {(a h+3 b d) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}-\frac {4 a f-x \left (2 x (a h+3 b d)+x^2 (3 a i+5 b e)+a g+7 b c\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 275
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
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+202 x^6}{\left (a+b x^4\right )^3} \, dx &=\frac {x \left (b c-a g+(b d-a h) x-(202 a-b e) x^2+b f 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 (606 a+5 b e) x^2-4 b^2 f 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-(202 a-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+2 (3 b d+a h) x+(606 a+5 b e) x^2\right )}{32 a^2 b \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 (606 a+5 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-(202 a-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+2 (3 b d+a h) x+(606 a+5 b e) x^2\right )}{32 a^2 b \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 (606 a+5 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-(202 a-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+2 (3 b d+a h) x+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {\int \frac {3 b (7 b c+a g)+b (606 a+5 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-(202 a-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+2 (3 b d+a h) x+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}-\frac {\left (606 a+5 b e-\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{64 a^2 b^2}+\frac {\left (606 a+5 b e+\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{64 a^2 b^2}+\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-(202 a-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+2 (3 b d+a h) x+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {(3 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}+\frac {\left (606 a+5 b e-\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \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}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {\left (606 a+5 b e-\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \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}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {\left (606 a+5 b e+\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b^2}+\frac {\left (606 a+5 b e+\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b^2}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(202 a-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+2 (3 b d+a h) x+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {(3 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}+\frac {\left (606 a+5 b e-\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}-\frac {\left (606 a+5 b e-\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}+\frac {\left (606 a+5 b e+\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}-\frac {\left (606 a+5 b e+\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}\\ &=\frac {x \left (b c-a g+(b d-a h) x-(202 a-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+2 (3 b d+a h) x+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac {(3 b d+a h) \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}-\frac {\left (606 a+5 b e+\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}+\frac {\left (606 a+5 b e+\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{9/4} b^{7/4}}+\frac {\left (606 a+5 b e-\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}-\frac {\left (606 a+5 b e-\frac {3 \sqrt {b} (7 b c+a g)}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{7/4}}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 473, normalized size = 1.02 \[ \frac {-\frac {32 a^{7/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{\left (a+b x^4\right )^2}+\frac {8 a^{3/4} b^{3/4} x (a g+a x (2 h+3 i x)+7 b c+b x (6 d+5 e x))}{a+b x^4}-2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (8 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i+24 \sqrt [4]{a} b^{5/4} d+5 \sqrt {2} \sqrt {a} b e+3 \sqrt {2} a \sqrt {b} g+21 \sqrt {2} b^{3/2} c\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-8 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i-24 \sqrt [4]{a} b^{5/4} d+5 \sqrt {2} \sqrt {a} b e+3 \sqrt {2} a \sqrt {b} g+21 \sqrt {2} b^{3/2} c\right )+\sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (3 a^{3/2} i+5 \sqrt {a} b e-3 a \sqrt {b} g-21 b^{3/2} c\right )+\sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-3 a^{3/2} i-5 \sqrt {a} b e+3 a \sqrt {b} g+21 b^{3/2} c\right )}{256 a^{11/4} b^{7/4}} \]
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 [A] time = 0.23, size = 661, normalized size = 1.43 \[ \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 (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 4 \, \sqrt {2} \sqrt {a b} a b h + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} 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 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 4 \, \sqrt {2} \sqrt {a b} a b h + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} 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 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} + \frac {3 \, a b i x^{7} + 5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 2 \, a b h x^{6} + 7 \, b^{2} c x^{5} + a b g x^{5} - a^{2} i x^{3} + 9 \, a b x^{3} e + 10 \, a b d x^{2} - 2 \, a^{2} h 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} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 716, normalized size = 1.55 \[ \frac {h \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{16 \sqrt {a b}\, a b}+\frac {3 d \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{16 \sqrt {a b}\, a^{2}}+\frac {3 \sqrt {2}\, i \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {3 \sqrt {2}\, i \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {3 \sqrt {2}\, i \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {5 \sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 \sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 \sqrt {2}\, e \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{128 a^{2} b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{128 a^{2} b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, g \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{256 a^{2} b}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{128 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{128 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{256 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 {\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 {f}{8 b}-\frac {\left (3 a g -11 b c \right ) x}{32 a b}}{\left (b \,x^{4}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.17, size = 497, normalized size = 1.07 \[ \frac {{\left (5 \, b^{2} e + 3 \, a b i\right )} x^{7} + 2 \, {\left (3 \, b^{2} d + a b h\right )} x^{6} + {\left (7 \, b^{2} c + a b g\right )} x^{5} + {\left (9 \, a b e - a^{2} i\right )} x^{3} - 4 \, a^{2} f + 2 \, {\left (5 \, a b d - a^{2} h\right )} x^{2} + {\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 {\sqrt {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 )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {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 )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 3 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} i - 24 \, \sqrt {a} b^{\frac {3}{2}} d - 8 \, a^{\frac {3}{2}} \sqrt {b} h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + 3 \, \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} i + 24 \, \sqrt {a} b^{\frac {3}{2}} d + 8 \, a^{\frac {3}{2}} \sqrt {b} h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.75, size = 2680, normalized size = 5.79 \[ \text {result too large to display} \]
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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