Optimal. Leaf size=349 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac {(5 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+4 a (2 b f-a j)}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{12 a b \left (a-b x^4\right )^3} \]
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Rubi [A] time = 0.52, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1858, 1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac {x \left (b (11 b c-a g)+2 b x (5 b d-a h)+3 b x^2 (3 b e-a i)\right )+4 a (2 b f-a j)}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac {(5 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{12 a b \left (a-b x^4\right )^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 275
Rule 1167
Rule 1854
Rule 1855
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+206 x^6+j x^7}{\left (a-b x^4\right )^4} \, dx &=\frac {x \left (b c+a g+(b d+a h) x+(206 a+b e) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}-\frac {\int \frac {-b (11 b c-a g)-2 b (5 b d-a h) x+3 b (206 a-3 b e) x^2-4 b (2 b f-a j) x^3}{\left (a-b x^4\right )^3} \, dx}{12 a b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(206 a+b e) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac {\int \frac {7 b (11 b c-a g)+12 b (5 b d-a h) x-15 b (206 a-3 b e) x^2}{\left (a-b x^4\right )^2} \, dx}{96 a^2 b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(206 a+b e) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c-a g)-24 b (5 b d-a h) x+15 b (206 a-3 b e) x^2}{a-b x^4} \, dx}{384 a^3 b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(206 a+b e) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac {\int \left (-\frac {24 b (5 b d-a h) x}{a-b x^4}+\frac {-21 b (11 b c-a g)+15 b (206 a-3 b e) x^2}{a-b x^4}\right ) \, dx}{384 a^3 b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(206 a+b e) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c-a g)+15 b (206 a-3 b e) x^2}{a-b x^4} \, dx}{384 a^3 b^2}+\frac {(5 b d-a h) \int \frac {x}{a-b x^4} \, dx}{16 a^3 b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(206 a+b e) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}-\frac {\left (5 (206 a-3 b e)-\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^3 b}-\frac {\left (5 (206 a-3 b e)+\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^3 b}+\frac {(5 b d-a h) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{32 a^3 b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(206 a+b e) x^2+(b f+a j) x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (206 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {4 a (2 b f-a j)+x \left (b (11 b c-a g)+2 b (5 b d-a h) x-3 b (206 a-3 b e) x^2\right )}{96 a^2 b^2 \left (a-b x^4\right )^2}+\frac {\left (5 (206 a-3 b e)+\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}-\frac {\left (5 (206 a-3 b e)-\frac {7 \sqrt {b} (11 b c-a g)}{\sqrt {a}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}+\frac {(5 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 439, normalized size = 1.26 \[ \frac {3 \sqrt [4]{a} \sqrt [4]{b} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i-40 \sqrt [4]{a} b^{5/4} d-15 \sqrt {a} b e+7 a \sqrt {b} g-77 b^{3/2} c\right )+3 \sqrt [4]{a} \sqrt [4]{b} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (8 a^{5/4} \sqrt [4]{b} h-5 a^{3/2} i-40 \sqrt [4]{a} b^{5/4} d+15 \sqrt {a} b e-7 a \sqrt {b} g+77 b^{3/2} c\right )+6 \sqrt [4]{a} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 a^{3/2} i-15 \sqrt {a} b e-7 a \sqrt {b} g+77 b^{3/2} c\right )-\frac {16 a^2 \left (12 a^2 j+a b x (g+x (2 h+3 i x))-b^2 x (11 c+x (10 d+9 e x))\right )}{\left (a-b x^4\right )^2}+\frac {128 a^3 \left (a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{\left (a-b x^4\right )^3}-\frac {4 a b x (7 a g+3 a x (4 h+5 i x)-77 b c-15 b x (4 d+3 e x))}{a-b x^4}-24 \sqrt {a} \sqrt {b} (a h-5 b d) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{1536 a^4 b^2} \]
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 = 759, normalized size = 2.17 \[ -\frac {5}{1024} \, 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^{3} 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^{3} b^{4}}\right )} - \frac {5}{1024} \, 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^{3} 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^{3} b^{4}}\right )} - \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 8 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h + 15 \, \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 )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g + 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 8 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h - 15 \, \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 )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 15 \, \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 )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} + \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 15 \, \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 )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} + \frac {15 \, a b^{3} i x^{11} - 45 \, b^{4} x^{11} e - 60 \, b^{4} d x^{10} + 12 \, a b^{3} h x^{10} - 77 \, b^{4} c x^{9} + 7 \, a b^{3} g x^{9} - 42 \, a^{2} b^{2} i x^{7} + 126 \, a b^{3} x^{7} e + 160 \, a b^{3} d x^{6} - 32 \, a^{2} b^{2} h x^{6} + 198 \, a b^{3} c x^{5} - 18 \, a^{2} b^{2} g x^{5} - 48 \, a^{3} b j x^{4} - 5 \, a^{3} b i x^{3} - 113 \, a^{2} b^{2} x^{3} e - 132 \, a^{2} b^{2} d x^{2} - 12 \, a^{3} b h x^{2} - 153 \, a^{2} b^{2} c x - 21 \, a^{3} b g x - 32 \, a^{3} b f + 16 \, a^{4} j}{384 \, {\left (b x^{4} - a\right )}^{3} a^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 538, normalized size = 1.54 \[ \frac {h \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{64 \sqrt {a b}\, a^{2} b}-\frac {5 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{64 \sqrt {a b}\, a^{3}}+\frac {5 i \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b^{2}}-\frac {5 i \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b^{2}}-\frac {15 e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {15 e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 a^{3} b}-\frac {7 \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 )}{512 a^{3} b}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 a^{4}}+\frac {77 \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 )}{512 a^{4}}+\frac {\frac {5 \left (a i -3 b e \right ) b \,x^{11}}{128 a^{3}}+\frac {\left (a h -5 b d \right ) b \,x^{10}}{32 a^{3}}+\frac {7 \left (a g -11 b c \right ) b \,x^{9}}{384 a^{3}}-\frac {7 \left (a i -3 b e \right ) x^{7}}{64 a^{2}}-\frac {\left (a h -5 b d \right ) x^{6}}{12 a^{2}}-\frac {j \,x^{4}}{8 b}-\frac {3 \left (a g -11 b c \right ) x^{5}}{64 a^{2}}-\frac {\left (5 a i +113 b e \right ) x^{3}}{384 a b}-\frac {\left (a h +11 b d \right ) x^{2}}{32 a b}-\frac {\left (7 a g +51 b c \right ) x}{128 a b}+\frac {a j -2 b f}{24 b^{2}}}{\left (b \,x^{4}-a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 463, normalized size = 1.33 \[ -\frac {15 \, {\left (3 \, b^{4} e - a b^{3} i\right )} x^{11} + 12 \, {\left (5 \, b^{4} d - a b^{3} h\right )} x^{10} + 7 \, {\left (11 \, b^{4} c - a b^{3} g\right )} x^{9} + 48 \, a^{3} b j x^{4} - 42 \, {\left (3 \, a b^{3} e - a^{2} b^{2} i\right )} x^{7} - 32 \, {\left (5 \, a b^{3} d - a^{2} b^{2} h\right )} x^{6} - 18 \, {\left (11 \, a b^{3} c - a^{2} b^{2} g\right )} x^{5} + 32 \, a^{3} b f - 16 \, a^{4} j + {\left (113 \, a^{2} b^{2} e + 5 \, a^{3} b i\right )} x^{3} + 12 \, {\left (11 \, a^{2} b^{2} d + a^{3} b h\right )} x^{2} + 3 \, {\left (51 \, a^{2} b^{2} c + 7 \, a^{3} b g\right )} x}{384 \, {\left (a^{3} b^{5} x^{12} - 3 \, a^{4} b^{4} x^{8} + 3 \, a^{5} b^{3} x^{4} - a^{6} b^{2}\right )}} + \frac {\frac {8 \, {\left (5 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {8 \, {\left (5 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g + 5 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (77 \, b^{\frac {3}{2}} c + 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g - 5 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{512 \, a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.40, size = 2764, normalized size = 7.92 \[ \text {result too large to display} \]
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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