Optimal. Leaf size=293 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac {(5 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {x \left (7 (11 b c-a g)+12 x (5 b d-a h)+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {x \left (2 x (5 b d-a h)-a g+11 b c+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac {x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]
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Rubi [A] time = 0.43, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1858, 1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac {(5 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} b^{3/2}}+\frac {x \left (2 x (5 b d-a h)-a g+11 b c+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac {x \left (7 (11 b c-a g)+12 x (5 b d-a h)+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]
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}{\left (a-b x^4\right )^4} \, dx &=\frac {x \left (b c+a g+(b d+a h) x+b e x^2+b f 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-9 b^2 e x^2-8 b^2 f 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+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {8 a f+x \left (11 b c-a g+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \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+45 b^2 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+b e x^2+b f 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+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \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-45 b^2 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+b e x^2+b f 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+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \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)-45 b^2 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+b e x^2+b f 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+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac {\int \frac {-21 b (11 b c-a g)-45 b^2 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+b e x^2+b f 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+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e-7 a g\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^{7/2} \sqrt {b}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e-7 a g\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^{7/2} \sqrt {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+b e x^2+b f 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+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e-7 a g\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{5/4}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e-7 a g\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{5/4}}+\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 = 0.49, size = 360, normalized size = 1.23 \[ \frac {-3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (-8 a^{5/4} h+15 \sqrt {a} b^{3/4} e+40 \sqrt [4]{a} b d-7 a \sqrt [4]{b} g+77 b^{5/4} c\right )+3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (8 a^{5/4} h+15 \sqrt {a} b^{3/4} e-40 \sqrt [4]{a} b d-7 a \sqrt [4]{b} g+77 b^{5/4} c\right )+\frac {128 a^{11/4} \sqrt {b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^3}+\frac {16 a^{7/4} \sqrt {b} x (-a (g+2 h x)+11 b c+b x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac {4 a^{3/4} \sqrt {b} x \left (-7 a g-12 a h x+77 b c+60 b d x+45 b e x^2\right )}{a-b x^4}+6 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )-24 \sqrt [4]{a} (a h-5 b d) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{1536 a^{15/4} b^{3/2}} \]
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 = 0.26, size = 501, normalized size = 1.71 \[ -\frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 8 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h + 15 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g + 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 8 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h - 15 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 15 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} + \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 15 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} - 12 \, a b^{2} h x^{10} + 77 \, b^{3} c x^{9} - 7 \, a b^{2} g x^{9} - 126 \, a b^{2} x^{7} e - 160 \, a b^{2} d x^{6} + 32 \, a^{2} b h x^{6} - 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 12 \, a^{3} h x^{2} + 153 \, a^{2} b c x + 21 \, a^{3} g x + 32 \, a^{3} f}{384 \, {\left (b x^{4} - a\right )}^{3} a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 434, normalized size = 1.48 \[ \frac {h \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{64 \sqrt {a b}\, a^{2} b}-\frac {5 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{64 \sqrt {a b}\, a^{3}}-\frac {15 e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {15 e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 a^{3} b}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 a^{3} b}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 a^{4}}+\frac {-\frac {15 b^{2} e \,x^{11}}{128 a^{3}}+\frac {\left (a h -5 b d \right ) b \,x^{10}}{32 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {7 \left (a g -11 b c \right ) b \,x^{9}}{384 a^{3}}-\frac {\left (a h -5 b d \right ) x^{6}}{12 a^{2}}-\frac {113 e \,x^{3}}{384 a}-\frac {3 \left (a g -11 b c \right ) x^{5}}{64 a^{2}}-\frac {\left (a h +11 b d \right ) x^{2}}{32 a b}-\frac {f}{12 b}-\frac {\left (7 a g +51 b c \right ) x}{128 a b}}{\left (b \,x^{4}-a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.18, size = 389, normalized size = 1.33 \[ -\frac {45 \, b^{3} e x^{11} - 126 \, a b^{2} e x^{7} + 12 \, {\left (5 \, b^{3} d - a b^{2} h\right )} x^{10} + 7 \, {\left (11 \, b^{3} c - a b^{2} g\right )} x^{9} + 113 \, a^{2} b e x^{3} - 32 \, {\left (5 \, a b^{2} d - a^{2} b h\right )} x^{6} - 18 \, {\left (11 \, a b^{2} c - a^{2} b g\right )} x^{5} + 32 \, a^{3} f + 12 \, {\left (11 \, a^{2} b d + a^{3} h\right )} x^{2} + 3 \, {\left (51 \, a^{2} b c + 7 \, a^{3} g\right )} x}{384 \, {\left (a^{3} b^{4} x^{12} - 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} - a^{6} b\right )}} + \frac {\frac {8 \, {\left (5 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {8 \, {\left (5 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (77 \, b^{\frac {3}{2}} c + 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{512 \, a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.99, size = 1747, normalized size = 5.96 \[ \text {result too large to display} \]
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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