Optimal. Leaf size=203 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac {(b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]
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Rubi [A] time = 0.27, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {1858, 1876, 275, 208, 1167, 205} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac {(b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 275
Rule 1167
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+193 x^6}{\left (a-b x^4\right )^2} \, dx &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \frac {-b (3 b c-a g)-2 b (b d-a h) x+b (579 a-b e) x^2}{a-b x^4} \, dx}{4 a b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \left (-\frac {2 b (b d-a h) x}{a-b x^4}+\frac {-b (3 b c-a g)+b (579 a-b e) x^2}{a-b x^4}\right ) \, dx}{4 a b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \frac {-b (3 b c-a g)+b (579 a-b e) x^2}{a-b x^4} \, dx}{4 a b^2}+\frac {(b d-a h) \int \frac {x}{a-b x^4} \, dx}{2 a b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac {\left (579 a-b e-\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{8 a b}-\frac {\left (579 a-b e+\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{8 a b}+\frac {(b d-a h) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{4 a b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac {\left (579 a-b e+\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{5/4} b^{7/4}}-\frac {\left (579 a-b e-\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{5/4} b^{7/4}}+\frac {(b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 302, normalized size = 1.49 \[ \frac {\frac {4 a^{3/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{a-b x^4}+\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d-\sqrt {a} b e+a \sqrt {b} g-3 b^{3/2} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d+\sqrt {a} b e-a \sqrt {b} g+3 b^{3/2} c\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-\sqrt {a} b e-a \sqrt {b} g+3 b^{3/2} c\right )-2 \sqrt [4]{a} \sqrt [4]{b} (a h-b d) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{16 a^{7/4} b^{7/4}} \]
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 [B] time = 0.20, size = 583, normalized size = 2.87 \[ -\frac {3}{32} \, 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 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 b^{4}}\right )} - \frac {3}{32} \, 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 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 b^{4}}\right )} - \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h + \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 )}{16 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g + 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h - \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 )}{16 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - \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 )}{32 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} + \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - \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 )}{32 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {a i x^{3} + b x^{3} e + b d x^{2} + a h x^{2} + b c x + a g x + a f}{4 \, {\left (b x^{4} - a\right )} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 409, normalized size = 2.01 \[ -\frac {d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{8 \sqrt {a b}\, a}+\frac {h \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{8 \sqrt {a b}\, b}-\frac {e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 a b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 a b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 a^{2}}+\frac {3 i \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {3 i \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {-\frac {\left (a i +b e \right ) x^{3}}{4 a b}-\frac {\left (a h +b d \right ) x^{2}}{4 a b}-\frac {f}{4 b}-\frac {\left (a g +b c \right ) x}{4 a b}}{b \,x^{4}-a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 260, normalized size = 1.28 \[ -\frac {{\left (b e + a i\right )} x^{3} + {\left (b d + a h\right )} x^{2} + a f + {\left (b c + a g\right )} x}{4 \, {\left (a b^{2} x^{4} - a^{2} b\right )}} + \frac {\frac {2 \, {\left (b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {2 \, {\left (b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (3 \, b^{\frac {3}{2}} c - \sqrt {a} b e - a \sqrt {b} g + 3 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (3 \, b^{\frac {3}{2}} c + \sqrt {a} b e - a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{16 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.67, size = 2611, normalized size = 12.86 \[ \text {result too large to display} \]
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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