Optimal. Leaf size=462 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(a h+5 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (a g+11 b c)+12 x (a h+5 b d)+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (2 x (a h+5 b d)+a g+11 b c+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{x \left (x (b d-a h)-a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]
[Out]
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Rubi [A] time = 1.32749, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.343 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(a h+5 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (a g+11 b c)+12 x (a h+5 b d)+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (2 x (a h+5 b d)+a g+11 b c+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{x \left (x (b d-a h)-a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^4)^4,x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a)**4,x)
[Out]
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Mathematica [A] time = 0.866516, size = 461, normalized size = 1. \[ \frac{-6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (16 a^{5/4} h+15 \sqrt{2} \sqrt{a} b^{3/4} e+80 \sqrt [4]{a} b d+7 \sqrt{2} a \sqrt [4]{b} g+77 \sqrt{2} b^{5/4} c\right )+6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-16 a^{5/4} h+15 \sqrt{2} \sqrt{a} b^{3/4} e-80 \sqrt [4]{a} b d+7 \sqrt{2} a \sqrt [4]{b} g+77 \sqrt{2} b^{5/4} c\right )-\frac{256 a^{11/4} \sqrt{b} (a (f+x (g+h x))-b x (c+x (d+e x)))}{\left (a+b x^4\right )^3}+\frac{32 a^{7/4} \sqrt{b} x (a (g+2 h x)+11 b c+b x (10 d+9 e x))}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} \sqrt{b} x \left (7 a g+12 a h x+77 b c+60 b d x+45 b e x^2\right )}{a+b x^4}-3 \sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )+3 \sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{3072 a^{15/4} b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(a + b*x^4)^4,x]
[Out]
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Maple [A] time = 0.02, size = 608, normalized size = 1.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^4,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^4,x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^4,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.225241, size = 703, normalized size = 1.52 \[ \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b^{2} d + 8 \, \sqrt{2} \sqrt{a b} a b h + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b^{2} d + 8 \, \sqrt{2} \sqrt{a b} a b h + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac{45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 12 \, a b^{2} h x^{10} + 77 \, b^{3} c x^{9} + 7 \, a b^{2} g x^{9} + 126 \, a b^{2} x^{7} e + 160 \, a b^{2} d x^{6} + 32 \, a^{2} b h x^{6} + 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} - 12 \, a^{3} h x^{2} + 153 \, a^{2} b c x - 21 \, a^{3} g x - 32 \, a^{3} f}{384 \,{\left (b x^{4} + a\right )}^{3} a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^4,x, algorithm="giac")
[Out]