Optimal. Leaf size=331 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (2 x (5 b d-a h)+3 x^2 (3 b e-a i)-a g+11 b c\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]
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Rubi [A] time = 1.27834, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (2 x (5 b d-a h)+3 x^2 (3 b e-a i)-a g+11 b c\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^4,x]
[Out]
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Rubi in Sympy [A] time = 155.188, size = 304, normalized size = 0.92 \[ \frac{x \left (a g + b c + b f x^{3} + x^{2} \left (a i + b e\right ) + x \left (a h + b d\right )\right )}{12 a b \left (a - b x^{4}\right )^{3}} + \frac{8 a f - x \left (a g - 11 b c + 3 x^{2} \left (a i - 3 b e\right ) + 2 x \left (a h - 5 b d\right )\right )}{96 a^{2} b \left (a - b x^{4}\right )^{2}} - \frac{x \left (7 a g - 77 b c + 15 x^{2} \left (2 a - 3 b e\right ) + 12 x \left (a h - 5 b d\right )\right )}{384 a^{3} b \left (a - b x^{4}\right )} - \frac{\left (a h - 5 b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} b^{\frac{3}{2}}} + \frac{\left (\sqrt{a} \left (10 a - 15 b e\right ) - 7 a \sqrt{b} g + 77 b^{\frac{3}{2}} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{7}{4}}} - \frac{\left (\sqrt{a} \left (10 a - 15 b e\right ) + 7 a \sqrt{b} g - 77 b^{\frac{3}{2}} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((i*x**6+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 = 0.742444, size = 422, normalized size = 1.27 \[ \frac{3 \sqrt [4]{a} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i-40 \sqrt [4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right )-3 \sqrt [4]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i+40 \sqrt [4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right )+6 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 a^{3/2} i-15 \sqrt{a} b e-7 a \sqrt{b} g+77 b^{3/2} c\right )+\frac{128 a^3 b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^3}-\frac{16 a^2 b^{3/4} x (a (g+x (2 h+3 i x))-b (11 c+x (10 d+9 e x)))}{\left (a-b x^4\right )^2}-\frac{4 a b^{3/4} x (7 a g+3 a x (4 h+5 i x)-77 b c-15 b x (4 d+3 e x))}{a-b x^4}-24 \sqrt{a} \sqrt [4]{b} (a h-5 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{1536 a^4 b^{7/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^4,x]
[Out]
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Maple [A] time = 0.02, size = 551, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((i*x^6+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((i*x^6 + 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((i*x^6 + 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((i*x**6+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 = 0.224947, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^4,x, algorithm="giac")
[Out]