Optimal. Leaf size=205 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{(a h+b d) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{(a j+b f) \log \left (a-b x^4\right )}{4 b^2}-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{i x^3}{3 b}-\frac{j x^4}{4 b} \]
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Rubi [A] time = 0.658252, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.196 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{(a h+b d) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{(a j+b f) \log \left (a-b x^4\right )}{4 b^2}-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{i x^3}{3 b}-\frac{j x^4}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{i x^{3}}{3 b} - \frac{\int g\, dx}{b} - \frac{\int ^{x^{2}} h\, dx}{2 b} - \frac{\int ^{x^{2}} x\, dx}{2 b} - \frac{\left (a + b f\right ) \log{\left (a - b x^{4} \right )}}{4 b^{2}} + \frac{\left (a h + b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{3}{2}}} - \frac{\left (\sqrt{a} \left (a i + b e\right ) - \sqrt{b} \left (a g + b c\right )\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{7}{4}}} + \frac{\left (\sqrt{a} \left (a i + b e\right ) + \sqrt{b} \left (a g + b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((j*x**7+i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)
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Mathematica [A] time = 1.03089, size = 318, normalized size = 1.55 \[ \frac{-\frac{3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{5/4} \sqrt [4]{b} h+a^{3/2} i+\sqrt [4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-a^{5/4} \sqrt [4]{b} h+a^{3/2} i-\sqrt [4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (a^{3/2} (-i)-\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{3 \sqrt [4]{b} (a h+b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{a}}-\frac{3 (a j+b f) \log \left (a-b x^4\right )}{\sqrt [4]{b}}-12 b^{3/4} g x-6 b^{3/4} h x^2-4 b^{3/4} i x^3-3 b^{3/4} j x^4}{12 b^{7/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4),x]
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Maple [B] time = 0.009, size = 393, normalized size = 1.9 \[ -{\frac{j{x}^{4}}{4\,b}}-{\frac{i{x}^{3}}{3\,b}}-{\frac{h{x}^{2}}{2\,b}}-{\frac{gx}{b}}+{\frac{g}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{g}{4\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{ah}{4\,b}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{d}{4}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{ai}{2\,{b}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{e}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{ai}{4\,{b}^{2}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{4\,b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\ln \left ( b{x}^{4}-a \right ) aj}{4\,{b}^{2}}}-{\frac{f\ln \left ( b{x}^{4}-a \right ) }{4\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x)
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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(-(j*x^7 + i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(j*x^7 + i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a),x, algorithm="fricas")
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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((j*x**7+i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)
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GIAC/XCAS [A] time = 0.230491, size = 828, normalized size = 4.04 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(j*x^7 + i*x^6 + h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a),x, algorithm="giac")
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