Optimal. Leaf size=402 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right )}{4 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right )}{4 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right )}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right )}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{(b d-a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{(b f-a j) \log \left (a+b x^4\right )}{4 b^2}+\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{i x^3}{3 b}+\frac{j x^4}{4 b} \]
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Rubi [A] time = 1.2483, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 12, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right )}{4 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right )}{4 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right )}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right )}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{(b d-a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{(b f-a j) \log \left (a+b x^4\right )}{4 b^2}+\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{i x^3}{3 b}+\frac{j x^4}{4 b} \]
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. \[ \frac{i x^{3}}{3 b} + \frac{\int g\, dx}{b} + \frac{\int ^{x^{2}} h\, dx}{2 b} + \frac{\int ^{x^{2}} x\, dx}{2 b} - \frac{\left (a - b f\right ) \log{\left (a + b x^{4} \right )}}{4 b^{2}} - \frac{\left (a h - b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{3}{2}}} - \frac{\sqrt{2} \left (\sqrt{a} \left (a i - b e\right ) - \sqrt{b} \left (a g - b c\right )\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} \left (a i - b e\right ) - \sqrt{b} \left (a g - b c\right )\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} \left (a i - b e\right ) + \sqrt{b} \left (a g - b c\right )\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} \left (a i - b e\right ) + \sqrt{b} \left (a g - b c\right )\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{7}{4}}} \]
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 = 0.626573, size = 445, normalized size = 1.11 \[ \frac{\frac{6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (2 a^{5/4} \sqrt [4]{b} h+\sqrt{2} a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d-\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g-\sqrt{2} b^{3/2} c\right )}{a^{3/4}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (2 a^{5/4} \sqrt [4]{b} h-\sqrt{2} a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e-\sqrt{2} a \sqrt{b} g+\sqrt{2} b^{3/2} c\right )}{a^{3/4}}-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{6 (b f-a j) \log \left (a+b x^4\right )}{\sqrt [4]{b}}+24 b^{3/4} g x+12 b^{3/4} h x^2+8 b^{3/4} i x^3+6 b^{3/4} j x^4}{24 b^{7/4}} \]
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 = 0.007, size = 627, normalized size = 1.6 \[ \text{result too large to display} \]
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")
[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((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)
[Out]
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GIAC/XCAS [A] time = 0.227175, size = 780, normalized size = 1.94 \[ \text{result too large to display} \]
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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