Optimal. Leaf size=534 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right )}{512 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right )}{512 \sqrt{2} a^{15/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right )}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right )}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{(a h+5 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{4 a (a j+2 b f)-x \left (b (a g+11 b c)+2 b x (a h+5 b d)+3 b x^2 (a i+3 b e)\right )}{96 a^2 b^2 \left (a+b x^4\right )^2}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{12 a b \left (a+b x^4\right )^3} \]
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Rubi [A] time = 1.86061, antiderivative size = 534, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right )}{512 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (7 \sqrt{b} (a g+11 b c)-5 \sqrt{a} (a i+3 b e)\right )}{512 \sqrt{2} a^{15/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right )}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (7 \sqrt{b} (a g+11 b c)+5 \sqrt{a} (a i+3 b e)\right )}{256 \sqrt{2} a^{15/4} b^{7/4}}+\frac{(a h+5 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (a g+11 b c)+12 x (a h+5 b d)+15 x^2 (a i+3 b e)\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{4 a (a j+2 b f)-x \left (b (a g+11 b c)+2 b x (a h+5 b d)+3 b x^2 (a i+3 b e)\right )}{96 a^2 b^2 \left (a+b x^4\right )^2}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)+x^3 (b f-a j)-a g+b c\right )}{12 a b \left (a+b x^4\right )^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(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((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)**4,x)
[Out]
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Mathematica [A] time = 1.12429, size = 555, normalized size = 1.04 \[ \frac{-6 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (16 a^{5/4} \sqrt [4]{b} h+5 \sqrt{2} a^{3/2} i+80 \sqrt [4]{a} b^{5/4} d+15 \sqrt{2} \sqrt{a} b e+7 \sqrt{2} a \sqrt{b} g+77 \sqrt{2} b^{3/2} c\right )+6 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-16 a^{5/4} \sqrt [4]{b} h+5 \sqrt{2} a^{3/2} i-80 \sqrt [4]{a} b^{5/4} d+15 \sqrt{2} \sqrt{a} b e+7 \sqrt{2} a \sqrt{b} g+77 \sqrt{2} b^{3/2} c\right )+3 \sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (5 a^{3/2} i+15 \sqrt{a} b e-7 a \sqrt{b} g-77 b^{3/2} c\right )+3 \sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 a^{3/2} i-15 \sqrt{a} b e+7 a \sqrt{b} g+77 b^{3/2} c\right )+\frac{8 a^{3/4} b x (7 a g+3 a x (4 h+5 i x)+77 b c+15 b x (4 d+3 e x))}{a+b x^4}+\frac{256 a^{11/4} \left (a^2 j-a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{\left (a+b x^4\right )^3}-\frac{32 a^{7/4} \left (12 a^2 j-a b x (g+x (2 h+3 i x))-b^2 x (11 c+x (10 d+9 e x))\right )}{\left (a+b x^4\right )^2}}{3072 a^{15/4} b^2} \]
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)^4,x]
[Out]
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Maple [A] time = 0.02, size = 784, normalized size = 1.5 \[ \text{result too large to display} \]
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)^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((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)^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((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)^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((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)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.228247, size = 1035, 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)^4,x, algorithm="giac")
[Out]