Optimal. Leaf size=395 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{(a h+b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{4 a b \left (a+b x^4\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.16444, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (a g+3 b c)-\sqrt{a} (3 a i+b e)\right )}{16 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (a g+3 b c)+\sqrt{a} (3 a i+b e)\right )}{8 \sqrt{2} a^{7/4} b^{7/4}}+\frac{(a h+b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{4 a b \left (a+b x^4\right )} \]
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)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 139.125, size = 364, normalized size = 0.92 \[ - \frac{x \left (a g - b c - b f x^{3} + x^{2} \left (a i - b e\right ) + x \left (a h - b d\right )\right )}{4 a b \left (a + b x^{4}\right )} + \frac{\left (a h + b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} b^{\frac{3}{2}}} + \frac{\sqrt{2} \left (\sqrt{a} \left (3 a i + b e\right ) - \sqrt{b} \left (a g + 3 b c\right )\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} \left (3 a i + b e\right ) - \sqrt{b} \left (a g + 3 b c\right )\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} \left (3 a i + b e\right ) + \sqrt{b} \left (a g + 3 b c\right )\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} \left (3 a i + b e\right ) + \sqrt{b} \left (a g + 3 b c\right )\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{7}{4}}} \]
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)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.715132, size = 415, normalized size = 1.05 \[ \frac{-\frac{8 a^{3/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{a+b x^4}-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 \sqrt{2} a^{3/2} i+4 \sqrt [4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-4 a^{5/4} \sqrt [4]{b} h+3 \sqrt{2} a^{3/2} i-4 \sqrt [4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g+3 \sqrt{2} b^{3/2} c\right )+\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (3 a^{3/2} i+\sqrt{a} b e-a \sqrt{b} g-3 b^{3/2} c\right )+\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-3 a^{3/2} i-\sqrt{a} b e+a \sqrt{b} g+3 b^{3/2} c\right )}{32 a^{7/4} b^{7/4}} \]
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)^2,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.016, size = 658, 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)^2,x)
[Out]
_______________________________________________________________________________________
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)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
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)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.23202, size = 795, normalized size = 2.01 \[ \text{result too large to display} \]
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)^2,x, algorithm="giac")
[Out]