Optimal. Leaf size=285 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{32 a^2 b^2 \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.924048, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (b (7 b c-a g)+2 b x (3 b d-a h)+b x^2 (5 b e-3 a i)\right )+4 a (b f-a j)}{32 a^2 b^2 \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{8 a b \left (a-b x^4\right )^2} \]
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)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 141.004, size = 267, normalized size = 0.94 \[ \frac{x \left (a g + b c + x^{3} \left (a j + b f\right ) + x^{2} \left (a i + b e\right ) + x \left (a h + b d\right )\right )}{8 a b \left (a - b x^{4}\right )^{2}} - \frac{4 a \left (a j - b f\right ) + x \left (b x^{2} \left (3 a i - 5 b e\right ) + 2 b x \left (a h - 3 b d\right ) + b \left (a g - 7 b c\right )\right )}{32 a^{2} b^{2} \left (a - b x^{4}\right )} - \frac{\left (a h - 3 b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} b^{\frac{3}{2}}} + \frac{\left (\sqrt{a} \left (6 a - 5 b e\right ) - 3 a \sqrt{b} g + 21 b^{\frac{3}{2}} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{7}{4}}} - \frac{\left (\sqrt{a} \left (6 a - 5 b e\right ) + 3 a \sqrt{b} g - 21 b^{\frac{3}{2}} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{7}{4}}} \]
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)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.542887, size = 380, normalized size = 1.33 \[ \frac{\sqrt [4]{b} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d-5 \sqrt{a} b e+3 a \sqrt{b} g-21 b^{3/2} c\right )+\sqrt [4]{b} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d+5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right )+2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right )+\frac{16 a^{7/4} \left (a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{\left (a-b x^4\right )^2}-\frac{4 a^{3/4} \left (8 a^2 j+a b x (g+x (2 h+3 i x))-b^2 x (7 c+x (6 d+5 e x))\right )}{a-b x^4}-4 \sqrt [4]{a} \sqrt{b} (a h-3 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{128 a^{11/4} b^2} \]
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)^3,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.017, size = 517, normalized size = 1.8 \[ \text{result too large to display} \]
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)^3,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(-(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)^3,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(-(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)^3,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((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)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.232313, size = 986, normalized size = 3.46 \[ \text{result too large to display} \]
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)^3,x, algorithm="giac")
[Out]