3.205 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a-b x^4\right )^4} \, dx\)

Optimal. Leaf size=331 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (2 x (5 b d-a h)+3 x^2 (3 b e-a i)-a g+11 b c\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^
3) + (x*(7*(11*b*c - a*g) + 12*(5*b*d - a*h)*x + 15*(3*b*e - a*i)*x^2))/(384*a^3
*b*(a - b*x^4)) + (8*a*f + x*(11*b*c - a*g + 2*(5*b*d - a*h)*x + 3*(3*b*e - a*i)
*x^2))/(96*a^2*b*(a - b*x^4)^2) + (((7*Sqrt[b]*(11*b*c - a*g))/Sqrt[a] - 5*(3*b*
e - a*i))*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/4)*b^(7/4)) + ((15*b*e + (7*Sq
rt[b]*(11*b*c - a*g))/Sqrt[a] - 5*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/
4)*b^(7/4)) + ((5*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*b^(3/2)
)

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Rubi [A]  time = 1.27834, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (2 x (5 b d-a h)+3 x^2 (3 b e-a i)-a g+11 b c\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^4,x]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^
3) + (x*(7*(11*b*c - a*g) + 12*(5*b*d - a*h)*x + 15*(3*b*e - a*i)*x^2))/(384*a^3
*b*(a - b*x^4)) + (8*a*f + x*(11*b*c - a*g + 2*(5*b*d - a*h)*x + 3*(3*b*e - a*i)
*x^2))/(96*a^2*b*(a - b*x^4)^2) + (((7*Sqrt[b]*(11*b*c - a*g))/Sqrt[a] - 5*(3*b*
e - a*i))*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/4)*b^(7/4)) + ((15*b*e + (7*Sq
rt[b]*(11*b*c - a*g))/Sqrt[a] - 5*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(13/
4)*b^(7/4)) + ((5*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*b^(3/2)
)

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Rubi in Sympy [A]  time = 155.188, size = 304, normalized size = 0.92 \[ \frac{x \left (a g + b c + b f x^{3} + x^{2} \left (a i + b e\right ) + x \left (a h + b d\right )\right )}{12 a b \left (a - b x^{4}\right )^{3}} + \frac{8 a f - x \left (a g - 11 b c + 3 x^{2} \left (a i - 3 b e\right ) + 2 x \left (a h - 5 b d\right )\right )}{96 a^{2} b \left (a - b x^{4}\right )^{2}} - \frac{x \left (7 a g - 77 b c + 15 x^{2} \left (2 a - 3 b e\right ) + 12 x \left (a h - 5 b d\right )\right )}{384 a^{3} b \left (a - b x^{4}\right )} - \frac{\left (a h - 5 b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} b^{\frac{3}{2}}} + \frac{\left (\sqrt{a} \left (10 a - 15 b e\right ) - 7 a \sqrt{b} g + 77 b^{\frac{3}{2}} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{7}{4}}} - \frac{\left (\sqrt{a} \left (10 a - 15 b e\right ) + 7 a \sqrt{b} g - 77 b^{\frac{3}{2}} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{256 a^{\frac{15}{4}} b^{\frac{7}{4}}} \]

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)**4,x)

[Out]

x*(a*g + b*c + b*f*x**3 + x**2*(a*i + b*e) + x*(a*h + b*d))/(12*a*b*(a - b*x**4)
**3) + (8*a*f - x*(a*g - 11*b*c + 3*x**2*(a*i - 3*b*e) + 2*x*(a*h - 5*b*d)))/(96
*a**2*b*(a - b*x**4)**2) - x*(7*a*g - 77*b*c + 15*x**2*(2*a - 3*b*e) + 12*x*(a*h
- 5*b*d))/(384*a**3*b*(a - b*x**4)) - (a*h - 5*b*d)*atanh(sqrt(b)*x**2/sqrt(a))
/(32*a**(7/2)*b**(3/2)) + (sqrt(a)*(10*a - 15*b*e) - 7*a*sqrt(b)*g + 77*b**(3/2)
*c)*atan(b**(1/4)*x/a**(1/4))/(256*a**(15/4)*b**(7/4)) - (sqrt(a)*(10*a - 15*b*e
) + 7*a*sqrt(b)*g - 77*b**(3/2)*c)*atanh(b**(1/4)*x/a**(1/4))/(256*a**(15/4)*b**
(7/4))

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Mathematica [A]  time = 0.742444, size = 422, normalized size = 1.27 \[ \frac{3 \sqrt [4]{a} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i-40 \sqrt [4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right )-3 \sqrt [4]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i+40 \sqrt [4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right )+6 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 a^{3/2} i-15 \sqrt{a} b e-7 a \sqrt{b} g+77 b^{3/2} c\right )+\frac{128 a^3 b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^3}-\frac{16 a^2 b^{3/4} x (a (g+x (2 h+3 i x))-b (11 c+x (10 d+9 e x)))}{\left (a-b x^4\right )^2}-\frac{4 a b^{3/4} x (7 a g+3 a x (4 h+5 i x)-77 b c-15 b x (4 d+3 e x))}{a-b x^4}-24 \sqrt{a} \sqrt [4]{b} (a h-5 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{1536 a^4 b^{7/4}} \]

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)^4,x]

[Out]

((-4*a*b^(3/4)*x*(-77*b*c + 7*a*g - 15*b*x*(4*d + 3*e*x) + 3*a*x*(4*h + 5*i*x)))
/(a - b*x^4) - (16*a^2*b^(3/4)*x*(-(b*(11*c + x*(10*d + 9*e*x))) + a*(g + x*(2*h
+ 3*i*x))))/(a - b*x^4)^2 + (128*a^3*b^(3/4)*(b*x*(c + x*(d + e*x)) + a*(f + x*
(g + x*(h + i*x)))))/(a - b*x^4)^3 + 6*a^(1/4)*(77*b^(3/2)*c - 15*Sqrt[a]*b*e -
7*a*Sqrt[b]*g + 5*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + 3*a^(1/4)*(-77*b^(3/2
)*c - 40*a^(1/4)*b^(5/4)*d - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g + 8*a^(5/4)*b^(1/4)*
h + 5*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] - 3*a^(1/4)*(-77*b^(3/2)*c + 40*a^(1/4
)*b^(5/4)*d - 15*Sqrt[a]*b*e + 7*a*Sqrt[b]*g - 8*a^(5/4)*b^(1/4)*h + 5*a^(3/2)*i
)*Log[a^(1/4) + b^(1/4)*x] - 24*Sqrt[a]*b^(1/4)*(-5*b*d + a*h)*Log[Sqrt[a] + Sqr
t[b]*x^2])/(1536*a^4*b^(7/4))

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Maple [A]  time = 0.02, size = 551, 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)^4,x)

[Out]

(5/128*(a*i-3*b*e)/a^3*b*x^11+1/32*(a*h-5*b*d)/a^3*b*x^10+7/384*(a*g-11*b*c)/a^3
*b*x^9-7/64/a^2*(a*i-3*b*e)*x^7-1/12/a^2*(a*h-5*b*d)*x^6-3/64/a^2*(a*g-11*b*c)*x
^5-1/384*(5*a*i+113*b*e)/a/b*x^3-1/32*(a*h+11*b*d)/a/b*x^2-1/128*(7*a*g+51*b*c)/
a/b*x-1/12*f/b)/(b*x^4-a)^3-7/256*(a/b)^(1/4)/a^3/b*arctan(x/(a/b)^(1/4))*g+77/2
56*c*(a/b)^(1/4)/a^4*arctan(x/(a/b)^(1/4))-7/512*(a/b)^(1/4)/a^3/b*ln((x+(a/b)^(
1/4))/(x-(a/b)^(1/4)))*g+77/512*c*(a/b)^(1/4)/a^4*ln((x+(a/b)^(1/4))/(x-(a/b)^(1
/4)))+1/64/(a^7*b^3)^(1/2)*ln((-a^4*b+x^2*(a^7*b^3)^(1/2))/(-a^4*b-x^2*(a^7*b^3)
^(1/2)))*a*h-5/64*b*d/(a^7*b^3)^(1/2)*ln((-a^4*b+x^2*(a^7*b^3)^(1/2))/(-a^4*b-x^
2*(a^7*b^3)^(1/2)))+5/256/a^2/b^2/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))*i-15/256*e/a
^3/b/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))-5/512/a^2/b^2/(a/b)^(1/4)*ln((x+(a/b)^(1/
4))/(x-(a/b)^(1/4)))*i+15/512*e/a^3/b/(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1
/4)))

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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((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]

Exception raised: ValueError

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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((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]

Exception raised: NotImplementedError

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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((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]

Timed out

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GIAC/XCAS [A]  time = 0.224947, size = 1, normalized size = 0. \[ \mathit{Done} \]

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)^4,x, algorithm="giac")

[Out]

Done