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

Optimal. Leaf size=184 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(4*a*b*(a - b*x^4)) + ((3*b*
c - Sqrt[a]*Sqrt[b]*e - a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(5/4)) +
((3*b*c + Sqrt[a]*Sqrt[b]*e - a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(5
/4)) + ((b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2))

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Rubi [A]  time = 0.462176, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(4*a*b*(a - b*x^4)) + ((3*b*
c - Sqrt[a]*Sqrt[b]*e - a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(5/4)) +
((3*b*c + Sqrt[a]*Sqrt[b]*e - a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(5
/4)) + ((b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2))

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Rubi in Sympy [A]  time = 79.1263, size = 167, normalized size = 0.91 \[ \frac{x \left (a g + b c + b e x^{2} + b f x^{3} + x \left (a h + b d\right )\right )}{4 a b \left (a - b x^{4}\right )} - \frac{\left (a h - b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} b^{\frac{3}{2}}} - \frac{\left (- \sqrt{a} \sqrt{b} e + a g - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{5}{4}}} - \frac{\left (\sqrt{a} \sqrt{b} e + a g - 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*g + b*c + b*e*x**2 + b*f*x**3 + x*(a*h + b*d))/(4*a*b*(a - b*x**4)) - (a*h
- b*d)*atanh(sqrt(b)*x**2/sqrt(a))/(4*a**(3/2)*b**(3/2)) - (-sqrt(a)*sqrt(b)*e +
a*g - 3*b*c)*atanh(b**(1/4)*x/a**(1/4))/(8*a**(7/4)*b**(5/4)) - (sqrt(a)*sqrt(b
)*e + a*g - 3*b*c)*atan(b**(1/4)*x/a**(1/4))/(8*a**(7/4)*b**(5/4))

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Mathematica [A]  time = 0.302911, size = 257, normalized size = 1.4 \[ \frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 a^{5/4} h-\sqrt{a} b^{3/4} e-2 \sqrt [4]{a} b d+a \sqrt [4]{b} g-3 b^{5/4} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (2 a^{5/4} h+\sqrt{a} b^{3/4} e-2 \sqrt [4]{a} b d-a \sqrt [4]{b} g+3 b^{5/4} c\right )+\frac{4 a^{3/4} \sqrt{b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{a-b x^4}-2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e+a g-3 b c\right )-2 \sqrt [4]{a} (a h-b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{16 a^{7/4} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((4*a^(3/4)*Sqrt[b]*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + h*x))))/(a - b*x^4) -
2*b^(1/4)*(-3*b*c + Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] + (-3*
b^(5/4)*c - 2*a^(1/4)*b*d - Sqrt[a]*b^(3/4)*e + a*b^(1/4)*g + 2*a^(5/4)*h)*Log[a
^(1/4) - b^(1/4)*x] + (3*b^(5/4)*c - 2*a^(1/4)*b*d + Sqrt[a]*b^(3/4)*e - a*b^(1/
4)*g + 2*a^(5/4)*h)*Log[a^(1/4) + b^(1/4)*x] - 2*a^(1/4)*(-(b*d) + a*h)*Log[Sqrt
[a] + Sqrt[b]*x^2])/(16*a^(7/4)*b^(3/2))

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Maple [B]  time = 0.013, size = 372, normalized size = 2. \[{\frac{1}{b{x}^{4}-a} \left ( -{\frac{e{x}^{3}}{4\,a}}-{\frac{ \left ( ah+bd \right ){x}^{2}}{4\,ab}}-{\frac{ \left ( ag+bc \right ) x}{4\,ab}}-{\frac{f}{4\,b}} \right ) }-{\frac{g}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{g}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{ah}{8}\ln \left ({1 \left ( -{a}^{2}b+{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) \left ( -{a}^{2}b-{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{3}{b}^{3}}}}}-{\frac{bd}{8}\ln \left ({1 \left ( -{a}^{2}b+{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) \left ( -{a}^{2}b-{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{3}{b}^{3}}}}}-{\frac{e}{8\,ab}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{16\,ab}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1/4/a*e*x^3-1/4*(a*h+b*d)/a/b*x^2-1/4*(a*g+b*c)/a/b*x-1/4*f/b)/(b*x^4-a)-1/8*(
a/b)^(1/4)/a/b*arctan(x/(a/b)^(1/4))*g+3/8*c/a^2*(a/b)^(1/4)*arctan(x/(a/b)^(1/4
))-1/16*(a/b)^(1/4)/a/b*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))*g+3/16*c/a^2*(a/b)^(
1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+1/8/(a^3*b^3)^(1/2)*ln((-a^2*b+x^2*(a^3
*b^3)^(1/2))/(-a^2*b-x^2*(a^3*b^3)^(1/2)))*a*h-1/8*b*d/(a^3*b^3)^(1/2)*ln((-a^2*
b+x^2*(a^3*b^3)^(1/2))/(-a^2*b-x^2*(a^3*b^3)^(1/2)))-1/8*e/a/b/(a/b)^(1/4)*arcta
n(x/(a/b)^(1/4))+1/16*e/a/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((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^2,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((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^2,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((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225356, size = 574, normalized size = 3.12 \[ -\frac{b x^{3} e + b d x^{2} + a h x^{2} + b c x + a g x + a f}{4 \,{\left (b x^{4} - a\right )} a b} - \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b^{2} d - 2 \, \sqrt{2} \sqrt{-a b} a b h - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (-a b^{3}\right )^{\frac{1}{4}} a b g - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b^{2} d - 2 \, \sqrt{2} \sqrt{-a b} a b h - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (-a b^{3}\right )^{\frac{1}{4}} a b g - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{1}{4}} a b g - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{1}{4}} a b g - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a)^2,x, algorithm="giac")

[Out]

-1/4*(b*x^3*e + b*d*x^2 + a*h*x^2 + b*c*x + a*g*x + a*f)/((b*x^4 - a)*a*b) - 1/1
6*sqrt(2)*(2*sqrt(2)*sqrt(-a*b)*b^2*d - 2*sqrt(2)*sqrt(-a*b)*a*b*h - 3*(-a*b^3)^
(1/4)*b^2*c + (-a*b^3)^(1/4)*a*b*g - (-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x +
sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^2*b^3) - 1/16*sqrt(2)*(2*sqrt(2)*sqrt(-a
*b)*b^2*d - 2*sqrt(2)*sqrt(-a*b)*a*b*h - 3*(-a*b^3)^(1/4)*b^2*c + (-a*b^3)^(1/4)
*a*b*g - (-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b
)^(1/4))/(a^2*b^3) + 1/32*sqrt(2)*(3*(-a*b^3)^(1/4)*b^2*c - (-a*b^3)^(1/4)*a*b*g
- (-a*b^3)^(3/4)*e)*ln(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^3) - 1
/32*sqrt(2)*(3*(-a*b^3)^(1/4)*b^2*c - (-a*b^3)^(1/4)*a*b*g - (-a*b^3)^(3/4)*e)*l
n(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^3)