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

Optimal. Leaf size=203 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.593013, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

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]

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

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Rubi in Sympy [A]  time = 86.02, size = 194, normalized size = 0.96 \[ \frac{x \left (a g + b c + b f x^{3} + x^{2} \left (a i + b e\right ) + x \left (a h + b d\right )\right )}{4 a b \left (a - b x^{4}\right )} - \frac{\left (a h - b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} b^{\frac{3}{2}}} - \frac{\left (\sqrt{a} \left (3 a i - b e\right ) + a \sqrt{b} g - 3 b^{\frac{3}{2}} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{7}{4}}} + \frac{\left (3 a^{\frac{3}{2}} i - \sqrt{a} b e - a \sqrt{b} g + 3 b^{\frac{3}{2}} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{7}{4}}} \]

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]

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

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Mathematica [A]  time = 0.44694, size = 302, normalized size = 1.49 \[ \frac{\frac{4 a^{3/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{a-b x^4}+\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d-\sqrt{a} b e+a \sqrt{b} g-3 b^{3/2} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d+\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right )-2 \sqrt [4]{a} \sqrt [4]{b} (a h-b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{16 a^{7/4} b^{7/4}} \]

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]

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

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Maple [B]  time = 0.013, size = 441, normalized size = 2.2 \[{\frac{1}{b{x}^{4}-a} \left ( -{\frac{ \left ( ai+be \right ){x}^{3}}{4\,ab}}-{\frac{ \left ( ah+bd \right ){x}^{2}}{4\,ab}}-{\frac{ \left ( ag+bc \right ) x}{4\,ab}}-{\frac{f}{4\,b}} \right ) }-{\frac{g}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{g}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{ah}{8}\ln \left ({1 \left ( -{a}^{2}b+{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) \left ( -{a}^{2}b-{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{3}{b}^{3}}}}}-{\frac{bd}{8}\ln \left ({1 \left ( -{a}^{2}b+{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) \left ( -{a}^{2}b-{x}^{2}\sqrt{{a}^{3}{b}^{3}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{3}{b}^{3}}}}}+{\frac{3\,i}{8\,{b}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{e}{8\,ab}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{3\,i}{16\,{b}^{2}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{16\,ab}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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]

(-1/4*(a*i+b*e)/a/b*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)))+3/8/b^2/(a/b)^(1/
4)*arctan(x/(a/b)^(1/4))*i-1/8*e/a/b/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))-3/16/b^2/
(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))*i+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((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]

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)^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((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]

Timed out

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GIAC/XCAS [A]  time = 0.229687, size = 848, normalized size = 4.18 \[ \text{result too large to display} \]

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]

-3/32*i*(2*sqrt(2)*(-a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4)
)/(-a/b)^(1/4))/(a*b^4) - sqrt(2)*(-a*b^3)^(3/4)*ln(x^2 + sqrt(2)*x*(-a/b)^(1/4)
+ sqrt(-a/b))/(a*b^4)) - 3/32*i*(2*sqrt(2)*(-a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2
*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a*b^4) + sqrt(2)*(-a*b^3)^(3/4)*ln(x^2
- sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a*b^4)) - 1/4*(a*i*x^3 + 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) - 1/16*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) - 1/16*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)*ln(x^2 - sqrt(2)*x*(-
a/b)^(1/4) + sqrt(-a/b))/(a^2*b^3)