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

Optimal. Leaf size=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]

(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*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2)) - ((Sqrt[b]*(3
*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8
*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*Arc
Tan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) - ((Sqrt[b]*(3
*b*c + a*g) - Sqrt[a]*(b*e + 3*a*i))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + S
qrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) - Sqrt[a]*(b
*e + 3*a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]
*a^(7/4)*b^(7/4))

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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]

(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*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*b^(3/2)) - ((Sqrt[b]*(3
*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8
*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*Arc
Tan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(7/4)) - ((Sqrt[b]*(3
*b*c + a*g) - Sqrt[a]*(b*e + 3*a*i))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + S
qrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(7/4)) + ((Sqrt[b]*(3*b*c + a*g) - Sqrt[a]*(b
*e + 3*a*i))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]
*a^(7/4)*b^(7/4))

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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]

-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)*atan(sqrt(b)*x**2/sqrt(a))/(4*a**(3/2)*b**(3/2)) + sqrt(2)*(sqrt
(a)*(3*a*i + b*e) - sqrt(b)*(a*g + 3*b*c))*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sq
rt(a)*sqrt(b) + b*x**2)/(32*a**(7/4)*b**(7/4)) - sqrt(2)*(sqrt(a)*(3*a*i + b*e)
- sqrt(b)*(a*g + 3*b*c))*log(sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x
**2)/(32*a**(7/4)*b**(7/4)) - sqrt(2)*(sqrt(a)*(3*a*i + b*e) + sqrt(b)*(a*g + 3*
b*c))*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(16*a**(7/4)*b**(7/4)) + sqrt(2)*(sq
rt(a)*(3*a*i + b*e) + sqrt(b)*(a*g + 3*b*c))*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4
))/(16*a**(7/4)*b**(7/4))

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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]

((-8*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*Sqrt[2]*b^(3/2)*c + 4*a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e
+ Sqrt[2]*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 - (S
qrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(3*Sqrt[2]*b^(3/2)*c - 4*a^(1/4)*b^(5/4)*d + Sqrt
[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g - 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*
i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-3*b^(3/2)*c + Sqrt[a]*b*e
- a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*
x^2] + Sqrt[2]*(3*b^(3/2)*c - Sqrt[a]*b*e + a*Sqrt[b]*g - 3*a^(3/2)*i)*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(7/4))

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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]

(-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)+1/16/b/a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*g+3/16*c/a^2*(
a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/16/b/a*(a/b)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(a/b)^(1/4)*x-1)*g+3/16*c/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/b)^(1/4)*x-1)+1/32/b/a*(a/b)^(1/4)*2^(1/2)*ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b
)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))*g+3/32*c/a^2*(a/b)^(1/4)*2^(1/
2)*ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(
1/2)))+1/4/(a^3*b^3)^(1/2)*arctan(x^2*(b/a)^(1/2))*a*h+1/4*b*d/(a^3*b^3)^(1/2)*a
rctan(x^2*(b/a)^(1/2))+3/32/b^2/(a/b)^(1/4)*2^(1/2)*ln((x^2-(a/b)^(1/4)*x*2^(1/2
)+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))*i+1/32*e/a/b/(a/b)^(1/4)
*2^(1/2)*ln((x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(
a/b)^(1/2)))+3/16/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*i+1/16
*e/a/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+3/16/b^2/(a/b)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*i+1/16*e/a/b/(a/b)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(a/b)^(1/4)*x-1)

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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 = 0.23202, 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]

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