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

Optimal. Leaf size=394 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{4 a f-x \left (a g+7 b c+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]

[Out]

(x*(b*c - a*g + b*d*x + b*e*x^2 + b*f*x^3))/(8*a*b*(a + b*x^4)^2) - (4*a*f - x*(
7*b*c + a*g + 6*b*d*x + 5*b*e*x^2))/(32*a^2*b*(a + b*x^4)) + (3*d*ArcTan[(Sqrt[b
]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]) - ((21*b*c + 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*
ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) + ((21*b*
c + 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sq
rt[2]*a^(11/4)*b^(5/4)) - ((21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] -
Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(5/4)) + ((21*
b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqr
t[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(5/4))

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Rubi [A]  time = 0.929769, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{4 a f-x \left (a g+7 b c+6 b d x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*c - a*g + b*d*x + b*e*x^2 + b*f*x^3))/(8*a*b*(a + b*x^4)^2) - (4*a*f - x*(
7*b*c + a*g + 6*b*d*x + 5*b*e*x^2))/(32*a^2*b*(a + b*x^4)) + (3*d*ArcTan[(Sqrt[b
]*x^2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]) - ((21*b*c + 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*
ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*b^(5/4)) + ((21*b*
c + 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(64*Sq
rt[2]*a^(11/4)*b^(5/4)) - ((21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] -
Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(5/4)) + ((21*
b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqr
t[b]*x^2])/(128*Sqrt[2]*a^(11/4)*b^(5/4))

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Rubi in Sympy [A]  time = 149.134, size = 386, normalized size = 0.98 \[ - \frac{x \left (a g - b c - b d x - b e x^{2} - b f x^{3}\right )}{8 a b \left (a + b x^{4}\right )^{2}} - \frac{4 a f - x \left (a g + 7 b c + 6 b d x + 5 b e x^{2}\right )}{32 a^{2} b \left (a + b x^{4}\right )} + \frac{3 d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{b}} - \frac{\sqrt{2} \left (- 5 \sqrt{a} \sqrt{b} e + 3 a g + 21 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{11}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (- 5 \sqrt{a} \sqrt{b} e + 3 a g + 21 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{256 a^{\frac{11}{4}} b^{\frac{5}{4}}} - \frac{\sqrt{2} \left (5 \sqrt{a} \sqrt{b} e + 3 a g + 21 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (5 \sqrt{a} \sqrt{b} e + 3 a g + 21 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{11}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a*g - b*c - b*d*x - b*e*x**2 - b*f*x**3)/(8*a*b*(a + b*x**4)**2) - (4*a*f -
x*(a*g + 7*b*c + 6*b*d*x + 5*b*e*x**2))/(32*a**2*b*(a + b*x**4)) + 3*d*atan(sqrt
(b)*x**2/sqrt(a))/(16*a**(5/2)*sqrt(b)) - sqrt(2)*(-5*sqrt(a)*sqrt(b)*e + 3*a*g
+ 21*b*c)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(256*a**(
11/4)*b**(5/4)) + sqrt(2)*(-5*sqrt(a)*sqrt(b)*e + 3*a*g + 21*b*c)*log(sqrt(2)*a*
*(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(256*a**(11/4)*b**(5/4)) - sqrt(2)
*(5*sqrt(a)*sqrt(b)*e + 3*a*g + 21*b*c)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(1
28*a**(11/4)*b**(5/4)) + sqrt(2)*(5*sqrt(a)*sqrt(b)*e + 3*a*g + 21*b*c)*atan(1 +
sqrt(2)*b**(1/4)*x/a**(1/4))/(128*a**(11/4)*b**(5/4))

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Mathematica [A]  time = 0.596171, size = 366, normalized size = 0.93 \[ \frac{-\frac{32 a^{7/4} \sqrt [4]{b} (a (f+g x)-b x (c+x (d+e x)))}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} \sqrt [4]{b} x (a g+7 b c+b x (6 d+5 e x))}{a+b x^4}-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (24 \sqrt [4]{a} b^{3/4} d+5 \sqrt{2} \sqrt{a} \sqrt{b} e+3 \sqrt{2} a g+21 \sqrt{2} b c\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-24 \sqrt [4]{a} b^{3/4} d+5 \sqrt{2} \sqrt{a} \sqrt{b} e+3 \sqrt{2} a g+21 \sqrt{2} b c\right )+\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g-21 b c\right )+\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{256 a^{11/4} b^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

((8*a^(3/4)*b^(1/4)*x*(7*b*c + a*g + b*x*(6*d + 5*e*x)))/(a + b*x^4) - (32*a^(7/
4)*b^(1/4)*(a*(f + g*x) - b*x*(c + x*(d + e*x))))/(a + b*x^4)^2 - 2*(21*Sqrt[2]*
b*c + 24*a^(1/4)*b^(3/4)*d + 5*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 3*Sqrt[2]*a*g)*ArcTan
[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(21*Sqrt[2]*b*c - 24*a^(1/4)*b^(3/4)*d + 5
*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 3*Sqrt[2]*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/
4)] + Sqrt[2]*(-21*b*c + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1
/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e + 3*a*g)*Lo
g[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(256*a^(11/4)*b^(5/4))

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Maple [A]  time = 0.018, size = 521, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(5/32/a^2*b*e*x^7+3/16/a^2*d*b*x^6+1/32*(a*g+7*b*c)/a^2*x^5+9/32/a*e*x^3+5/16*d/
a*x^2-1/32*(3*a*g-11*b*c)/a/b*x-1/8*f/b)/(b*x^4+a)^2+3/128*(a/b)^(1/4)/a^2/b*2^(
1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*g+21/128*c/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(a/b)^(1/4)*x-1)+3/256*(a/b)^(1/4)/a^2/b*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+21/256*c/a^3*(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)))+3/128*(a/b)^(1/4)/a^2/b*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+
1)*g+21/128*c/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+3/16*b*d/(
a^5*b^3)^(1/2)*arctan(x^2*(b/a)^(1/2))+5/256*e/a^2/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)))+5/1
28*e/a^2/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+5/128*e/a^2/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((g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^3,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((g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 + a)^3,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((g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**4+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225624, size = 562, normalized size = 1.43 \[ \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b^{3}} + \frac{5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 7 \, b^{2} c x^{5} + a b g x^{5} + 9 \, a b x^{3} e + 10 \, a b d x^{2} + 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*b^2*d + 21*(a*b^3)^(1/4)*b^2*c + 3*(a*b^3)^(
1/4)*a*b*g + 5*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(
a/b)^(1/4))/(a^3*b^3) + 1/128*sqrt(2)*(12*sqrt(2)*sqrt(a*b)*b^2*d + 21*(a*b^3)^(
1/4)*b^2*c + 3*(a*b^3)^(1/4)*a*b*g + 5*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x
- sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3*b^3) + 1/256*sqrt(2)*(21*(a*b^3)^(1/4)*
b^2*c + 3*(a*b^3)^(1/4)*a*b*g - 5*(a*b^3)^(3/4)*e)*ln(x^2 + sqrt(2)*x*(a/b)^(1/4
) + sqrt(a/b))/(a^3*b^3) - 1/256*sqrt(2)*(21*(a*b^3)^(1/4)*b^2*c + 3*(a*b^3)^(1/
4)*a*b*g - 5*(a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b
^3) + 1/32*(5*b^2*x^7*e + 6*b^2*d*x^6 + 7*b^2*c*x^5 + a*b*g*x^5 + 9*a*b*x^3*e +
10*a*b*d*x^2 + 11*a*b*c*x - 3*a^2*g*x - 4*a^2*f)/((b*x^4 + a)^2*a^2*b)