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

Optimal. Leaf size=437 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (7 (a g+11 b c)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (a g+11 b c+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]

[Out]

(x*(b*c - a*g + b*d*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a + b*x^4)^3) + (x*(7*(11*b
*c + a*g) + 60*b*d*x + 45*b*e*x^2))/(384*a^3*b*(a + b*x^4)) - (8*a*f - x*(11*b*c
+ a*g + 10*b*d*x + 9*b*e*x^2))/(96*a^2*b*(a + b*x^4)^2) + (5*d*ArcTan[(Sqrt[b]*
x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b]) - ((77*b*c + 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*A
rcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^(5/4)) + ((77*b*
c + 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*
Sqrt[2]*a^(15/4)*b^(5/4)) - ((77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(5/4)) + ((
77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x +
Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(5/4))

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Rubi [A]  time = 1.09223, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{512 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{256 \sqrt{2} a^{15/4} b^{5/4}}+\frac{5 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} \sqrt{b}}+\frac{x \left (7 (a g+11 b c)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a+b x^4\right )}-\frac{8 a f-x \left (a g+11 b c+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a+b x^4\right )^2}+\frac{x \left (-a g+b c+b d x+b e x^2+b f x^3\right )}{12 a b \left (a+b x^4\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*c - a*g + b*d*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a + b*x^4)^3) + (x*(7*(11*b
*c + a*g) + 60*b*d*x + 45*b*e*x^2))/(384*a^3*b*(a + b*x^4)) - (8*a*f - x*(11*b*c
+ a*g + 10*b*d*x + 9*b*e*x^2))/(96*a^2*b*(a + b*x^4)^2) + (5*d*ArcTan[(Sqrt[b]*
x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b]) - ((77*b*c + 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*A
rcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^(5/4)) + ((77*b*
c + 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*
Sqrt[2]*a^(15/4)*b^(5/4)) - ((77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(5/4)) + ((
77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x +
Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(5/4))

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Rubi in Sympy [A]  time = 177.693, size = 428, normalized size = 0.98 \[ - \frac{x \left (a g - b c - b d x - b e x^{2} - b f x^{3}\right )}{12 a b \left (a + b x^{4}\right )^{3}} - \frac{8 a f - x \left (a g + 11 b c + 10 b d x + 9 b e x^{2}\right )}{96 a^{2} b \left (a + b x^{4}\right )^{2}} + \frac{x \left (7 a g + 77 b c + 60 b d x + 45 b e x^{2}\right )}{384 a^{3} b \left (a + b x^{4}\right )} + \frac{5 d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{32 a^{\frac{7}{2}} \sqrt{b}} - \frac{\sqrt{2} \left (- 15 \sqrt{a} \sqrt{b} e + 7 a g + 77 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{1024 a^{\frac{15}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (- 15 \sqrt{a} \sqrt{b} e + 7 a g + 77 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{1024 a^{\frac{15}{4}} b^{\frac{5}{4}}} - \frac{\sqrt{2} \left (15 \sqrt{a} \sqrt{b} e + 7 a g + 77 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{512 a^{\frac{15}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (15 \sqrt{a} \sqrt{b} e + 7 a g + 77 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{512 a^{\frac{15}{4}} b^{\frac{5}{4}}} \]

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)**4,x)

[Out]

-x*(a*g - b*c - b*d*x - b*e*x**2 - b*f*x**3)/(12*a*b*(a + b*x**4)**3) - (8*a*f -
x*(a*g + 11*b*c + 10*b*d*x + 9*b*e*x**2))/(96*a**2*b*(a + b*x**4)**2) + x*(7*a*
g + 77*b*c + 60*b*d*x + 45*b*e*x**2)/(384*a**3*b*(a + b*x**4)) + 5*d*atan(sqrt(b
)*x**2/sqrt(a))/(32*a**(7/2)*sqrt(b)) - sqrt(2)*(-15*sqrt(a)*sqrt(b)*e + 7*a*g +
77*b*c)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(1024*a**(
15/4)*b**(5/4)) + sqrt(2)*(-15*sqrt(a)*sqrt(b)*e + 7*a*g + 77*b*c)*log(sqrt(2)*a
**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(1024*a**(15/4)*b**(5/4)) - sqrt(
2)*(15*sqrt(a)*sqrt(b)*e + 7*a*g + 77*b*c)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))
/(512*a**(15/4)*b**(5/4)) + sqrt(2)*(15*sqrt(a)*sqrt(b)*e + 7*a*g + 77*b*c)*atan
(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(512*a**(15/4)*b**(5/4))

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Mathematica [A]  time = 0.694736, size = 411, normalized size = 0.94 \[ \frac{-\frac{256 a^{11/4} \sqrt [4]{b} (a (f+g x)-b x (c+x (d+e x)))}{\left (a+b x^4\right )^3}+\frac{32 a^{7/4} \sqrt [4]{b} x (a g+11 b c+b x (10 d+9 e x))}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} \sqrt [4]{b} x (7 a g+77 b c+15 b x (4 d+3 e x))}{a+b x^4}-6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (80 \sqrt [4]{a} b^{3/4} d+15 \sqrt{2} \sqrt{a} \sqrt{b} e+7 \sqrt{2} a g+77 \sqrt{2} b c\right )+6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-80 \sqrt [4]{a} b^{3/4} d+15 \sqrt{2} \sqrt{a} \sqrt{b} e+7 \sqrt{2} a g+77 \sqrt{2} b c\right )-3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )+3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-15 \sqrt{a} \sqrt{b} e+7 a g+77 b c\right )}{3072 a^{15/4} b^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

((8*a^(3/4)*b^(1/4)*x*(77*b*c + 7*a*g + 15*b*x*(4*d + 3*e*x)))/(a + b*x^4) + (32
*a^(7/4)*b^(1/4)*x*(11*b*c + a*g + b*x*(10*d + 9*e*x)))/(a + b*x^4)^2 - (256*a^(
11/4)*b^(1/4)*(a*(f + g*x) - b*x*(c + x*(d + e*x))))/(a + b*x^4)^3 - 6*(77*Sqrt[
2]*b*c + 80*a^(1/4)*b^(3/4)*d + 15*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 7*Sqrt[2]*a*g)*Ar
cTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 6*(77*Sqrt[2]*b*c - 80*a^(1/4)*b^(3/4)*d
+ 15*Sqrt[2]*Sqrt[a]*Sqrt[b]*e + 7*Sqrt[2]*a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/
a^(1/4)] - 3*Sqrt[2]*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e + 7*a*g)*Log[Sqrt[a] - Sqrt[
2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 3*Sqrt[2]*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e +
7*a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(3072*a^(15/4)*b
^(5/4))

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Maple [A]  time = 0.021, size = 562, 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)^4,x)

[Out]

(15/128*e/a^3*b^2*x^11+5/32*d/a^3*b^2*x^10+7/384*(a*g+11*b*c)/a^3*b*x^9+21/64/a^
2*b*e*x^7+5/12/a^2*d*b*x^6+3/64/a^2*(a*g+11*b*c)*x^5+113/384/a*e*x^3+11/32*d/a*x
^2-1/128*(7*a*g-51*b*c)/a/b*x-1/12*f/b)/(b*x^4+a)^3+7/512*(a/b)^(1/4)/a^3/b*2^(1
/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*g+77/512*c*(a/b)^(1/4)/a^4*2^(1/2)*arctan(2^
(1/2)/(a/b)^(1/4)*x-1)+7/1024*(a/b)^(1/4)/a^3/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+77/1024*c*(a/b)^(1
/4)/a^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)))+7/512*(a/b)^(1/4)/a^3/b*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x
+1)*g+77/512*c*(a/b)^(1/4)/a^4*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+5/32*b*d/
(a^7*b^3)^(1/2)*arctan(x^2*(b/a)^(1/2))+15/1024*e/a^3/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)))+
15/512*e/a^3/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+15/512*e/a^3/
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)^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((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((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 = 0.222268, size = 629, normalized size = 1.44 \[ \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac{45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 7 \, a b^{2} g x^{9} + 126 \, a b^{2} x^{7} e + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 21 \, a^{3} g x - 32 \, a^{3} f}{384 \,{\left (b x^{4} + a\right )}^{3} a^{3} b} \]

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)^4,x, algorithm="giac")

[Out]

1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 77*(a*b^3)^(1/4)*b^2*c + 7*(a*b^3)^(
1/4)*a*b*g + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/
(a/b)^(1/4))/(a^4*b^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 77*(a*b^3)^
(1/4)*b^2*c + 7*(a*b^3)^(1/4)*a*b*g + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*
x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^4*b^3) + 1/1024*sqrt(2)*(77*(a*b^3)^(1/
4)*b^2*c + 7*(a*b^3)^(1/4)*a*b*g - 15*(a*b^3)^(3/4)*e)*ln(x^2 + sqrt(2)*x*(a/b)^
(1/4) + sqrt(a/b))/(a^4*b^3) - 1/1024*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c + 7*(a*b^3
)^(1/4)*a*b*g - 15*(a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/
(a^4*b^3) + 1/384*(45*b^3*x^11*e + 60*b^3*d*x^10 + 77*b^3*c*x^9 + 7*a*b^2*g*x^9
+ 126*a*b^2*x^7*e + 160*a*b^2*d*x^6 + 198*a*b^2*c*x^5 + 18*a^2*b*g*x^5 + 113*a^2
*b*x^3*e + 132*a^2*b*d*x^2 + 153*a^2*b*c*x - 21*a^3*g*x - 32*a^3*f)/((b*x^4 + a)
^3*a^3*b)