3.71 \(\int \frac{c+d x^4}{\left (a+b x^4\right )^2} \, dx\)

Optimal. Leaf size=245 \[ -\frac{(a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{5/4}}-\frac{(a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{x (b c-a d)}{4 a b \left (a+b x^4\right )} \]

[Out]

((b*c - a*d)*x)/(4*a*b*(a + b*x^4)) - ((3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)
*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(5/4)) + ((3*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b
^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(5/4)) - ((3*b*c + a*d)*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(5/4)) + ((3*b*c
+ a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/
4)*b^(5/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.316828, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{(a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{5/4}}-\frac{(a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{x (b c-a d)}{4 a b \left (a+b x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*x)/(4*a*b*(a + b*x^4)) - ((3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)
*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(5/4)) + ((3*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b
^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(5/4)) - ((3*b*c + a*d)*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(5/4)) + ((3*b*c
+ a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/
4)*b^(5/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 62.0599, size = 226, normalized size = 0.92 \[ - \frac{x \left (a d - b c\right )}{4 a b \left (a + b x^{4}\right )} - \frac{\sqrt{2} \left (a d + 3 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (a d + 3 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{5}{4}}} - \frac{\sqrt{2} \left (a d + 3 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (a d + 3 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a*d - b*c)/(4*a*b*(a + b*x**4)) - sqrt(2)*(a*d + 3*b*c)*log(-sqrt(2)*a**(1/4
)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(32*a**(7/4)*b**(5/4)) + sqrt(2)*(a*d + 3
*b*c)*log(sqrt(2)*a**(1/4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(32*a**(7/4)*b**
(5/4)) - sqrt(2)*(a*d + 3*b*c)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(16*a**(7/4
)*b**(5/4)) + sqrt(2)*(a*d + 3*b*c)*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(16*a*
*(7/4)*b**(5/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.30402, size = 212, normalized size = 0.87 \[ \frac{-\frac{8 a^{3/4} \sqrt [4]{b} x (a d-b c)}{a+b x^4}-\sqrt{2} (a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+\sqrt{2} (a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-2 \sqrt{2} (a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \sqrt{2} (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{32 a^{7/4} b^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-8*a^(3/4)*b^(1/4)*(-(b*c) + a*d)*x)/(a + b*x^4) - 2*Sqrt[2]*(3*b*c + a*d)*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*(3*b*c + a*d)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*x)/a^(1/4)] - Sqrt[2]*(3*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(3*b*c + a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(5/4))

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 295, normalized size = 1.2 \[ -{\frac{ \left ( ad-bc \right ) x}{4\,ab \left ( b{x}^{4}+a \right ) }}+{\frac{\sqrt{2}d}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}d}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{\sqrt{2}d}{32\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}c}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^4+c)/(b*x^4+a)^2,x)

[Out]

-1/4*(a*d-b*c)/a/b*x/(b*x^4+a)+1/16/a/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)
^(1/4)*x+1)*d+3/16/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*c+1/1
6/a/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*d+3/16/a^2*(a/b)^(1/4)
*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c+1/32/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)))*d+3/
32/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)))*c

_______________________________________________________________________________________

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((d*x^4 + c)/(b*x^4 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.229541, size = 838, normalized size = 3.42 \[ -\frac{4 \,{\left (a b^{2} x^{4} + a^{2} b\right )} \left (-\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} b \left (-\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}}}{{\left (3 \, b c + a d\right )} x +{\left (3 \, b c + a d\right )} \sqrt{\frac{a^{4} b^{2} \sqrt{-\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}} +{\left (9 \, b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2}}{9 \, b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}}}}\right ) -{\left (a b^{2} x^{4} + a^{2} b\right )} \left (-\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (a^{2} b \left (-\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} +{\left (3 \, b c + a d\right )} x\right ) +{\left (a b^{2} x^{4} + a^{2} b\right )} \left (-\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (-a^{2} b \left (-\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} +{\left (3 \, b c + a d\right )} x\right ) - 4 \,{\left (b c - a d\right )} x}{16 \,{\left (a b^{2} x^{4} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^4 + c)/(b*x^4 + a)^2,x, algorithm="fricas")

[Out]

-1/16*(4*(a*b^2*x^4 + a^2*b)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^
2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4)*arctan(a^2*b*(-(81*b^4*c^4 + 108*
a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4)/((
3*b*c + a*d)*x + (3*b*c + a*d)*sqrt((a^4*b^2*sqrt(-(81*b^4*c^4 + 108*a*b^3*c^3*d
+ 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5)) + (9*b^2*c^2 + 6*a*
b*c*d + a^2*d^2)*x^2)/(9*b^2*c^2 + 6*a*b*c*d + a^2*d^2)))) - (a*b^2*x^4 + a^2*b)
*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4
)/(a^7*b^5))^(1/4)*log(a^2*b*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^
2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4) + (3*b*c + a*d)*x) + (a*b^2*x^4 +
a^2*b)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 +
a^4*d^4)/(a^7*b^5))^(1/4)*log(-a^2*b*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^
2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4) + (3*b*c + a*d)*x) - 4*(b
*c - a*d)*x)/(a*b^2*x^4 + a^2*b)

_______________________________________________________________________________________

Sympy [A]  time = 3.40875, size = 112, normalized size = 0.46 \[ - \frac{x \left (a d - b c\right )}{4 a^{2} b + 4 a b^{2} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{5} + a^{4} d^{4} + 12 a^{3} b c d^{3} + 54 a^{2} b^{2} c^{2} d^{2} + 108 a b^{3} c^{3} d + 81 b^{4} c^{4}, \left ( t \mapsto t \log{\left (\frac{16 t a^{2} b}{a d + 3 b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**4+c)/(b*x**4+a)**2,x)

[Out]

-x*(a*d - b*c)/(4*a**2*b + 4*a*b**2*x**4) + RootSum(65536*_t**4*a**7*b**5 + a**4
*d**4 + 12*a**3*b*c*d**3 + 54*a**2*b**2*c**2*d**2 + 108*a*b**3*c**3*d + 81*b**4*
c**4, Lambda(_t, _t*log(16*_t*a**2*b/(a*d + 3*b*c) + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220218, size = 359, normalized size = 1.47 \[ \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b c + \left (a b^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b c + \left (a b^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b c + \left (a b^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{2}} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b c + \left (a b^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{2}} + \frac{b c x - a d x}{4 \,{\left (b x^{4} + a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b*c + (a*b^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x +
sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^2) + 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b*c
+ (a*b^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4)
)/(a^2*b^2) + 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b*c + (a*b^3)^(1/4)*a*d)*ln(x^2 + sq
rt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^2) - 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b*c +
(a*b^3)^(1/4)*a*d)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^2) + 1/4*
(b*c*x - a*d*x)/((b*x^4 + a)*a*b)