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

Optimal. Leaf size=291 \[ -\frac{(b c-a d) (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(b c-a d) (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{9/4}}+\frac{x (b c-a d)^2}{4 a b^2 \left (a+b x^4\right )}+\frac{d^2 x}{b^2} \]

[Out]

(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(4*a*b^2*(a + b*x^4)) - ((b*c - a*d)*(3*b*c + 5*
a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(9/4)) + ((b*
c - a*d)*(3*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(
7/4)*b^(9/4)) - ((b*c - a*d)*(3*b*c + 5*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^(9/4)) + ((b*c - a*d)*(3*b*c + 5*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^(
9/4))

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Rubi [A]  time = 0.724132, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{(b c-a d) (5 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(b c-a d) (5 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(b c-a d) (5 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{9/4}}+\frac{x (b c-a d)^2}{4 a b^2 \left (a+b x^4\right )}+\frac{d^2 x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(4*a*b^2*(a + b*x^4)) - ((b*c - a*d)*(3*b*c + 5*
a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(9/4)) + ((b*
c - a*d)*(3*b*c + 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(
7/4)*b^(9/4)) - ((b*c - a*d)*(3*b*c + 5*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^(9/4)) + ((b*c - a*d)*(3*b*c + 5*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^(
9/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \int \frac{1}{b^{2}}\, dx + \frac{x \left (a d - b c\right )^{2}}{4 a b^{2} \left (a + b x^{4}\right )} + \frac{\sqrt{2} \left (a d - b c\right ) \left (5 a d + 3 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{9}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (5 a d + 3 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{9}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (5 a d + 3 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{9}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (5 a d + 3 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*Integral(b**(-2), x) + x*(a*d - b*c)**2/(4*a*b**2*(a + b*x**4)) + sqrt(2)*(
a*d - b*c)*(5*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**(9/4)) - sqrt(2)*(a*d - b*c)*(5*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**(9/4)) + sqrt(2)*(a
*d - b*c)*(5*a*d + 3*b*c)*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(16*a**(7/4)*b**
(9/4)) - sqrt(2)*(a*d - b*c)*(5*a*d + 3*b*c)*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4
))/(16*a**(7/4)*b**(9/4))

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Mathematica [A]  time = 0.299828, size = 297, normalized size = 1.02 \[ \frac{\frac{\sqrt{2} \left (5 a^2 d^2-2 a b c d-3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{\sqrt{2} \left (-5 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{2 \sqrt{2} \left (5 a^2 d^2-2 a b c d-3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{2 \sqrt{2} \left (-5 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac{8 \sqrt [4]{b} x (b c-a d)^2}{a \left (a+b x^4\right )}+32 \sqrt [4]{b} d^2 x}{32 b^{9/4}} \]

Antiderivative was successfully verified.

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

[Out]

(32*b^(1/4)*d^2*x + (8*b^(1/4)*(b*c - a*d)^2*x)/(a*(a + b*x^4)) + (2*Sqrt[2]*(-3
*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/
4) + (2*Sqrt[2]*(3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*
x)/a^(1/4)])/a^(7/4) + (Sqrt[2]*(-3*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (Sqrt[2]*(3*b^2*c^2 + 2*a
*b*c*d - 5*a^2*d^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7
/4))/(32*b^(9/4))

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Maple [B]  time = 0.014, size = 475, normalized size = 1.6 \[{\frac{{d}^{2}x}{{b}^{2}}}+{\frac{ax{d}^{2}}{4\,{b}^{2} \left ( b{x}^{4}+a \right ) }}-{\frac{cxd}{2\,b \left ( b{x}^{4}+a \right ) }}+{\frac{x{c}^{2}}{4\,a \left ( b{x}^{4}+a \right ) }}-{\frac{5\,\sqrt{2}{d}^{2}}{16\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}cd}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}{c}^{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}{d}^{2}}{16\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{\sqrt{2}cd}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}{c}^{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}{d}^{2}}{32\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}cd}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}{c}^{2}}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^4+c)^2/(b*x^4+a)^2,x)

[Out]

d^2*x/b^2+1/4/b^2*x*a/(b*x^4+a)*d^2-1/2/b*x/(b*x^4+a)*c*d+1/4*x/a/(b*x^4+a)*c^2-
5/16/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*d^2+1/8/b/a*(a/b)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*c*d+3/16/a^2*(a/b)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(a/b)^(1/4)*x+1)*c^2-5/16/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a
/b)^(1/4)*x-1)*d^2+1/8/b/a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c
*d+3/16/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^2-5/32/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)))*d^2+1/16/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)))*c*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^2

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245341, size = 1592, normalized size = 5.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(16*a*b*d^2*x^5 + 4*(a*b^3*x^4 + a^2*b^2)*(-(81*b^8*c^8 + 216*a*b^7*c^7*d -
324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*
c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*b^9))^(1/4)
*arctan(-a^2*b^2*(-(81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3
*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6
- 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*b^9))^(1/4)/((3*b^2*c^2 + 2*a*b*c*d - 5*a
^2*d^2)*x + (3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*sqrt((a^4*b^4*sqrt(-(81*b^8*c^8
+ 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*
d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^
8)/(a^7*b^9)) + (9*b^4*c^4 + 12*a*b^3*c^3*d - 26*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^
3 + 25*a^4*d^4)*x^2)/(9*b^4*c^4 + 12*a*b^3*c^3*d - 26*a^2*b^2*c^2*d^2 - 20*a^3*b
*c*d^3 + 25*a^4*d^4)))) - (a*b^3*x^4 + a^2*b^2)*(-(81*b^8*c^8 + 216*a*b^7*c^7*d
- 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3
*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*b^9))^(1/4
)*log(a^2*b^2*(-(81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^
5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1
000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*b^9))^(1/4) - (3*b^2*c^2 + 2*a*b*c*d - 5*a^2
*d^2)*x) + (a*b^3*x^4 + a^2*b^2)*(-(81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c
^6*d^2 - 984*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*
a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^7*b^9))^(1/4)*log(-a^2*b^2*
(-(81*b^8*c^8 + 216*a*b^7*c^7*d - 324*a^2*b^6*c^6*d^2 - 984*a^3*b^5*c^5*d^3 + 64
6*a^4*b^4*c^4*d^4 + 1640*a^5*b^3*c^3*d^5 - 900*a^6*b^2*c^2*d^6 - 1000*a^7*b*c*d^
7 + 625*a^8*d^8)/(a^7*b^9))^(1/4) - (3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*x) + 4*(
b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x)/(a*b^3*x^4 + a^2*b^2)

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Sympy [A]  time = 7.23865, size = 219, normalized size = 0.75 \[ \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{4 a^{2} b^{2} + 4 a b^{3} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{9} + 625 a^{8} d^{8} - 1000 a^{7} b c d^{7} - 900 a^{6} b^{2} c^{2} d^{6} + 1640 a^{5} b^{3} c^{3} d^{5} + 646 a^{4} b^{4} c^{4} d^{4} - 984 a^{3} b^{5} c^{5} d^{3} - 324 a^{2} b^{6} c^{6} d^{2} + 216 a b^{7} c^{7} d + 81 b^{8} c^{8}, \left ( t \mapsto t \log{\left (- \frac{16 t a^{2} b^{2}}{5 a^{2} d^{2} - 2 a b c d - 3 b^{2} c^{2}} + x \right )} \right )\right )} + \frac{d^{2} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(4*a**2*b**2 + 4*a*b**3*x**4) + RootSum(65
536*_t**4*a**7*b**9 + 625*a**8*d**8 - 1000*a**7*b*c*d**7 - 900*a**6*b**2*c**2*d*
*6 + 1640*a**5*b**3*c**3*d**5 + 646*a**4*b**4*c**4*d**4 - 984*a**3*b**5*c**5*d**
3 - 324*a**2*b**6*c**6*d**2 + 216*a*b**7*c**7*d + 81*b**8*c**8, Lambda(_t, _t*lo
g(-16*_t*a**2*b**2/(5*a**2*d**2 - 2*a*b*c*d - 3*b**2*c**2) + x))) + d**2*x/b**2

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GIAC/XCAS [A]  time = 0.222079, size = 508, normalized size = 1.75 \[ \frac{d^{2} x}{b^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a b c d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a b c d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a b c d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 2 \, \left (a b^{3}\right )^{\frac{1}{4}} a b c d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{32 \, a^{2} b^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{4 \,{\left (b x^{4} + a\right )} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^2*x/b^2 + 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^2*c^2 + 2*(a*b^3)^(1/4)*a*b*c*d - 5*
(a*b^3)^(1/4)*a^2*d^2)*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)*(3*(a*b^3)^(1/4)*b^2*c^2 + 2*(a*b^3)^(1/4)*a*b*c*d -
5*(a*b^3)^(1/4)*a^2*d^2)*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^2 + 2*(a*b^3)^(1/4)*a*b*c*
d - 5*(a*b^3)^(1/4)*a^2*d^2)*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^2 + 2*(a*b^3)^(1/4)*a*b*c*d - 5*(a*b^3)
^(1/4)*a^2*d^2)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^3) + 1/4*(b^2
*c^2*x - 2*a*b*c*d*x + a^2*d^2*x)/((b*x^4 + a)*a*b^2)