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

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

[Out]

(b^2*x)/d^2 + ((b*c - a*d)^2*x)/(4*c*d^2*(c + d*x^4)) + ((b*c - a*d)*(5*b*c + 3*
a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(8*Sqrt[2]*c^(7/4)*d^(9/4)) - ((b*
c - a*d)*(5*b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(8*Sqrt[2]*c^(
7/4)*d^(9/4)) + ((b*c - a*d)*(5*b*c + 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/
4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*d^(9/4)) - ((b*c - a*d)*(5*b*c + 3*a*d)
*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*d^(
9/4))

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Rubi [A]  time = 0.722594, 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 \[ \frac{(b c-a d) (3 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} d^{9/4}}-\frac{(b c-a d) (3 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} d^{9/4}}+\frac{(b c-a d) (3 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} d^{9/4}}-\frac{(b c-a d) (3 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} d^{9/4}}+\frac{x (b c-a d)^2}{4 c d^2 \left (c+d x^4\right )}+\frac{b^2 x}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

(b^2*x)/d^2 + ((b*c - a*d)^2*x)/(4*c*d^2*(c + d*x^4)) + ((b*c - a*d)*(5*b*c + 3*
a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(8*Sqrt[2]*c^(7/4)*d^(9/4)) - ((b*
c - a*d)*(5*b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(8*Sqrt[2]*c^(
7/4)*d^(9/4)) + ((b*c - a*d)*(5*b*c + 3*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/
4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*d^(9/4)) - ((b*c - a*d)*(5*b*c + 3*a*d)
*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(16*Sqrt[2]*c^(7/4)*d^(
9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(b**2, x)/d**2 + x*(a*d - b*c)**2/(4*c*d**2*(c + d*x**4)) - sqrt(2)*(a*d
- b*c)*(3*a*d + 5*b*c)*log(-sqrt(2)*c**(1/4)*d**(1/4)*x + sqrt(c) + sqrt(d)*x**
2)/(32*c**(7/4)*d**(9/4)) + sqrt(2)*(a*d - b*c)*(3*a*d + 5*b*c)*log(sqrt(2)*c**(
1/4)*d**(1/4)*x + sqrt(c) + sqrt(d)*x**2)/(32*c**(7/4)*d**(9/4)) - sqrt(2)*(a*d
- b*c)*(3*a*d + 5*b*c)*atan(1 - sqrt(2)*d**(1/4)*x/c**(1/4))/(16*c**(7/4)*d**(9/
4)) + sqrt(2)*(a*d - b*c)*(3*a*d + 5*b*c)*atan(1 + sqrt(2)*d**(1/4)*x/c**(1/4))/
(16*c**(7/4)*d**(9/4))

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Mathematica [A]  time = 0.282857, size = 298, normalized size = 1.02 \[ \frac{\frac{\sqrt{2} \left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{7/4}}-\frac{\sqrt{2} \left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{c^{7/4}}+\frac{2 \sqrt{2} \left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{c^{7/4}}-\frac{2 \sqrt{2} \left (-3 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{c^{7/4}}+\frac{8 \sqrt [4]{d} x (b c-a d)^2}{c \left (c+d x^4\right )}+32 b^2 \sqrt [4]{d} x}{32 d^{9/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.002, size = 475, normalized size = 1.6 \[{\frac{{b}^{2}x}{{d}^{2}}}+{\frac{x{a}^{2}}{4\,c \left ( d{x}^{4}+c \right ) }}-{\frac{axb}{2\,d \left ( d{x}^{4}+c \right ) }}+{\frac{cx{b}^{2}}{4\,{d}^{2} \left ( d{x}^{4}+c \right ) }}+{\frac{3\,\sqrt{2}{a}^{2}}{16\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}ab}{8\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}{b}^{2}}{16\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}{a}^{2}}{32\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}ab}{16\,cd}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{5\,\sqrt{2}{b}^{2}}{32\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}{a}^{2}}{16\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}ab}{8\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}{b}^{2}}{16\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2/(d*x^4+c)^2,x)

[Out]

b^2*x/d^2+1/4/c*x/(d*x^4+c)*a^2-1/2/d*x/(d*x^4+c)*a*b+1/4/d^2*c*x/(d*x^4+c)*b^2+
3/16/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*a^2+1/8/d/c*(c/d)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*a*b-5/16/d^2*(c/d)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(c/d)^(1/4)*x-1)*b^2+3/32/c^2*(c/d)^(1/4)*2^(1/2)*ln((x^2+(c/d)^(1/
4)*x*2^(1/2)+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2)))*a^2+1/16/d/c*
(c/d)^(1/4)*2^(1/2)*ln((x^2+(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*
x*2^(1/2)+(c/d)^(1/2)))*a*b-5/32/d^2*(c/d)^(1/4)*2^(1/2)*ln((x^2+(c/d)^(1/4)*x*2
^(1/2)+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*x*2^(1/2)+(c/d)^(1/2)))*b^2+3/16/c^2*(c/d)^
(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*a^2+1/8/d/c*(c/d)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(c/d)^(1/4)*x+1)*a*b-5/16/d^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c
/d)^(1/4)*x+1)*b^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((b*x^4 + a)^2/(d*x^4 + c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256329, size = 1592, normalized size = 5.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(16*b^2*c*d*x^5 - 4*(c*d^3*x^4 + c^2*d^2)*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d
- 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^
3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)
*arctan(-c^2*d^2*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*
a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^
6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)/((5*b^2*c^2 - 2*a*b*c*d - 3*a
^2*d^2)*x + (5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*sqrt((c^4*d^4*sqrt(-(625*b^8*c^8
- 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c
^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^
8)/(c^7*d^9)) + (25*b^4*c^4 - 20*a*b^3*c^3*d - 26*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d
^3 + 9*a^4*d^4)*x^2)/(25*b^4*c^4 - 20*a*b^3*c^3*d - 26*a^2*b^2*c^2*d^2 + 12*a^3*
b*c*d^3 + 9*a^4*d^4)))) + (c*d^3*x^4 + c^2*d^2)*(-(625*b^8*c^8 - 1000*a*b^7*c^7*
d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b
^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4
)*log(c^2*d^2*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3
*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 +
216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4) - (5*b^2*c^2 - 2*a*b*c*d - 3*a^2
*d^2)*x) - (c*d^3*x^4 + c^2*d^2)*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6
*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 32
4*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)*log(-c^2*d^2*
(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 +
646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d
^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4) - (5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*x) + 4*(
5*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/(c*d^3*x^4 + c^2*d^2)

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Sympy [A]  time = 6.90248, size = 219, normalized size = 0.75 \[ \frac{b^{2} x}{d^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{4 c^{2} d^{2} + 4 c d^{3} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} c^{7} d^{9} + 81 a^{8} d^{8} + 216 a^{7} b c d^{7} - 324 a^{6} b^{2} c^{2} d^{6} - 984 a^{5} b^{3} c^{3} d^{5} + 646 a^{4} b^{4} c^{4} d^{4} + 1640 a^{3} b^{5} c^{5} d^{3} - 900 a^{2} b^{6} c^{6} d^{2} - 1000 a b^{7} c^{7} d + 625 b^{8} c^{8}, \left ( t \mapsto t \log{\left (\frac{16 t c^{2} d^{2}}{3 a^{2} d^{2} + 2 a b c d - 5 b^{2} c^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x/d**2 + x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(4*c**2*d**2 + 4*c*d**3*x**4
) + RootSum(65536*_t**4*c**7*d**9 + 81*a**8*d**8 + 216*a**7*b*c*d**7 - 324*a**6*
b**2*c**2*d**6 - 984*a**5*b**3*c**3*d**5 + 646*a**4*b**4*c**4*d**4 + 1640*a**3*b
**5*c**5*d**3 - 900*a**2*b**6*c**6*d**2 - 1000*a*b**7*c**7*d + 625*b**8*c**8, La
mbda(_t, _t*log(16*_t*c**2*d**2/(3*a**2*d**2 + 2*a*b*c*d - 5*b**2*c**2) + x)))

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GIAC/XCAS [A]  time = 0.222355, size = 508, normalized size = 1.75 \[ \frac{b^{2} x}{d^{2}} - \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{16 \, c^{2} d^{3}} - \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{16 \, c^{2} d^{3}} - \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{32 \, c^{2} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{32 \, c^{2} d^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{4 \,{\left (d x^{4} + c\right )} c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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