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

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

[Out]

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

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Rubi [A]  time = 0.312072, 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 \[ -\frac{(3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )}{16 \sqrt{2} c^{7/4} d^{5/4}}-\frac{(3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{8 \sqrt{2} c^{7/4} d^{5/4}}+\frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{8 \sqrt{2} c^{7/4} d^{5/4}}-\frac{x (b c-a d)}{4 c d \left (c+d x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 59.1342, size = 226, normalized size = 0.92 \[ \frac{x \left (a d - b c\right )}{4 c d \left (c + d x^{4}\right )} - \frac{\sqrt{2} \left (3 a d + b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{32 c^{\frac{7}{4}} d^{\frac{5}{4}}} + \frac{\sqrt{2} \left (3 a d + b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x + \sqrt{c} + \sqrt{d} x^{2} \right )}}{32 c^{\frac{7}{4}} d^{\frac{5}{4}}} - \frac{\sqrt{2} \left (3 a d + b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{16 c^{\frac{7}{4}} d^{\frac{5}{4}}} + \frac{\sqrt{2} \left (3 a d + b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}} \right )}}{16 c^{\frac{7}{4}} d^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.290427, size = 212, normalized size = 0.87 \[ \frac{-\frac{8 c^{3/4} \sqrt [4]{d} x (b c-a d)}{c+d x^4}-\sqrt{2} (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+\sqrt{2} (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-2 \sqrt{2} (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt{2} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{32 c^{7/4} d^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 295, normalized size = 1.2 \[{\frac{ \left ( ad-bc \right ) x}{4\,cd \left ( d{x}^{4}+c \right ) }}+{\frac{3\,\sqrt{2}a}{16\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}b}{16\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}a}{32\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}b}{32\,cd}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}a}{16\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}b}{16\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)/(d*x^4+c)^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240921, size = 838, normalized size = 3.42 \[ -\frac{4 \,{\left (c d^{2} x^{4} + c^{2} d\right )} \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{c^{2} d \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}}}{{\left (b c + 3 \, a d\right )} x +{\left (b c + 3 \, a d\right )} \sqrt{\frac{c^{4} d^{2} \sqrt{-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}} +{\left (b^{2} c^{2} + 6 \, a b c d + 9 \, a^{2} d^{2}\right )} x^{2}}{b^{2} c^{2} + 6 \, a b c d + 9 \, a^{2} d^{2}}}}\right ) -{\left (c d^{2} x^{4} + c^{2} d\right )} \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} \log \left (c^{2} d \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} +{\left (b c + 3 \, a d\right )} x\right ) +{\left (c d^{2} x^{4} + c^{2} d\right )} \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} \log \left (-c^{2} d \left (-\frac{b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{c^{7} d^{5}}\right )^{\frac{1}{4}} +{\left (b c + 3 \, a d\right )} x\right ) + 4 \,{\left (b c - a d\right )} x}{16 \,{\left (c d^{2} x^{4} + c^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.20672, size = 112, normalized size = 0.46 \[ \frac{x \left (a d - b c\right )}{4 c^{2} d + 4 c d^{2} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} c^{7} d^{5} + 81 a^{4} d^{4} + 108 a^{3} b c d^{3} + 54 a^{2} b^{2} c^{2} d^{2} + 12 a b^{3} c^{3} d + b^{4} c^{4}, \left ( t \mapsto t \log{\left (\frac{16 t c^{2} d}{3 a d + b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219457, size = 359, normalized size = 1.47 \[ \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{16 \, c^{2} d^{2}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{16 \, c^{2} d^{2}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{32 \, c^{2} d^{2}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b c + 3 \, \left (c d^{3}\right )^{\frac{1}{4}} a d\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{c}{d}\right )^{\frac{1}{4}} + \sqrt{\frac{c}{d}}\right )}{32 \, c^{2} d^{2}} - \frac{b c x - a d x}{4 \,{\left (d x^{4} + c\right )} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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