3.68 \(\int \frac{\left (c+d x^4\right )^4}{\left (a+b x^4\right )^2} \, dx\)
Optimal. Leaf size=357 \[ -\frac{(b c-a d)^3 (13 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}-\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left (a+b x^4\right )}+\frac{2 d^3 x^5 (2 b c-a d)}{5 b^3}+\frac{d^4 x^9}{9 b^2} \]
[Out]
(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^5)/(5*b
^3) + (d^4*x^9)/(9*b^2) + ((b*c - a*d)^4*x)/(4*a*b^4*(a + b*x^4)) - ((b*c - a*d)
^3*(3*b*c + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*
b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/
4)])/(8*Sqrt[2]*a^(7/4)*b^(17/4)) - ((b*c - a*d)^3*(3*b*c + 13*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^(17/4)) + ((b*
c - a*d)^3*(3*b*c + 13*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^(17/4))
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Rubi [A] time = 0.718939, antiderivative size = 357, 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
\[ -\frac{(b c-a d)^3 (13 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{17/4}}-\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{(b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{17/4}}+\frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{x (b c-a d)^4}{4 a b^4 \left (a+b x^4\right )}+\frac{2 d^3 x^5 (2 b c-a d)}{5 b^3}+\frac{d^4 x^9}{9 b^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^4)^4/(a + b*x^4)^2,x]
[Out]
(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^5)/(5*b
^3) + (d^4*x^9)/(9*b^2) + ((b*c - a*d)^4*x)/(4*a*b^4*(a + b*x^4)) - ((b*c - a*d)
^3*(3*b*c + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*
b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/
4)])/(8*Sqrt[2]*a^(7/4)*b^(17/4)) - ((b*c - a*d)^3*(3*b*c + 13*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^(17/4)) + ((b*
c - a*d)^3*(3*b*c + 13*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^(17/4))
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ d^{2} \left (3 a^{2} d^{2} - 8 a b c d + 6 b^{2} c^{2}\right ) \int \frac{1}{b^{4}}\, dx + \frac{d^{4} x^{9}}{9 b^{2}} - \frac{2 d^{3} x^{5} \left (a d - 2 b c\right )}{5 b^{3}} + \frac{x \left (a d - b c\right )^{4}}{4 a b^{4} \left (a + b x^{4}\right )} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \left (13 a d + 3 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{17}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \left (13 a d + 3 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{32 a^{\frac{7}{4}} b^{\frac{17}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \left (13 a d + 3 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{17}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \left (13 a d + 3 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{16 a^{\frac{7}{4}} b^{\frac{17}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**4/(b*x**4+a)**2,x)
[Out]
d**2*(3*a**2*d**2 - 8*a*b*c*d + 6*b**2*c**2)*Integral(b**(-4), x) + d**4*x**9/(9
*b**2) - 2*d**3*x**5*(a*d - 2*b*c)/(5*b**3) + x*(a*d - b*c)**4/(4*a*b**4*(a + b*
x**4)) + sqrt(2)*(a*d - b*c)**3*(13*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**(17/4)) - sqrt(2)*(a*d - b*c)**3*(13
*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**(17/4)) + sqrt(2)*(a*d - b*c)**3*(13*a*d + 3*b*c)*atan(1 - sqrt(2)*b**(1
/4)*x/a**(1/4))/(16*a**(7/4)*b**(17/4)) - sqrt(2)*(a*d - b*c)**3*(13*a*d + 3*b*c
)*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(16*a**(7/4)*b**(17/4))
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Mathematica [A] time = 0.501044, size = 341, normalized size = 0.96 \[ \frac{\frac{45 \sqrt{2} (a d-b c)^3 (13 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{45 \sqrt{2} (b c-a d)^3 (13 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{90 \sqrt{2} (a d-b c)^3 (13 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{90 \sqrt{2} (b c-a d)^3 (13 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+1440 \sqrt [4]{b} d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )+576 b^{5/4} d^3 x^5 (2 b c-a d)+\frac{360 \sqrt [4]{b} x (b c-a d)^4}{a \left (a+b x^4\right )}+160 b^{9/4} d^4 x^9}{1440 b^{17/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^4)^4/(a + b*x^4)^2,x]
[Out]
(1440*b^(1/4)*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x + 576*b^(5/4)*d^3*(2*b*c
- a*d)*x^5 + 160*b^(9/4)*d^4*x^9 + (360*b^(1/4)*(b*c - a*d)^4*x)/(a*(a + b*x^4)
) + (90*Sqrt[2]*(-(b*c) + a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)
/a^(1/4)])/a^(7/4) + (90*Sqrt[2]*(b*c - a*d)^3*(3*b*c + 13*a*d)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (45*Sqrt[2]*(-(b*c) + a*d)^3*(3*b*c + 13*a*d)
*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (45*Sqrt[2]*(
b*c - a*d)^3*(3*b*c + 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*
x^2])/a^(7/4))/(1440*b^(17/4))
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Maple [B] time = 0.017, size = 885, normalized size = 2.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^4+c)^4/(b*x^4+a)^2,x)
[Out]
1/9*d^4*x^9/b^2-8*d^3/b^3*a*c*x+3*d^4/b^4*a^2*x+6*d^2/b^2*c^2*x-1/b^3*x*a^2/(b*x
^4+a)*c*d^3+3/2/b^2*x*a/(b*x^4+a)*c^2*d^2-13/16/b^4*a^2*(a/b)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(a/b)^(1/4)*x+1)*d^4-15/8/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b
)^(1/4)*x+1)*c^2*d^2-13/16/b^4*a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4
)*x-1)*d^4-15/8/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^2*d^2-
13/32/b^4*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)))*d^4-15/16/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)))*c^2*
d^2-1/b*x/(b*x^4+a)*c^3*d+3/16/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4
)*x+1)*c^4+3/16/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^4+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^4+1/4/b^4*x*a^3/(b*x^4+a)*d^4+9/4/b^3*a*(a/b)^(1/
4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*c*d^3+1/4/b/a*(a/b)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(a/b)^(1/4)*x+1)*c^3*d+9/4/b^3*a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/b)^(1/4)*x-1)*c*d^3-2/5*d^4/b^3*x^5*a+4/5*d^3/b^2*x^5*c+1/4*x/a/(b*x^4+a)*c^4
+1/4/b/a*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^3*d+9/8/b^3*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+1/8/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^3*d
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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)^4/(b*x^4 + a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.2572, size = 3083, normalized size = 8.64 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^4/(b*x^4 + a)^2,x, algorithm="fricas")
[Out]
1/720*(80*a*b^3*d^4*x^13 + 16*(36*a*b^3*c*d^3 - 13*a^2*b^2*d^4)*x^9 + 144*(30*a*
b^3*c^2*d^2 - 36*a^2*b^2*c*d^3 + 13*a^3*b*d^4)*x^5 + 180*(a*b^5*x^4 + a^2*b^4)*(
-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13
*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*
d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9
+ 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*
d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d
^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4)*arctan(-a^2*b^4*(-(81*b^16*c^16 + 432*a
*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*
c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c
^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6
*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^
3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)
/(a^7*b^17))^(1/4)/((3*b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 + 36*a^3*b*c
*d^3 - 13*a^4*d^4)*x + (3*b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 + 36*a^3*
b*c*d^3 - 13*a^4*d^4)*sqrt((a^4*b^8*sqrt(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 23
76*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*
a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a
^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*
a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577
784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17)) + (9
*b^8*c^8 + 24*a*b^7*c^7*d - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c^5*d^3 + 1110*a^4*
b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 - 936*a^7*b*c*d^7 + 16
9*a^8*d^8)*x^2)/(9*b^8*c^8 + 24*a*b^7*c^7*d - 164*a^2*b^6*c^6*d^2 - 24*a^3*b^5*c
^5*d^3 + 1110*a^4*b^4*c^4*d^4 - 2264*a^5*b^3*c^3*d^5 + 2076*a^6*b^2*c^2*d^6 - 93
6*a^7*b*c*d^7 + 169*a^8*d^8)))) - 45*(a*b^5*x^4 + a^2*b^4)*(-(81*b^16*c^16 + 432
*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^1
2*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9
*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c
^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*
b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^1
6)/(a^7*b^17))^(1/4)*log(a^2*b^4*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*
b^14*c^14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^1
1*c^11*d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*
c^8*d^8 - 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^
5*c^5*d^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^1
4*b^2*c^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4) - (3*
b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*a^4*d^4)*x) +
45*(a*b^5*x^4 + a^2*b^4)*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^
14*d^2 - 8304*a^3*b^13*c^13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*
d^5 - 434808*a^6*b^10*c^10*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8
- 4810608*a^9*b^7*c^7*d^9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d
^11 + 8923164*a^12*b^4*c^4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c
^2*d^14 - 316368*a^15*b*c*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4)*log(-a^2*b^4
*(-(81*b^16*c^16 + 432*a*b^15*c^15*d - 2376*a^2*b^14*c^14*d^2 - 8304*a^3*b^13*c^
13*d^3 + 45724*a^4*b^12*c^12*d^4 + 20400*a^5*b^11*c^11*d^5 - 434808*a^6*b^10*c^1
0*d^6 + 772112*a^7*b^9*c^9*d^7 + 617958*a^8*b^8*c^8*d^8 - 4810608*a^9*b^7*c^7*d^
9 + 9723912*a^10*b^6*c^6*d^10 - 11486160*a^11*b^5*c^5*d^11 + 8923164*a^12*b^4*c^
4*d^12 - 4651504*a^13*b^3*c^3*d^13 + 1577784*a^14*b^2*c^2*d^14 - 316368*a^15*b*c
*d^15 + 28561*a^16*d^16)/(a^7*b^17))^(1/4) - (3*b^4*c^4 + 4*a*b^3*c^3*d - 30*a^2
*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 13*a^4*d^4)*x) + 180*(b^4*c^4 - 4*a*b^3*c^3*d +
30*a^2*b^2*c^2*d^2 - 36*a^3*b*c*d^3 + 13*a^4*d^4)*x)/(a*b^5*x^4 + a^2*b^4)
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Sympy [A] time = 89.2013, size = 466, normalized size = 1.31 \[ \frac{x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{4 a^{2} b^{4} + 4 a b^{5} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{17} + 28561 a^{16} d^{16} - 316368 a^{15} b c d^{15} + 1577784 a^{14} b^{2} c^{2} d^{14} - 4651504 a^{13} b^{3} c^{3} d^{13} + 8923164 a^{12} b^{4} c^{4} d^{12} - 11486160 a^{11} b^{5} c^{5} d^{11} + 9723912 a^{10} b^{6} c^{6} d^{10} - 4810608 a^{9} b^{7} c^{7} d^{9} + 617958 a^{8} b^{8} c^{8} d^{8} + 772112 a^{7} b^{9} c^{9} d^{7} - 434808 a^{6} b^{10} c^{10} d^{6} + 20400 a^{5} b^{11} c^{11} d^{5} + 45724 a^{4} b^{12} c^{12} d^{4} - 8304 a^{3} b^{13} c^{13} d^{3} - 2376 a^{2} b^{14} c^{14} d^{2} + 432 a b^{15} c^{15} d + 81 b^{16} c^{16}, \left ( t \mapsto t \log{\left (- \frac{16 t a^{2} b^{4}}{13 a^{4} d^{4} - 36 a^{3} b c d^{3} + 30 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - 3 b^{4} c^{4}} + x \right )} \right )\right )} + \frac{d^{4} x^{9}}{9 b^{2}} - \frac{x^{5} \left (2 a d^{4} - 4 b c d^{3}\right )}{5 b^{3}} + \frac{x \left (3 a^{2} d^{4} - 8 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**4+c)**4/(b*x**4+a)**2,x)
[Out]
x*(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*
c**4)/(4*a**2*b**4 + 4*a*b**5*x**4) + RootSum(65536*_t**4*a**7*b**17 + 28561*a**
16*d**16 - 316368*a**15*b*c*d**15 + 1577784*a**14*b**2*c**2*d**14 - 4651504*a**1
3*b**3*c**3*d**13 + 8923164*a**12*b**4*c**4*d**12 - 11486160*a**11*b**5*c**5*d**
11 + 9723912*a**10*b**6*c**6*d**10 - 4810608*a**9*b**7*c**7*d**9 + 617958*a**8*b
**8*c**8*d**8 + 772112*a**7*b**9*c**9*d**7 - 434808*a**6*b**10*c**10*d**6 + 2040
0*a**5*b**11*c**11*d**5 + 45724*a**4*b**12*c**12*d**4 - 8304*a**3*b**13*c**13*d*
*3 - 2376*a**2*b**14*c**14*d**2 + 432*a*b**15*c**15*d + 81*b**16*c**16, Lambda(_
t, _t*log(-16*_t*a**2*b**4/(13*a**4*d**4 - 36*a**3*b*c*d**3 + 30*a**2*b**2*c**2*
d**2 - 4*a*b**3*c**3*d - 3*b**4*c**4) + x))) + d**4*x**9/(9*b**2) - x**5*(2*a*d*
*4 - 4*b*c*d**3)/(5*b**3) + x*(3*a**2*d**4 - 8*a*b*c*d**3 + 6*b**2*c**2*d**2)/b*
*4
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GIAC/XCAS [A] time = 0.220568, size = 867, normalized size = 2.43 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^4/(b*x^4 + a)^2,x, algorithm="giac")
[Out]
1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)
^(1/4)*a^2*b^2*c^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4
)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^5) + 1/16*s
qrt(2)*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1/4)
*a^2*b^2*c^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*arct
an(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^5) + 1/32*sqrt(2)
*(3*(a*b^3)^(1/4)*b^4*c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1/4)*a^2*b
^2*c^2*d^2 + 36*(a*b^3)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*ln(x^2 + s
qrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^5) - 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^4*
c^4 + 4*(a*b^3)^(1/4)*a*b^3*c^3*d - 30*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 + 36*(a*b^3
)^(1/4)*a^3*b*c*d^3 - 13*(a*b^3)^(1/4)*a^4*d^4)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) +
sqrt(a/b))/(a^2*b^5) + 1/4*(b^4*c^4*x - 4*a*b^3*c^3*d*x + 6*a^2*b^2*c^2*d^2*x -
4*a^3*b*c*d^3*x + a^4*d^4*x)/((b*x^4 + a)*a*b^4) + 1/45*(5*b^16*d^4*x^9 + 36*b^
16*c*d^3*x^5 - 18*a*b^15*d^4*x^5 + 270*b^16*c^2*d^2*x - 360*a*b^15*c*d^3*x + 135
*a^2*b^14*d^4*x)/b^18