3.61 \(\int \frac{\left (c+d x^4\right )^4}{a+b x^4} \, dx\)
Optimal. Leaf size=332 \[ -\frac{(b c-a d)^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-a d)^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^2 x^5 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{5 b^3}+\frac{d^3 x^9 (4 b c-a d)}{9 b^2}+\frac{d^4 x^{13}}{13 b} \]
[Out]
(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*
a*b*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^3*(4*b*c - a*d)*x^9)/(9*b^2) + (d^4*x^13)/(
13*b) - ((b*c - a*d)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/
4)*b^(17/4)) + ((b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2
]*a^(3/4)*b^(17/4)) - ((b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + S
qrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - a*d)^4*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4))
_______________________________________________________________________________________
Rubi [A] time = 0.581279, antiderivative size = 332, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368
\[ -\frac{(b c-a d)^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-a d)^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{17/4}}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^2 x^5 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{5 b^3}+\frac{d^3 x^9 (4 b c-a d)}{9 b^2}+\frac{d^4 x^{13}}{13 b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^4)^4/(a + b*x^4),x]
[Out]
(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*
a*b*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^3*(4*b*c - a*d)*x^9)/(9*b^2) + (d^4*x^13)/(
13*b) - ((b*c - a*d)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/
4)*b^(17/4)) + ((b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2
]*a^(3/4)*b^(17/4)) - ((b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + S
qrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - a*d)^4*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(17/4))
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{d^{4} x^{13}}{13 b} - \frac{d^{3} x^{9} \left (a d - 4 b c\right )}{9 b^{2}} + \frac{d^{2} x^{5} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right )}{5 b^{3}} - \frac{\left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right ) \int d\, dx}{b^{4}} - \frac{\sqrt{2} \left (a d - b c\right )^{4} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{17}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{4} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{17}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{4} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{17}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{4} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{17}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**4+c)**4/(b*x**4+a),x)
[Out]
d**4*x**13/(13*b) - d**3*x**9*(a*d - 4*b*c)/(9*b**2) + d**2*x**5*(a**2*d**2 - 4*
a*b*c*d + 6*b**2*c**2)/(5*b**3) - (a*d - 2*b*c)*(a**2*d**2 - 2*a*b*c*d + 2*b**2*
c**2)*Integral(d, x)/b**4 - sqrt(2)*(a*d - b*c)**4*log(-sqrt(2)*a**(1/4)*b**(1/4
)*x + sqrt(a) + sqrt(b)*x**2)/(8*a**(3/4)*b**(17/4)) + sqrt(2)*(a*d - b*c)**4*lo
g(sqrt(2)*a**(1/4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(8*a**(3/4)*b**(17/4)) -
sqrt(2)*(a*d - b*c)**4*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3/4)*b**(17
/4)) + sqrt(2)*(a*d - b*c)**4*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3/4)*
b**(17/4))
_______________________________________________________________________________________
Mathematica [A] time = 0.329157, size = 322, normalized size = 0.97 \[ \frac{-\frac{585 \sqrt{2} (b c-a d)^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}+\frac{585 \sqrt{2} (b c-a d)^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/4}}-\frac{1170 \sqrt{2} (b c-a d)^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{1170 \sqrt{2} (b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}+936 b^{5/4} d^2 x^5 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )+4680 \sqrt [4]{b} d x \left (-a^3 d^3+4 a^2 b c d^2-6 a b^2 c^2 d+4 b^3 c^3\right )+520 b^{9/4} d^3 x^9 (4 b c-a d)+360 b^{13/4} d^4 x^{13}}{4680 b^{17/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^4)^4/(a + b*x^4),x]
[Out]
(4680*b^(1/4)*d*(4*b^3*c^3 - 6*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x + 936*b^
(5/4)*d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^5 + 520*b^(9/4)*d^3*(4*b*c - a*d)*
x^9 + 360*b^(13/4)*d^4*x^13 - (1170*Sqrt[2]*(b*c - a*d)^4*ArcTan[1 - (Sqrt[2]*b^
(1/4)*x)/a^(1/4)])/a^(3/4) + (1170*Sqrt[2]*(b*c - a*d)^4*ArcTan[1 + (Sqrt[2]*b^(
1/4)*x)/a^(1/4)])/a^(3/4) - (585*Sqrt[2]*(b*c - a*d)^4*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4) + (585*Sqrt[2]*(b*c - a*d)^4*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(3/4))/(4680*b^(17/4))
_______________________________________________________________________________________
Maple [B] time = 0.008, size = 837, 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),x)
[Out]
1/13*d^4*x^13/b+1/4*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*c^4-4/
5*d^3/b^2*x^5*a*c+1/4*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^4-
1/9*d^4/b^2*x^9*a+4/9*d^3/b*x^9*c+1/5*d^4/b^3*x^5*a^2+6/5*d^2/b*x^5*c^2+4*d^3/b^
3*a^2*c*x+3/2/b^2*(a/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^2*d^2-
1/2/b^3*(a/b)^(1/4)*a^2*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^3-6*d^2/b^2*a*c^2*x+1/8*(a/b)^(1/4)/a*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-d^4/b^4*a^3*x+4*d/b*c^3*x+3/4/b^2*(a/b)^(1/4)*a*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^3*(a/b)^(1/4)*a^2*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*c*d^3+3/2/b^2*(a
/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*c^2*d^2-1/b^3*(a/b)^(1/4)*a^
2*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c*d^3+1/8/b^4*(a/b)^(1/4)*a^3*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^4-1/2/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)))*c^3*d-1/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/
2)/(a/b)^(1/4)*x+1)*c^3*d+1/4/b^4*(a/b)^(1/4)*a^3*2^(1/2)*arctan(2^(1/2)/(a/b)^(
1/4)*x+1)*d^4-1/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^3*d+1/4/
b^4*(a/b)^(1/4)*a^3*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*d^4
_______________________________________________________________________________________
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),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.258891, size = 2939, normalized size = 8.85 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^4/(b*x^4 + a),x, algorithm="fricas")
[Out]
1/2340*(180*b^3*d^4*x^13 + 260*(4*b^3*c*d^3 - a*b^2*d^4)*x^9 + 468*(6*b^3*c^2*d^
2 - 4*a*b^2*c*d^3 + a^2*b*d^4)*x^5 - 2340*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d +
120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^
5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8
*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^
11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16
*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4)*arctan(a*b^4*(-(b^16*c^16 - 16*a*b
^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*
d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 +
12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^
11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*
c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4)/((b^4*c^4 - 4*a*b^3*c
^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x + (b^4*c^4 - 4*a*b^3*c^3*d
+ 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt((a^2*b^8*sqrt(-(b^16*c^16 -
16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820*a^4*b^1
2*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9
*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 -
4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13 + 120*a^
14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17)) + (b^8*c^8 - 8*a*b^7
*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b
^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)*x^2)/(b^8*c^8 - 8*a*b
^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5
*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)))) + 585*b^4*(-(b^1
6*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*d^3 + 1820
*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 - 11440*a^7
*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^10*b^6*c^6
*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3*c^3*d^13
+ 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1/4)*log(a*
b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^13*c^13*
d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^10*d^6 -
11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 + 8008*a^
10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a^13*b^3
*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^17))^(1
/4) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x)
- 585*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 - 560*a^3*b^1
3*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a^6*b^10*c^1
0*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7*c^7*d^9 +
8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d^12 - 560*a
^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^16)/(a^3*b^
17))^(1/4)*log(-a*b^4*(-(b^16*c^16 - 16*a*b^15*c^15*d + 120*a^2*b^14*c^14*d^2 -
560*a^3*b^13*c^13*d^3 + 1820*a^4*b^12*c^12*d^4 - 4368*a^5*b^11*c^11*d^5 + 8008*a
^6*b^10*c^10*d^6 - 11440*a^7*b^9*c^9*d^7 + 12870*a^8*b^8*c^8*d^8 - 11440*a^9*b^7
*c^7*d^9 + 8008*a^10*b^6*c^6*d^10 - 4368*a^11*b^5*c^5*d^11 + 1820*a^12*b^4*c^4*d
^12 - 560*a^13*b^3*c^3*d^13 + 120*a^14*b^2*c^2*d^14 - 16*a^15*b*c*d^15 + a^16*d^
16)/(a^3*b^17))^(1/4) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c
*d^3 + a^4*d^4)*x) + 2340*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d
^4)*x)/b^4
_______________________________________________________________________________________
Sympy [A] time = 9.62074, size = 430, normalized size = 1.3 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{17} + a^{16} d^{16} - 16 a^{15} b c d^{15} + 120 a^{14} b^{2} c^{2} d^{14} - 560 a^{13} b^{3} c^{3} d^{13} + 1820 a^{12} b^{4} c^{4} d^{12} - 4368 a^{11} b^{5} c^{5} d^{11} + 8008 a^{10} b^{6} c^{6} d^{10} - 11440 a^{9} b^{7} c^{7} d^{9} + 12870 a^{8} b^{8} c^{8} d^{8} - 11440 a^{7} b^{9} c^{9} d^{7} + 8008 a^{6} b^{10} c^{10} d^{6} - 4368 a^{5} b^{11} c^{11} d^{5} + 1820 a^{4} b^{12} c^{12} d^{4} - 560 a^{3} b^{13} c^{13} d^{3} + 120 a^{2} b^{14} c^{14} d^{2} - 16 a b^{15} c^{15} d + b^{16} c^{16}, \left ( t \mapsto t \log{\left (\frac{4 t a b^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )} \right )\right )} + \frac{d^{4} x^{13}}{13 b} - \frac{x^{9} \left (a d^{4} - 4 b c d^{3}\right )}{9 b^{2}} + \frac{x^{5} \left (a^{2} d^{4} - 4 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{5 b^{3}} - \frac{x \left (a^{3} d^{4} - 4 a^{2} b c d^{3} + 6 a b^{2} c^{2} d^{2} - 4 b^{3} c^{3} d\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**4+c)**4/(b*x**4+a),x)
[Out]
RootSum(256*_t**4*a**3*b**17 + a**16*d**16 - 16*a**15*b*c*d**15 + 120*a**14*b**2
*c**2*d**14 - 560*a**13*b**3*c**3*d**13 + 1820*a**12*b**4*c**4*d**12 - 4368*a**1
1*b**5*c**5*d**11 + 8008*a**10*b**6*c**6*d**10 - 11440*a**9*b**7*c**7*d**9 + 128
70*a**8*b**8*c**8*d**8 - 11440*a**7*b**9*c**9*d**7 + 8008*a**6*b**10*c**10*d**6
- 4368*a**5*b**11*c**11*d**5 + 1820*a**4*b**12*c**12*d**4 - 560*a**3*b**13*c**13
*d**3 + 120*a**2*b**14*c**14*d**2 - 16*a*b**15*c**15*d + b**16*c**16, Lambda(_t,
_t*log(4*_t*a*b**4/(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) + x))) + d**4*x**13/(13*b) - x**9*(a*d**4 - 4*b*c*d**3)/
(9*b**2) + x**5*(a**2*d**4 - 4*a*b*c*d**3 + 6*b**2*c**2*d**2)/(5*b**3) - x*(a**3
*d**4 - 4*a**2*b*c*d**3 + 6*a*b**2*c**2*d**2 - 4*b**3*c**3*d)/b**4
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.219731, size = 833, normalized size = 2.51 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^4 + c)^4/(b*x^4 + a),x, algorithm="giac")
[Out]
1/4*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/
4)*a^2*b^2*c^2*d^2 - 4*(a*b^3)^(1/4)*a^3*b*c*d^3 + (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*b^5) + 1/4*sqrt(2)*((a*
b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d
^2 - 4*(a*b^3)^(1/4)*a^3*b*c*d^3 + (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*b^5) + 1/8*sqrt(2)*((a*b^3)^(1/4)*b^4*c
^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d + 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4*(a*b^3)^(
1/4)*a^3*b*c*d^3 + (a*b^3)^(1/4)*a^4*d^4)*ln(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(
a/b))/(a*b^5) - 1/8*sqrt(2)*((a*b^3)^(1/4)*b^4*c^4 - 4*(a*b^3)^(1/4)*a*b^3*c^3*d
+ 6*(a*b^3)^(1/4)*a^2*b^2*c^2*d^2 - 4*(a*b^3)^(1/4)*a^3*b*c*d^3 + (a*b^3)^(1/4)
*a^4*d^4)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^5) + 1/585*(45*b^12*d
^4*x^13 + 260*b^12*c*d^3*x^9 - 65*a*b^11*d^4*x^9 + 702*b^12*c^2*d^2*x^5 - 468*a*
b^11*c*d^3*x^5 + 117*a^2*b^10*d^4*x^5 + 2340*b^12*c^3*d*x - 3510*a*b^11*c^2*d^2*
x + 2340*a^2*b^10*c*d^3*x - 585*a^3*b^9*d^4*x)/b^13