3.62 \(\int \frac{\left (c+d x^4\right )^3}{a+b x^4} \, dx\)

Optimal. Leaf size=288 \[ -\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{13/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{d^3 x^9}{9 b} \]

[Out]

(d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x)/b^3 + (d^2*(3*b*c - a*d)*x^5)/(5*b^2) +
(d^3*x^9)/(9*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqr
t[2]*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])
/(2*Sqrt[2]*a^(3/4)*b^(13/4)) - ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(13/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.450354, antiderivative size = 288, 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 \[ -\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{13/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{d^3 x^9}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^4)^3/(a + b*x^4),x]

[Out]

(d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x)/b^3 + (d^2*(3*b*c - a*d)*x^5)/(5*b^2) +
(d^3*x^9)/(9*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqr
t[2]*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])
/(2*Sqrt[2]*a^(3/4)*b^(13/4)) - ((b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(13/4))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{3} x^{9}}{9 b} - \frac{d^{2} x^{5} \left (a d - 3 b c\right )}{5 b^{2}} + \frac{\left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right ) \int d\, dx}{b^{3}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{13}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{13}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{13}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**4+c)**3/(b*x**4+a),x)

[Out]

d**3*x**9/(9*b) - d**2*x**5*(a*d - 3*b*c)/(5*b**2) + (a**2*d**2 - 3*a*b*c*d + 3*
b**2*c**2)*Integral(d, x)/b**3 + sqrt(2)*(a*d - b*c)**3*log(-sqrt(2)*a**(1/4)*b*
*(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(8*a**(3/4)*b**(13/4)) - sqrt(2)*(a*d - b*c)*
*3*log(sqrt(2)*a**(1/4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(8*a**(3/4)*b**(13/
4)) + sqrt(2)*(a*d - b*c)**3*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3/4)*b
**(13/4)) - sqrt(2)*(a*d - b*c)**3*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(
3/4)*b**(13/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.234732, size = 271, normalized size = 0.94 \[ \frac{-72 a^{3/4} b^{5/4} d^2 x^5 (a d-3 b c)+40 a^{3/4} b^{9/4} d^3 x^9+360 a^{3/4} \sqrt [4]{b} d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )-45 \sqrt{2} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+45 \sqrt{2} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-90 \sqrt{2} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+90 \sqrt{2} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{360 a^{3/4} b^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^4)^3/(a + b*x^4),x]

[Out]

(360*a^(3/4)*b^(1/4)*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x - 72*a^(3/4)*b^(5/4)*
d^2*(-3*b*c + a*d)*x^5 + 40*a^(3/4)*b^(9/4)*d^3*x^9 - 90*Sqrt[2]*(b*c - a*d)^3*A
rcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 90*Sqrt[2]*(b*c - a*d)^3*ArcTan[1 + (Sq
rt[2]*b^(1/4)*x)/a^(1/4)] - 45*Sqrt[2]*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/
4)*b^(1/4)*x + Sqrt[b]*x^2] + 45*Sqrt[2]*(b*c - a*d)^3*Log[Sqrt[a] + Sqrt[2]*a^(
1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(360*a^(3/4)*b^(13/4))

_______________________________________________________________________________________

Maple [B]  time = 0.002, size = 627, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^4+c)^3/(b*x^4+a),x)

[Out]

1/9*d^3*x^9/b-1/5*d^3/b^2*x^5*a+3/5*d^2/b*x^5*c+d^3/b^3*a^2*x-3*d^2/b^2*a*c*x+3*
d/b*c^2*x-1/4/b^3*(a/b)^(1/4)*a^2*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*d^3+3/
4/b^2*(a/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*c*d^2-3/4/b*(a/b)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)*c^2*d+1/4*(a/b)^(1/4)/a*2^(1/2)*arct
an(2^(1/2)/(a/b)^(1/4)*x+1)*c^3-1/4/b^3*(a/b)^(1/4)*a^2*2^(1/2)*arctan(2^(1/2)/(
a/b)^(1/4)*x-1)*d^3+3/4/b^2*(a/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1
)*c*d^2-3/4/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^2*d+1/4*(a/b
)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)*c^3-1/8/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)))*d^3+3/8/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*d^2-3/8/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^2*d+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^3

_______________________________________________________________________________________

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)^3/(b*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.250542, size = 2198, normalized size = 7.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^4 + c)^3/(b*x^4 + a),x, algorithm="fricas")

[Out]

1/180*(20*b^2*d^3*x^9 + 36*(3*b^2*c*d^2 - a*b*d^3)*x^5 + 180*b^3*(-(b^12*c^12 -
12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*
d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*
b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^
12*d^12)/(a^3*b^13))^(1/4)*arctan(-a*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^
2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^
5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^
3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/
4)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x + (b^3*c^3 - 3*a*b^2*c
^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt((a^2*b^6*sqrt(-(b^12*c^12 - 12*a*b^11*c^11*
d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b
^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 2
20*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b
^13)) + (b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*
a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*x^2)/(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^
2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^
6)))) - 45*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*
b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 -
792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^
2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4)*log(a*b^3*(-(b^12*c^12
- 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^
8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^
8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 +
a^12*d^12)/(a^3*b^13))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^
3)*x) + 45*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*
b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 -
792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^
2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4)*log(-a*b^3*(-(b^12*c^12
- 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c
^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a
^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 +
a^12*d^12)/(a^3*b^13))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d
^3)*x) + 180*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x)/b^3

_______________________________________________________________________________________

Sympy [A]  time = 6.1807, size = 301, normalized size = 1.05 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{13} + a^{12} d^{12} - 12 a^{11} b c d^{11} + 66 a^{10} b^{2} c^{2} d^{10} - 220 a^{9} b^{3} c^{3} d^{9} + 495 a^{8} b^{4} c^{4} d^{8} - 792 a^{7} b^{5} c^{5} d^{7} + 924 a^{6} b^{6} c^{6} d^{6} - 792 a^{5} b^{7} c^{7} d^{5} + 495 a^{4} b^{8} c^{8} d^{4} - 220 a^{3} b^{9} c^{9} d^{3} + 66 a^{2} b^{10} c^{10} d^{2} - 12 a b^{11} c^{11} d + b^{12} c^{12}, \left ( t \mapsto t \log{\left (- \frac{4 t a b^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )} \right )\right )} + \frac{d^{3} x^{9}}{9 b} - \frac{x^{5} \left (a d^{3} - 3 b c d^{2}\right )}{5 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**4+c)**3/(b*x**4+a),x)

[Out]

RootSum(256*_t**4*a**3*b**13 + a**12*d**12 - 12*a**11*b*c*d**11 + 66*a**10*b**2*
c**2*d**10 - 220*a**9*b**3*c**3*d**9 + 495*a**8*b**4*c**4*d**8 - 792*a**7*b**5*c
**5*d**7 + 924*a**6*b**6*c**6*d**6 - 792*a**5*b**7*c**7*d**5 + 495*a**4*b**8*c**
8*d**4 - 220*a**3*b**9*c**9*d**3 + 66*a**2*b**10*c**10*d**2 - 12*a*b**11*c**11*d
+ b**12*c**12, Lambda(_t, _t*log(-4*_t*a*b**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*
a*b**2*c**2*d - b**3*c**3) + x))) + d**3*x**9/(9*b) - x**5*(a*d**3 - 3*b*c*d**2)
/(5*b**2) + x*(a**2*d**3 - 3*a*b*c*d**2 + 3*b**2*c**2*d)/b**3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218108, size = 649, normalized size = 2.25 \[ \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, a b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, a b^{4}} + \frac{5 \, b^{8} d^{3} x^{9} + 27 \, b^{8} c d^{2} x^{5} - 9 \, a b^{7} d^{3} x^{5} + 135 \, b^{8} c^{2} d x - 135 \, a b^{7} c d^{2} x + 45 \, a^{2} b^{6} d^{3} x}{45 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^4 + c)^3/(b*x^4 + a),x, algorithm="giac")

[Out]

1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/
4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^
(1/4))/(a/b)^(1/4))/(a*b^4) + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/
4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(1/2
*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^4) + 1/8*sqrt(2)*((a*b^3)
^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*
b^3)^(1/4)*a^3*d^3)*ln(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^4) - 1/8*sq
rt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2
*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*ln(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a
*b^4) + 1/45*(5*b^8*d^3*x^9 + 27*b^8*c*d^2*x^5 - 9*a*b^7*d^3*x^5 + 135*b^8*c^2*d
*x - 135*a*b^7*c*d^2*x + 45*a^2*b^6*d^3*x)/b^9