\(\int \frac {(c+d x^3)^4}{a+b x^3} \, dx\) [29]

Optimal result

Integrand size = 19, antiderivative size = 252 \[ \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^3 (4 b c-a d) x^7}{7 b^2}+\frac {d^4 x^{10}}{10 b}-\frac {(b c-a d)^4 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{13/3}}+\frac {(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac {(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}} \] Output:

d*(-a*d+2*b*c)*(a^2*d^2-2*a*b*c*d+2*b^2*c^2)*x/b^4+1/4*d^2*(a^2*d^2-4*a*b* 
c*d+6*b^2*c^2)*x^4/b^3+1/7*d^3*(-a*d+4*b*c)*x^7/b^2+1/10*d^4*x^10/b-1/3*(- 
a*d+b*c)^4*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(2/ 
3)/b^(13/3)+1/3*(-a*d+b*c)^4*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(13/3)-1/6*(- 
a*d+b*c)^4*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(13/3)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx=\frac {420 \sqrt [3]{b} d \left (4 b^3 c^3-6 a b^2 c^2 d+4 a^2 b c d^2-a^3 d^3\right ) x+105 b^{4/3} d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4+60 b^{7/3} d^3 (4 b c-a d) x^7+42 b^{10/3} d^4 x^{10}+\frac {140 \sqrt {3} (b c-a d)^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3}}+\frac {140 (b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac {70 (b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}}{420 b^{13/3}} \] Input:

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

Output:

(420*b^(1/3)*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 + 1 
05*b^(4/3)*d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^4 + 60*b^(7/3)*d^3*(4*b 
*c - a*d)*x^7 + 42*b^(10/3)*d^4*x^10 + (140*Sqrt[3]*(b*c - a*d)^4*ArcTan[( 
-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(2/3) + (140*(b*c - a*d)^4*L 
og[a^(1/3) + b^(1/3)*x])/a^(2/3) - (70*(b*c - a*d)^4*Log[a^(2/3) - a^(1/3) 
*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(420*b^(13/3))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {915, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 915

\(\displaystyle \int \left (\frac {d (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac {d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{b^3}+\frac {a^4 d^4-4 a^3 b c d^3+6 a^2 b^2 c^2 d^2-4 a b^3 c^3 d+b^4 c^4}{b^4 \left (a+b x^3\right )}+\frac {d^3 x^6 (4 b c-a d)}{b^2}+\frac {d^4 x^9}{b}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (b c-a d)^4}{\sqrt {3} a^{2/3} b^{13/3}}-\frac {(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac {(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}+\frac {d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac {d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{4 b^3}+\frac {d^3 x^7 (4 b c-a d)}{7 b^2}+\frac {d^4 x^{10}}{10 b}\)

Input:

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

Output:

(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^4)/(4*b^3) + (d^3*(4*b*c - a*d)*x^7)/(7*b^2) + 
(d^4*x^10)/(10*b) - ((b*c - a*d)^4*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3] 
*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(13/3)) + ((b*c - a*d)^4*Log[a^(1/3) + b^(1 
/3)*x])/(3*a^(2/3)*b^(13/3)) - ((b*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3 
)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(13/3))
 

Defintions of rubi rules used

rule 915

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Int[PolynomialDivide[(a + b*x^n)^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a 
, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, 
0] && GeQ[p, -q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.03 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {d^{4} x^{10}}{10 b}-\frac {d^{4} x^{7} a}{7 b^{2}}+\frac {4 d^{3} c \,x^{7}}{7 b}-\frac {d^{3} a c \,x^{4}}{b^{2}}+\frac {3 d^{2} c^{2} x^{4}}{2 b}+\frac {d^{4} a^{2} x^{4}}{4 b^{3}}-\frac {d^{4} a^{3} x}{b^{4}}+\frac {4 d^{3} a^{2} c x}{b^{3}}-\frac {6 d^{2} a \,c^{2} x}{b^{2}}+\frac {4 d \,c^{3} x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{5}}\)	\(201\)
default	\(-\frac {d \left (-\frac {d^{3} x^{10} b^{3}}{10}+\frac {\left (\left (a d -2 b c \right ) b^{2} d^{2}-2 b^{3} c \,d^{2}\right ) x^{7}}{7}+\frac {\left (2 \left (a d -2 b c \right ) b^{2} c d -d b \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right )\right ) x^{4}}{4}+\left (a d -2 b c \right ) \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x \right )}{b^{4}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{b^{4}}\)	\(272\)

Input:

int((d*x^3+c)^4/(b*x^3+a),x,method=_RETURNVERBOSE)
 

Output:

1/10*d^4*x^10/b-1/7*d^4/b^2*x^7*a+4/7*d^3/b*c*x^7-d^3/b^2*a*c*x^4+3/2*d^2/ 
b*c^2*x^4+1/4*d^4/b^3*a^2*x^4-d^4/b^4*a^3*x+4*d^3/b^3*a^2*c*x-6*d^2/b^2*a* 
c^2*x+4*d/b*c^3*x+1/3/b^5*sum((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)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 873, normalized size of antiderivative = 3.46 \[ \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/420*(42*a^2*b^4*d^4*x^10 + 60*(4*a^2*b^4*c*d^3 - a^3*b^3*d^4)*x^7 + 105 
*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^4 + 210*sqrt(1/3)*( 
a*b^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b* 
d^4)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3 
*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^( 
1/3)/b))/(b*x^3 + a)) - 70*(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)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^ 
2*b)^(1/3)*a) + 140*(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)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 420*(4*a^2*b^ 
4*c^3*d - 6*a^3*b^3*c^2*d^2 + 4*a^4*b^2*c*d^3 - a^5*b*d^4)*x)/(a^2*b^5), 1 
/420*(42*a^2*b^4*d^4*x^10 + 60*(4*a^2*b^4*c*d^3 - a^3*b^3*d^4)*x^7 + 105*( 
6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^4 + 420*sqrt(1/3)*(a* 
b^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^ 
4)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/ 
3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 70*(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)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2 
/3)*x + (a^2*b)^(1/3)*a) + 140*(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)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 42 
0*(4*a^2*b^4*c^3*d - 6*a^3*b^3*c^2*d^2 + 4*a^4*b^2*c*d^3 - a^5*b*d^4)*x)/( 
a^2*b^5)]
 

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.47 \[ \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx=x^{7} \left (- \frac {a d^{4}}{7 b^{2}} + \frac {4 c d^{3}}{7 b}\right ) + x^{4} \left (\frac {a^{2} d^{4}}{4 b^{3}} - \frac {a c d^{3}}{b^{2}} + \frac {3 c^{2} d^{2}}{2 b}\right ) + x \left (- \frac {a^{3} d^{4}}{b^{4}} + \frac {4 a^{2} c d^{3}}{b^{3}} - \frac {6 a c^{2} d^{2}}{b^{2}} + \frac {4 c^{3} d}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{2} b^{13} - a^{12} d^{12} + 12 a^{11} b c d^{11} - 66 a^{10} b^{2} c^{2} d^{10} + 220 a^{9} b^{3} c^{3} d^{9} - 495 a^{8} b^{4} c^{4} d^{8} + 792 a^{7} b^{5} c^{5} d^{7} - 924 a^{6} b^{6} c^{6} d^{6} + 792 a^{5} b^{7} c^{7} d^{5} - 495 a^{4} b^{8} c^{8} d^{4} + 220 a^{3} b^{9} c^{9} d^{3} - 66 a^{2} b^{10} c^{10} d^{2} + 12 a b^{11} c^{11} d - b^{12} c^{12}, \left ( t \mapsto t \log {\left (\frac {3 t a b^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )} \right )\right )} + \frac {d^{4} x^{10}}{10 b} \] Input:

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

Output:

x**7*(-a*d**4/(7*b**2) + 4*c*d**3/(7*b)) + x**4*(a**2*d**4/(4*b**3) - a*c* 
d**3/b**2 + 3*c**2*d**2/(2*b)) + x*(-a**3*d**4/b**4 + 4*a**2*c*d**3/b**3 - 
 6*a*c**2*d**2/b**2 + 4*c**3*d/b) + RootSum(27*_t**3*a**2*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(3*_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**10/(10*b)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.44 \[ \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx=\frac {14 \, b^{3} d^{4} x^{10} + 20 \, {\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{7} + 35 \, {\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{4} + 140 \, {\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x}{140 \, b^{4}} + \frac {\sqrt {3} {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:

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

Output:

1/140*(14*b^3*d^4*x^10 + 20*(4*b^3*c*d^3 - a*b^2*d^4)*x^7 + 35*(6*b^3*c^2* 
d^2 - 4*a*b^2*c*d^3 + a^2*b*d^4)*x^4 + 140*(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 + 1/3*sqrt(3)*(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)*arctan(1/3*sqrt(3)*(2*x - (a 
/b)^(1/3))/(a/b)^(1/3))/(b^5*(a/b)^(2/3)) - 1/6*(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)*log(x^2 - x*(a/b)^(1/3) + (a 
/b)^(2/3))/(b^5*(a/b)^(2/3)) + 1/3*(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)*log(x + (a/b)^(1/3))/(b^5*(a/b)^(2/3))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.55 \[ \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3}} - \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3}} - \frac {{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{10}} + \frac {14 \, b^{9} d^{4} x^{10} + 80 \, b^{9} c d^{3} x^{7} - 20 \, a b^{8} d^{4} x^{7} + 210 \, b^{9} c^{2} d^{2} x^{4} - 140 \, a b^{8} c d^{3} x^{4} + 35 \, a^{2} b^{7} d^{4} x^{4} + 560 \, b^{9} c^{3} d x - 840 \, a b^{8} c^{2} d^{2} x + 560 \, a^{2} b^{7} c d^{3} x - 140 \, a^{3} b^{6} d^{4} x}{140 \, b^{10}} \] Input:

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

Output:

-1/3*sqrt(3)*(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)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2) 
^(2/3)*b^3) - 1/6*(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)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*b 
^3) - 1/3*(b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 
+ a^4*b^6*d^4)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^10) + 1/140*(1 
4*b^9*d^4*x^10 + 80*b^9*c*d^3*x^7 - 20*a*b^8*d^4*x^7 + 210*b^9*c^2*d^2*x^4 
 - 140*a*b^8*c*d^3*x^4 + 35*a^2*b^7*d^4*x^4 + 560*b^9*c^3*d*x - 840*a*b^8* 
c^2*d^2*x + 560*a^2*b^7*c*d^3*x - 140*a^3*b^6*d^4*x)/b^10
 

Mupad [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx=x\,\left (\frac {4\,c^3\,d}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{b}+\frac {6\,c^2\,d^2}{b}\right )}{b}\right )-x^7\,\left (\frac {a\,d^4}{7\,b^2}-\frac {4\,c\,d^3}{7\,b}\right )+x^4\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{4\,b}+\frac {3\,c^2\,d^2}{2\,b}\right )+\frac {d^4\,x^{10}}{10\,b}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^4}{3\,a^{2/3}\,b^{13/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^4}{a^{2/3}\,b^{13/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^4}{3\,a^{2/3}\,b^{13/3}} \] Input:

int((c + d*x^3)^4/(a + b*x^3),x)
 

Output:

x*((4*c^3*d)/b - (a*((a*((a*d^4)/b^2 - (4*c*d^3)/b))/b + (6*c^2*d^2)/b))/b 
) - x^7*((a*d^4)/(7*b^2) - (4*c*d^3)/(7*b)) + x^4*((a*((a*d^4)/b^2 - (4*c* 
d^3)/b))/(4*b) + (3*c^2*d^2)/(2*b)) + (d^4*x^10)/(10*b) + (log(b^(1/3)*x + 
 a^(1/3))*(a*d - b*c)^4)/(3*a^(2/3)*b^(13/3)) + (log(3^(1/2)*a^(1/3)*1i + 
2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/6 - 1/6)*(a*d - b*c)^4)/(a^(2/3)*b^(1 
3/3)) - (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 
 1/2)*(a*d - b*c)^4)/(3*a^(2/3)*b^(13/3))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.25 \[ \int \frac {\left (c+d x^3\right )^4}{a+b x^3} \, dx =\text {Too large to display} \] Input:

int((d*x^3+c)^4/(b*x^3+a),x)
 

Output:

( - 140*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)) 
)*a**4*d**4 + 560*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3 
)*sqrt(3)))*a**3*b*c*d**3 - 840*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/ 
3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2*c**2*d**2 + 560*a**(1/3)*sqrt(3)*atan( 
(a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**3*c**3*d - 140*a**(1/3) 
*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**4*c**4 - 70 
*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*d**4 + 
280*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b*c* 
d**3 - 420*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a* 
*2*b**2*c**2*d**2 + 280*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**( 
2/3)*x**2)*a*b**3*c**3*d - 70*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x 
+ b**(2/3)*x**2)*b**4*c**4 + 140*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**4* 
d**4 - 560*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**3*b*c*d**3 + 840*a**(1/3 
)*log(a**(1/3) + b**(1/3)*x)*a**2*b**2*c**2*d**2 - 560*a**(1/3)*log(a**(1/ 
3) + b**(1/3)*x)*a*b**3*c**3*d + 140*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*b 
**4*c**4 - 420*b**(1/3)*a**4*d**4*x + 1680*b**(1/3)*a**3*b*c*d**3*x + 105* 
b**(1/3)*a**3*b*d**4*x**4 - 2520*b**(1/3)*a**2*b**2*c**2*d**2*x - 420*b**( 
1/3)*a**2*b**2*c*d**3*x**4 - 60*b**(1/3)*a**2*b**2*d**4*x**7 + 1680*b**(1/ 
3)*a*b**3*c**3*d*x + 630*b**(1/3)*a*b**3*c**2*d**2*x**4 + 240*b**(1/3)*a*b 
**3*c*d**3*x**7 + 42*b**(1/3)*a*b**3*d**4*x**10)/(420*b**(1/3)*a*b**4)