Integrand size = 19, antiderivative size = 267 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {d^3 (2 b c-a d) x^4}{2 b^3}+\frac {d^4 x^7}{7 b^2}+\frac {(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}-\frac {2 (b c-a d)^3 (b c+5 a d) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{13/3}}+\frac {2 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac {(b c-a d)^3 (b c+5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}} \] Output:
d^2*(3*a^2*d^2-8*a*b*c*d+6*b^2*c^2)*x/b^4+1/2*d^3*(-a*d+2*b*c)*x^4/b^3+1/7 *d^4*x^7/b^2+1/3*(-a*d+b*c)^4*x/a/b^4/(b*x^3+a)-2/9*(-a*d+b*c)^3*(5*a*d+b* c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(5/3)/b^(13 /3)+2/9*(-a*d+b*c)^3*(5*a*d+b*c)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(13/3)-1/ 9*(-a*d+b*c)^3*(5*a*d+b*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/ 3)/b^(13/3)
Time = 0.17 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=\frac {126 \sqrt [3]{b} d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x+63 b^{4/3} d^3 (2 b c-a d) x^4+18 b^{7/3} d^4 x^7+\frac {42 \sqrt [3]{b} (b c-a d)^4 x}{a \left (a+b x^3\right )}+\frac {28 \sqrt {3} (b c-a d)^3 (b c+5 a d) \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3}}+\frac {28 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {14 (-b c+a d)^3 (b c+5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{126 b^{13/3}} \] Input:
Integrate[(c + d*x^3)^4/(a + b*x^3)^2,x]
Output:
(126*b^(1/3)*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x + 63*b^(4/3)*d^3*(2 *b*c - a*d)*x^4 + 18*b^(7/3)*d^4*x^7 + (42*b^(1/3)*(b*c - a*d)^4*x)/(a*(a + b*x^3)) + (28*Sqrt[3]*(b*c - a*d)^3*(b*c + 5*a*d)*ArcTan[(-a^(1/3) + 2*b ^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(5/3) + (28*(b*c - a*d)^3*(b*c + 5*a*d)*Lo g[a^(1/3) + b^(1/3)*x])/a^(5/3) + (14*(-(b*c) + a*d)^3*(b*c + 5*a*d)*Log[a ^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(126*b^(13/3))
Time = 0.75 (sec) , antiderivative size = 267, 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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 915 |
| \(\displaystyle \int \left (\frac {d^2 \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac {4 b d x^3 (b c-a d)^3+(3 a d+b c) (b c-a d)^3}{b^4 \left (a+b x^3\right )^2}+\frac {2 d^3 x^3 (2 b c-a d)}{b^3}+\frac {d^4 x^6}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (b c-a d)^3 (5 a d+b c)}{3 \sqrt {3} a^{5/3} b^{13/3}}-\frac {(b c-a d)^3 (5 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}+\frac {2 (b c-a d)^3 (5 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}+\frac {d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac {x (b c-a d)^4}{3 a b^4 \left (a+b x^3\right )}+\frac {d^3 x^4 (2 b c-a d)}{2 b^3}+\frac {d^4 x^7}{7 b^2}\) |
Input:
Int[(c + d*x^3)^4/(a + b*x^3)^2,x]
Output:
(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (d^3*(2*b*c - a*d)*x^4)/ (2*b^3) + (d^4*x^7)/(7*b^2) + ((b*c - a*d)^4*x)/(3*a*b^4*(a + b*x^3)) - (2 *(b*c - a*d)^3*(b*c + 5*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/ 3))])/(3*Sqrt[3]*a^(5/3)*b^(13/3)) + (2*(b*c - a*d)^3*(b*c + 5*a*d)*Log[a^ (1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(13/3)) - ((b*c - a*d)^3*(b*c + 5*a*d)*Lo g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9*a^(5/3)*b^(13/3))
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.92 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(\frac {d^{4} x^{7}}{7 b^{2}}-\frac {d^{4} a \,x^{4}}{2 b^{3}}+\frac {d^{3} c \,x^{4}}{b^{2}}+\frac {3 d^{4} a^{2} x}{b^{4}}-\frac {8 d^{3} a c x}{b^{3}}+\frac {6 d^{2} c^{2} x}{b^{2}}+\frac {\left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) x}{3 a \,b^{4} \left (b \,x^{3}+a \right )}-\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (5 d^{4} a^{4}-14 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-2 a \,b^{3} c^{3} d -c^{4} b^{4}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 b^{5} a}\) | \(218\) |
| default | \(\frac {d^{2} \left (\frac {1}{7} b^{2} d^{2} x^{7}-\frac {1}{2} a b \,d^{2} x^{4}+b^{2} c d \,x^{4}+3 a^{2} d^{2} x -8 a b c d x +6 b^{2} c^{2} x \right )}{b^{4}}-\frac {-\frac {\left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) x}{3 a \left (b \,x^{3}+a \right )}+\frac {2 \left (5 d^{4} a^{4}-14 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-2 a \,b^{3} c^{3} d -c^{4} b^{4}\right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a}}{b^{4}}\) | \(281\) |
Input:
int((d*x^3+c)^4/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
1/7*d^4*x^7/b^2-1/2*d^4/b^3*a*x^4+d^3/b^2*c*x^4+3*d^4/b^4*a^2*x-8*d^3/b^3* a*c*x+6*d^2/b^2*c^2*x+1/3*(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)/a*x/b^4/(b*x^3+a)-2/9/b^5/a*sum((5*a^4*d^4-14*a^3*b*c*d^3+ 12*a^2*b^2*c^2*d^2-2*a*b^3*c^3*d-b^4*c^4)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a ))
Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (224) = 448\).
Time = 0.10 (sec) , antiderivative size = 1316, normalized size of antiderivative = 4.93 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x^3+c)^4/(b*x^3+a)^2,x, algorithm="fricas")
Output:
[1/126*(18*a^3*b^4*d^4*x^10 + 9*(14*a^3*b^4*c*d^3 - 5*a^4*b^3*d^4)*x^7 + 6 3*(12*a^3*b^4*c^2*d^2 - 14*a^4*b^3*c*d^3 + 5*a^5*b^2*d^4)*x^4 - 42*sqrt(1/ 3)*(a^2*b^5*c^4 + 2*a^3*b^4*c^3*d - 12*a^4*b^3*c^2*d^2 + 14*a^5*b^2*c*d^3 - 5*a^6*b*d^4 + (a*b^6*c^4 + 2*a^2*b^5*c^3*d - 12*a^3*b^4*c^2*d^2 + 14*a^4 *b^3*c*d^3 - 5*a^5*b^2*d^4)*x^3)*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)) - 14*(a*b^4*c^4 + 2*a ^2*b^3*c^3*d - 12*a^3*b^2*c^2*d^2 + 14*a^4*b*c*d^3 - 5*a^5*d^4 + (b^5*c^4 + 2*a*b^4*c^3*d - 12*a^2*b^3*c^2*d^2 + 14*a^3*b^2*c*d^3 - 5*a^4*b*d^4)*x^3 )*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*( a*b^4*c^4 + 2*a^2*b^3*c^3*d - 12*a^3*b^2*c^2*d^2 + 14*a^4*b*c*d^3 - 5*a^5* d^4 + (b^5*c^4 + 2*a*b^4*c^3*d - 12*a^2*b^3*c^2*d^2 + 14*a^3*b^2*c*d^3 - 5 *a^4*b*d^4)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 42*(a^2*b^5* c^4 - 4*a^3*b^4*c^3*d + 24*a^4*b^3*c^2*d^2 - 28*a^5*b^2*c*d^3 + 10*a^6*b*d ^4)*x)/(a^3*b^6*x^3 + a^4*b^5), 1/126*(18*a^3*b^4*d^4*x^10 + 9*(14*a^3*b^4 *c*d^3 - 5*a^4*b^3*d^4)*x^7 + 63*(12*a^3*b^4*c^2*d^2 - 14*a^4*b^3*c*d^3 + 5*a^5*b^2*d^4)*x^4 + 84*sqrt(1/3)*(a^2*b^5*c^4 + 2*a^3*b^4*c^3*d - 12*a^4* b^3*c^2*d^2 + 14*a^5*b^2*c*d^3 - 5*a^6*b*d^4 + (a*b^6*c^4 + 2*a^2*b^5*c^3* d - 12*a^3*b^4*c^2*d^2 + 14*a^4*b^3*c*d^3 - 5*a^5*b^2*d^4)*x^3)*sqrt(-(-a^ 2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(-a^2*b)^(2/3)*x + (-a^2*b)^(1/3)*a)*...
Time = 2.26 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.52 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=x^{4} \left (- \frac {a d^{4}}{2 b^{3}} + \frac {c d^{3}}{b^{2}}\right ) + x \left (\frac {3 a^{2} d^{4}}{b^{4}} - \frac {8 a c d^{3}}{b^{3}} + \frac {6 c^{2} d^{2}}{b^{2}}\right ) + \frac {x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{3 a^{2} b^{4} + 3 a b^{5} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{13} + 1000 a^{12} d^{12} - 8400 a^{11} b c d^{11} + 30720 a^{10} b^{2} c^{2} d^{10} - 63472 a^{9} b^{3} c^{3} d^{9} + 79848 a^{8} b^{4} c^{4} d^{8} - 60192 a^{7} b^{5} c^{5} d^{7} + 22848 a^{6} b^{6} c^{6} d^{6} + 288 a^{5} b^{7} c^{7} d^{5} - 3528 a^{4} b^{8} c^{8} d^{4} + 752 a^{3} b^{9} c^{9} d^{3} + 192 a^{2} b^{10} c^{10} d^{2} - 48 a b^{11} c^{11} d - 8 b^{12} c^{12}, \left ( t \mapsto t \log {\left (- \frac {9 t a^{2} b^{4}}{10 a^{4} d^{4} - 28 a^{3} b c d^{3} + 24 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - 2 b^{4} c^{4}} + x \right )} \right )\right )} + \frac {d^{4} x^{7}}{7 b^{2}} \] Input:
integrate((d*x**3+c)**4/(b*x**3+a)**2,x)
Output:
x**4*(-a*d**4/(2*b**3) + c*d**3/b**2) + x*(3*a**2*d**4/b**4 - 8*a*c*d**3/b **3 + 6*c**2*d**2/b**2) + 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)/(3*a**2*b**4 + 3*a*b**5*x**3) + Root Sum(729*_t**3*a**5*b**13 + 1000*a**12*d**12 - 8400*a**11*b*c*d**11 + 30720 *a**10*b**2*c**2*d**10 - 63472*a**9*b**3*c**3*d**9 + 79848*a**8*b**4*c**4* d**8 - 60192*a**7*b**5*c**5*d**7 + 22848*a**6*b**6*c**6*d**6 + 288*a**5*b* *7*c**7*d**5 - 3528*a**4*b**8*c**8*d**4 + 752*a**3*b**9*c**9*d**3 + 192*a* *2*b**10*c**10*d**2 - 48*a*b**11*c**11*d - 8*b**12*c**12, Lambda(_t, _t*lo g(-9*_t*a**2*b**4/(10*a**4*d**4 - 28*a**3*b*c*d**3 + 24*a**2*b**2*c**2*d** 2 - 4*a*b**3*c**3*d - 2*b**4*c**4) + x))) + d**4*x**7/(7*b**2)
Time = 0.11 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.49 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x}{3 \, {\left (a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac {2 \, b^{2} d^{4} x^{7} + 7 \, {\left (2 \, b^{2} c d^{3} - a b d^{4}\right )} x^{4} + 14 \, {\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x}{14 \, b^{4}} + \frac {2 \, \sqrt {3} {\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((d*x^3+c)^4/(b*x^3+a)^2,x, algorithm="maxima")
Output:
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 )*x/(a*b^5*x^3 + a^2*b^4) + 1/14*(2*b^2*d^4*x^7 + 7*(2*b^2*c*d^3 - a*b*d^4 )*x^4 + 14*(6*b^2*c^2*d^2 - 8*a*b*c*d^3 + 3*a^2*d^4)*x)/b^4 + 2/9*sqrt(3)* (b^4*c^4 + 2*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^3 - 5*a^4*d^4 )*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^5*(a/b)^(2/3)) - 1/9*(b^4*c^4 + 2*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^3 - 5*a ^4*d^4)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^5*(a/b)^(2/3)) + 2/9*( b^4*c^4 + 2*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^3 - 5*a^4*d^4) *log(x + (a/b)^(1/3))/(a*b^5*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.54 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=-\frac {2 \, \sqrt {3} {\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{3}} - \frac {{\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{3}} - \frac {2 \, {\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{4}} + \frac {b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{3 \, {\left (b x^{3} + a\right )} a b^{4}} + \frac {2 \, b^{12} d^{4} x^{7} + 14 \, b^{12} c d^{3} x^{4} - 7 \, a b^{11} d^{4} x^{4} + 84 \, b^{12} c^{2} d^{2} x - 112 \, a b^{11} c d^{3} x + 42 \, a^{2} b^{10} d^{4} x}{14 \, b^{14}} \] Input:
integrate((d*x^3+c)^4/(b*x^3+a)^2,x, algorithm="giac")
Output:
-2/9*sqrt(3)*(b^4*c^4 + 2*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^ 3 - 5*a^4*d^4)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a* b^2)^(2/3)*a*b^3) - 1/9*(b^4*c^4 + 2*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 14 *a^3*b*c*d^3 - 5*a^4*d^4)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^ 2)^(2/3)*a*b^3) - 2/9*(b^4*c^4 + 2*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 14*a ^3*b*c*d^3 - 5*a^4*d^4)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^4) + 1/3*(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^3 + a)*a*b^4) + 1/14*(2*b^12*d^4*x^7 + 14*b^12*c*d^3*x ^4 - 7*a*b^11*d^4*x^4 + 84*b^12*c^2*d^2*x - 112*a*b^11*c*d^3*x + 42*a^2*b^ 10*d^4*x)/b^14
Time = 0.79 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=x\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^4}{b^3}-\frac {4\,c\,d^3}{b^2}\right )}{b}-\frac {a^2\,d^4}{b^4}+\frac {6\,c^2\,d^2}{b^2}\right )-x^4\,\left (\frac {a\,d^4}{2\,b^3}-\frac {c\,d^3}{b^2}\right )+\frac {d^4\,x^7}{7\,b^2}+\frac {x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{3\,a\,\left (b^5\,x^3+a\,b^4\right )}-\frac {2\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (5\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{13/3}}+\frac {2\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (5\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{13/3}}-\frac {2\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (5\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{13/3}} \] Input:
int((c + d*x^3)^4/(a + b*x^3)^2,x)
Output:
x*((2*a*((2*a*d^4)/b^3 - (4*c*d^3)/b^2))/b - (a^2*d^4)/b^4 + (6*c^2*d^2)/b ^2) - x^4*((a*d^4)/(2*b^3) - (c*d^3)/b^2) + (d^4*x^7)/(7*b^2) + (x*(a^4*d^ 4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/(3*a*(a* b^4 + b^5*x^3)) - (2*log(b^(1/3)*x + a^(1/3))*(a*d - b*c)^3*(5*a*d + b*c)) /(9*a^(5/3)*b^(13/3)) + (2*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)^3*(5*a*d + b*c))/(9*a^(5/3)*b^(13/3)) - (2*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)^3*(5*a*d + b*c))/(9*a^(5/3)*b^(13/3))
Time = 0.21 (sec) , antiderivative size = 1074, normalized size of antiderivative = 4.02 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
int((d*x^3+c)^4/(b*x^3+a)^2,x)
Output:
(140*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a **5*d**4 - 392*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*a**4*b*c*d**3 + 140*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)* x)/(a**(1/3)*sqrt(3)))*a**4*b*d**4*x**3 + 336*a**(1/3)*sqrt(3)*atan((a**(1 /3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**2*c**2*d**2 - 392*a**(1/3) *sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b**2*c*d* *3*x**3 - 56*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqr t(3)))*a**2*b**3*c**3*d + 336*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3) *x)/(a**(1/3)*sqrt(3)))*a**2*b**3*c**2*d**2*x**3 - 28*a**(1/3)*sqrt(3)*ata n((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4*c**4 - 56*a**(1/3)* sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4*c**3*d*x **3 - 28*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3) ))*b**5*c**4*x**3 + 70*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2 /3)*x**2)*a**5*d**4 - 196*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b* *(2/3)*x**2)*a**4*b*c*d**3 + 70*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)* x + b**(2/3)*x**2)*a**4*b*d**4*x**3 + 168*a**(1/3)*log(a**(2/3) - b**(1/3) *a**(1/3)*x + b**(2/3)*x**2)*a**3*b**2*c**2*d**2 - 196*a**(1/3)*log(a**(2/ 3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b**2*c*d**3*x**3 - 28*a**(1 /3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b**3*c**3*d + 168*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*...