Integrand size = 19, antiderivative size = 203 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx=\frac {d^2 x}{b^2}+\frac {(b c-a d)^2 x}{3 a b^2 \left (a+b x^3\right )}-\frac {2 (b c-a d) (b c+2 a d) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{7/3}}+\frac {2 (b c-a d) (b c+2 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{7/3}}-\frac {(b c-a d) (b c+2 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{7/3}} \] Output:
d^2*x/b^2+1/3*(-a*d+b*c)^2*x/a/b^2/(b*x^3+a)-2/9*(-a*d+b*c)*(2*a*d+b*c)*ar ctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(5/3)/b^(7/3)+2/ 9*(-a*d+b*c)*(2*a*d+b*c)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(7/3)-1/9*(-a*d+b *c)*(2*a*d+b*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/b^(7/3)
Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.01 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx=\frac {9 \sqrt [3]{b} d^2 x+\frac {3 \sqrt [3]{b} (b c-a d)^2 x}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \left (b^2 c^2+a b c d-2 a^2 d^2\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \left (b^2 c^2+a b c d-2 a^2 d^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {\left (b^2 c^2+a b c d-2 a^2 d^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{9 b^{7/3}} \] Input:
Integrate[(c + d*x^3)^2/(a + b*x^3)^2,x]
Output:
(9*b^(1/3)*d^2*x + (3*b^(1/3)*(b*c - a*d)^2*x)/(a*(a + b*x^3)) - (2*Sqrt[3 ]*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[ 3]])/a^(5/3) + (2*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*Log[a^(1/3) + b^(1/3)*x] )/a^(5/3) - ((b^2*c^2 + a*b*c*d - 2*a^2*d^2)*Log[a^(2/3) - a^(1/3)*b^(1/3) *x + b^(2/3)*x^2])/a^(5/3))/(9*b^(7/3))
Time = 0.69 (sec) , antiderivative size = 203, 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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 915 |
| \(\displaystyle \int \left (\frac {-a^2 d^2+2 b d x^3 (b c-a d)+b^2 c^2}{b^2 \left (a+b x^3\right )^2}+\frac {d^2}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (b c-a d) (2 a d+b c)}{3 \sqrt {3} a^{5/3} b^{7/3}}-\frac {(b c-a d) (2 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{7/3}}+\frac {2 (b c-a d) (2 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{7/3}}+\frac {x (b c-a d)^2}{3 a b^2 \left (a+b x^3\right )}+\frac {d^2 x}{b^2}\) |
Input:
Int[(c + d*x^3)^2/(a + b*x^3)^2,x]
Output:
(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(3*a*b^2*(a + b*x^3)) - (2*(b*c - a*d)*(b* c + 2*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a ^(5/3)*b^(7/3)) + (2*(b*c - a*d)*(b*c + 2*a*d)*Log[a^(1/3) + b^(1/3)*x])/( 9*a^(5/3)*b^(7/3)) - ((b*c - a*d)*(b*c + 2*a*d)*Log[a^(2/3) - a^(1/3)*b^(1 /3)*x + b^(2/3)*x^2])/(9*a^(5/3)*b^(7/3))
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 = 0.89 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(\frac {d^{2} x}{b^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{3 a \,b^{2} \left (b \,x^{3}+a \right )}-\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (2 a^{2} d^{2}-a b c d -b^{2} c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 b^{3} a}\) | \(101\) |
| default | \(\frac {d^{2} x}{b^{2}}-\frac {-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{3 a \left (b \,x^{3}+a \right )}+\frac {2 \left (2 a^{2} d^{2}-a b c d -b^{2} c^{2}\right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a}}{b^{2}}\) | \(170\) |
Input:
int((d*x^3+c)^2/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
d^2*x/b^2+1/3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a*x/b^2/(b*x^3+a)-2/9/b^3/a*sum( (2*a^2*d^2-a*b*c*d-b^2*c^2)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (164) = 328\).
Time = 0.09 (sec) , antiderivative size = 768, normalized size of antiderivative = 3.78 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((d*x^3+c)^2/(b*x^3+a)^2,x, algorithm="fricas")
Output:
[1/9*(9*a^3*b^2*d^2*x^4 - 3*sqrt(1/3)*(a^2*b^3*c^2 + a^3*b^2*c*d - 2*a^4*b *d^2 + (a*b^4*c^2 + a^2*b^3*c*d - 2*a^3*b^2*d^2)*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)) - (a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^2) *x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 2*(a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^2) *x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 3*(a^2*b^3*c^2 - 2*a^3* b^2*c*d + 4*a^4*b*d^2)*x)/(a^3*b^4*x^3 + a^4*b^3), 1/9*(9*a^3*b^2*d^2*x^4 + 6*sqrt(1/3)*(a^2*b^3*c^2 + a^3*b^2*c*d - 2*a^4*b*d^2 + (a*b^4*c^2 + a^2* b^3*c*d - 2*a^3*b^2*d^2)*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)*sqrt(-(-a^2*b)^(1/3)/b)/a^2) - (a*b^2 *c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^2)*x^3)*(- a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 2*(a*b^2 *c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^2)*x^3)*(- a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 3*(a^2*b^3*c^2 - 2*a^3*b^2*c*d + 4*a^4*b*d^2)*x)/(a^3*b^4*x^3 + a^4*b^3)]
Time = 0.81 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx=\frac {x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 a^{2} b^{2} + 3 a b^{3} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{7} + 64 a^{6} d^{6} - 96 a^{5} b c d^{5} - 48 a^{4} b^{2} c^{2} d^{4} + 88 a^{3} b^{3} c^{3} d^{3} + 24 a^{2} b^{4} c^{4} d^{2} - 24 a b^{5} c^{5} d - 8 b^{6} c^{6}, \left ( t \mapsto t \log {\left (- \frac {9 t a^{2} b^{2}}{4 a^{2} d^{2} - 2 a b c d - 2 b^{2} c^{2}} + x \right )} \right )\right )} + \frac {d^{2} x}{b^{2}} \] Input:
integrate((d*x**3+c)**2/(b*x**3+a)**2,x)
Output:
x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(3*a**2*b**2 + 3*a*b**3*x**3) + Root Sum(729*_t**3*a**5*b**7 + 64*a**6*d**6 - 96*a**5*b*c*d**5 - 48*a**4*b**2*c **2*d**4 + 88*a**3*b**3*c**3*d**3 + 24*a**2*b**4*c**4*d**2 - 24*a*b**5*c** 5*d - 8*b**6*c**6, Lambda(_t, _t*log(-9*_t*a**2*b**2/(4*a**2*d**2 - 2*a*b* c*d - 2*b**2*c**2) + x))) + d**2*x/b**2
Time = 0.11 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{3 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {d^{2} x}{b^{2}} + \frac {2 \, \sqrt {3} {\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((d*x^3+c)^2/(b*x^3+a)^2,x, algorithm="maxima")
Output:
1/3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/(a*b^3*x^3 + a^2*b^2) + d^2*x/b^2 + 2/9*sqrt(3)*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*arctan(1/3*sqrt(3)*(2*x - (a/b )^(1/3))/(a/b)^(1/3))/(a*b^3*(a/b)^(2/3)) - 1/9*(b^2*c^2 + a*b*c*d - 2*a^2 *d^2)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^3*(a/b)^(2/3)) + 2/9*(b^ 2*c^2 + a*b*c*d - 2*a^2*d^2)*log(x + (a/b)^(1/3))/(a*b^3*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.12 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx=\frac {d^{2} x}{b^{2}} - \frac {2 \, \sqrt {3} {\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {2 \, {\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{2}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{3 \, {\left (b x^{3} + a\right )} a b^{2}} \] Input:
integrate((d*x^3+c)^2/(b*x^3+a)^2,x, algorithm="giac")
Output:
d^2*x/b^2 - 2/9*sqrt(3)*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*arctan(1/3*sqrt(3) *(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a*b) - 1/9*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2 /3)*a*b) - 2/9*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*(-a/b)^(1/3)*log(abs(x - (- a/b)^(1/3)))/(a^2*b^2) + 1/3*(b^2*c^2*x - 2*a*b*c*d*x + a^2*d^2*x)/((b*x^3 + a)*a*b^2)
Time = 0.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx=\frac {d^2\,x}{b^2}+\frac {x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{3\,a\,\left (b^3\,x^3+a\,b^2\right )}-\frac {2\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (a\,d-b\,c\right )\,\left (2\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{7/3}}-\frac {2\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )\,\left (2\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{7/3}}+\frac {2\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )\,\left (2\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{7/3}} \] Input:
int((c + d*x^3)^2/(a + b*x^3)^2,x)
Output:
(d^2*x)/b^2 + (x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(3*a*(a*b^2 + b^3*x^3)) - (2*log(b^(1/3)*x + a^(1/3))*(a*d - b*c)*(2*a*d + b*c))/(9*a^(5/3)*b^(7/3 )) - (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)*(2*a*d + b*c))/(9*a^(5/3)*b^(7/3)) + (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)*(2*a*d + b*c))/(9*a^(5/3)*b^(7/3))
Time = 0.22 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.83 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
int((d*x^3+c)^2/(b*x^3+a)^2,x)
Output:
(4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a** 3*d**2 - 2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt( 3)))*a**2*b*c*d + 4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1 /3)*sqrt(3)))*a**2*b*d**2*x**3 - 2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b** (1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**2*c**2 - 2*a**(1/3)*sqrt(3)*atan((a**(1/ 3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**2*c*d*x**3 - 2*a**(1/3)*sqrt(3 )*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**3*c**2*x**3 + 2*a* *(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*d**2 - a** (1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*c*d + 2*a **(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*d**2*x* *3 - a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**2*c **2 - a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**2* c*d*x**3 - a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b* *3*c**2*x**3 - 4*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**3*d**2 + 2*a**(1/3 )*log(a**(1/3) + b**(1/3)*x)*a**2*b*c*d - 4*a**(1/3)*log(a**(1/3) + b**(1/ 3)*x)*a**2*b*d**2*x**3 + 2*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a*b**2*c**2 + 2*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a*b**2*c*d*x**3 + 2*a**(1/3)*log( a**(1/3) + b**(1/3)*x)*b**3*c**2*x**3 + 12*b**(1/3)*a**3*d**2*x - 6*b**(1/ 3)*a**2*b*c*d*x + 9*b**(1/3)*a**2*b*d**2*x**4 + 3*b**(1/3)*a*b**2*c**2*x)/ (9*b**(1/3)*a**2*b**2*(a + b*x**3))