Integrand size = 19, antiderivative size = 203 \[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx=\frac {b^2 x}{d^2}+\frac {(b c-a d)^2 x}{3 c d^2 \left (c+d x^3\right )}+\frac {2 (b c-a d) (2 b c+a d) \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} d^{7/3}}-\frac {2 (b c-a d) (2 b c+a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{7/3}}+\frac {(b c-a d) (2 b c+a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} d^{7/3}} \] Output:
b^2*x/d^2+1/3*(-a*d+b*c)^2*x/c/d^2/(d*x^3+c)+2/9*(-a*d+b*c)*(a*d+2*b*c)*ar ctan(1/3*(c^(1/3)-2*d^(1/3)*x)*3^(1/2)/c^(1/3))*3^(1/2)/c^(5/3)/d^(7/3)-2/ 9*(-a*d+b*c)*(a*d+2*b*c)*ln(c^(1/3)+d^(1/3)*x)/c^(5/3)/d^(7/3)+1/9*(-a*d+b *c)*(a*d+2*b*c)*ln(c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/c^(5/3)/d^(7/3)
Time = 0.16 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx=\frac {9 b^2 \sqrt [3]{d} x+\frac {3 \sqrt [3]{d} (b c-a d)^2 x}{c \left (c+d x^3\right )}+\frac {2 \sqrt {3} \left (2 b^2 c^2-a b c d-a^2 d^2\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{5/3}}-\frac {2 \left (2 b^2 c^2-a b c d-a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3}}+\frac {\left (2 b^2 c^2-a b c d-a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3}}}{9 d^{7/3}} \] Input:
Integrate[(a + b*x^3)^2/(c + d*x^3)^2,x]
Output:
(9*b^2*d^(1/3)*x + (3*d^(1/3)*(b*c - a*d)^2*x)/(c*(c + d*x^3)) + (2*Sqrt[3 ]*(2*b^2*c^2 - a*b*c*d - a^2*d^2)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[ 3]])/c^(5/3) - (2*(2*b^2*c^2 - a*b*c*d - a^2*d^2)*Log[c^(1/3) + d^(1/3)*x] )/c^(5/3) + ((2*b^2*c^2 - a*b*c*d - a^2*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3) *x + d^(2/3)*x^2])/c^(5/3))/(9*d^(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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 915 |
| \(\displaystyle \int \left (\frac {b^2}{d^2}-\frac {-a^2 d^2+2 b d x^3 (b c-a d)+b^2 c^2}{d^2 \left (c+d x^3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (b c-a d) (a d+2 b c) \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} d^{7/3}}+\frac {(b c-a d) (a d+2 b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} d^{7/3}}-\frac {2 (b c-a d) (a d+2 b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{7/3}}+\frac {x (b c-a d)^2}{3 c d^2 \left (c+d x^3\right )}+\frac {b^2 x}{d^2}\) |
Input:
Int[(a + b*x^3)^2/(c + d*x^3)^2,x]
Output:
(b^2*x)/d^2 + ((b*c - a*d)^2*x)/(3*c*d^2*(c + d*x^3)) + (2*(b*c - a*d)*(2* b*c + a*d)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(3*Sqrt[3]*c ^(5/3)*d^(7/3)) - (2*(b*c - a*d)*(2*b*c + a*d)*Log[c^(1/3) + d^(1/3)*x])/( 9*c^(5/3)*d^(7/3)) + ((b*c - a*d)*(2*b*c + a*d)*Log[c^(2/3) - c^(1/3)*d^(1 /3)*x + d^(2/3)*x^2])/(9*c^(5/3)*d^(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 = 0.88 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(\frac {b^{2} x}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{3 c \,d^{2} \left (d \,x^{3}+c \right )}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+c \right )}{\sum }\frac {\left (a^{2} d^{2}+a b c d -2 b^{2} c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 d^{3} c}\) | \(99\) |
| default | \(\frac {b^{2} x}{d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{3 c \left (d \,x^{3}+c \right )}+\frac {2 \left (a^{2} d^{2}+a b c d -2 b^{2} c^{2}\right ) \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )}{3 c}}{d^{2}}\) | \(167\) |
Input:
int((b*x^3+a)^2/(d*x^3+c)^2,x,method=_RETURNVERBOSE)
Output:
b^2*x/d^2+1/3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/c*x/d^2/(d*x^3+c)+2/9/d^3/c*sum( (a^2*d^2+a*b*c*d-2*b^2*c^2)/_R^2*ln(x-_R),_R=RootOf(_Z^3*d+c))
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (164) = 328\).
Time = 0.09 (sec) , antiderivative size = 771, normalized size of antiderivative = 3.80 \[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((b*x^3+a)^2/(d*x^3+c)^2,x, algorithm="fricas")
Output:
[1/9*(9*b^2*c^3*d^2*x^4 - 3*sqrt(1/3)*(2*b^2*c^4*d - a*b*c^3*d^2 - a^2*c^2 *d^3 + (2*b^2*c^3*d^2 - a*b*c^2*d^3 - a^2*c*d^4)*x^3)*sqrt(-(c^2*d)^(1/3)/ d)*log((2*c*d*x^3 - 3*(c^2*d)^(1/3)*c*x - c^2 + 3*sqrt(1/3)*(2*c*d*x^2 + ( c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt(-(c^2*d)^(1/3)/d))/(d*x^3 + c)) + ( 2*b^2*c^3 - a*b*c^2*d - a^2*c*d^2 + (2*b^2*c^2*d - a*b*c*d^2 - a^2*d^3)*x^ 3)*(c^2*d)^(2/3)*log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) - 2*(2*b ^2*c^3 - a*b*c^2*d - a^2*c*d^2 + (2*b^2*c^2*d - a*b*c*d^2 - a^2*d^3)*x^3)* (c^2*d)^(2/3)*log(c*d*x + (c^2*d)^(2/3)) + 3*(4*b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3)*x)/(c^3*d^4*x^3 + c^4*d^3), 1/9*(9*b^2*c^3*d^2*x^4 - 6*sqrt (1/3)*(2*b^2*c^4*d - a*b*c^3*d^2 - a^2*c^2*d^3 + (2*b^2*c^3*d^2 - a*b*c^2* d^3 - a^2*c*d^4)*x^3)*sqrt((c^2*d)^(1/3)/d)*arctan(sqrt(1/3)*(2*(c^2*d)^(2 /3)*x - (c^2*d)^(1/3)*c)*sqrt((c^2*d)^(1/3)/d)/c^2) + (2*b^2*c^3 - a*b*c^2 *d - a^2*c*d^2 + (2*b^2*c^2*d - a*b*c*d^2 - a^2*d^3)*x^3)*(c^2*d)^(2/3)*lo g(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) - 2*(2*b^2*c^3 - a*b*c^2*d - a^2*c*d^2 + (2*b^2*c^2*d - a*b*c*d^2 - a^2*d^3)*x^3)*(c^2*d)^(2/3)*log(c *d*x + (c^2*d)^(2/3)) + 3*(4*b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3)*x)/( c^3*d^4*x^3 + c^4*d^3)]
Time = 0.63 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx=\frac {b^{2} x}{d^{2}} + \frac {x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 c^{2} d^{2} + 3 c d^{3} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} c^{5} d^{7} - 8 a^{6} d^{6} - 24 a^{5} b c d^{5} + 24 a^{4} b^{2} c^{2} d^{4} + 88 a^{3} b^{3} c^{3} d^{3} - 48 a^{2} b^{4} c^{4} d^{2} - 96 a b^{5} c^{5} d + 64 b^{6} c^{6}, \left ( t \mapsto t \log {\left (\frac {9 t c^{2} d^{2}}{2 a^{2} d^{2} + 2 a b c d - 4 b^{2} c^{2}} + x \right )} \right )\right )} \] Input:
integrate((b*x**3+a)**2/(d*x**3+c)**2,x)
Output:
b**2*x/d**2 + x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(3*c**2*d**2 + 3*c*d** 3*x**3) + RootSum(729*_t**3*c**5*d**7 - 8*a**6*d**6 - 24*a**5*b*c*d**5 + 2 4*a**4*b**2*c**2*d**4 + 88*a**3*b**3*c**3*d**3 - 48*a**2*b**4*c**4*d**2 - 96*a*b**5*c**5*d + 64*b**6*c**6, Lambda(_t, _t*log(9*_t*c**2*d**2/(2*a**2* d**2 + 2*a*b*c*d - 4*b**2*c**2) + x)))
Time = 0.11 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{3 \, {\left (c d^{3} x^{3} + c^{2} d^{2}\right )}} + \frac {b^{2} x}{d^{2}} - \frac {2 \, \sqrt {3} {\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{9 \, c d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{9 \, c d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{9 \, c d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} \] Input:
integrate((b*x^3+a)^2/(d*x^3+c)^2,x, algorithm="maxima")
Output:
1/3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/(c*d^3*x^3 + c^2*d^2) + b^2*x/d^2 - 2/9*sqrt(3)*(2*b^2*c^2 - a*b*c*d - a^2*d^2)*arctan(1/3*sqrt(3)*(2*x - (c/d )^(1/3))/(c/d)^(1/3))/(c*d^3*(c/d)^(2/3)) + 1/9*(2*b^2*c^2 - a*b*c*d - a^2 *d^2)*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3))/(c*d^3*(c/d)^(2/3)) - 2/9*(2* b^2*c^2 - a*b*c*d - a^2*d^2)*log(x + (c/d)^(1/3))/(c*d^3*(c/d)^(2/3))
Time = 0.13 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx=\frac {b^{2} x}{d^{2}} + \frac {2 \, \sqrt {3} {\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-c d^{2}\right )^{\frac {2}{3}} c d} + \frac {{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{9 \, \left (-c d^{2}\right )^{\frac {2}{3}} c d} + \frac {2 \, {\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{9 \, c^{2} d^{2}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{3 \, {\left (d x^{3} + c\right )} c d^{2}} \] Input:
integrate((b*x^3+a)^2/(d*x^3+c)^2,x, algorithm="giac")
Output:
b^2*x/d^2 + 2/9*sqrt(3)*(2*b^2*c^2 - a*b*c*d - a^2*d^2)*arctan(1/3*sqrt(3) *(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/((-c*d^2)^(2/3)*c*d) + 1/9*(2*b^2*c^2 - a*b*c*d - a^2*d^2)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/((-c*d^2)^(2 /3)*c*d) + 2/9*(2*b^2*c^2 - a*b*c*d - a^2*d^2)*(-c/d)^(1/3)*log(abs(x - (- c/d)^(1/3)))/(c^2*d^2) + 1/3*(b^2*c^2*x - 2*a*b*c*d*x + a^2*d^2*x)/((d*x^3 + c)*c*d^2)
Time = 0.70 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx=\frac {b^2\,x}{d^2}+\frac {x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{3\,c\,\left (d^3\,x^3+c\,d^2\right )}+\frac {2\,\ln \left (d^{1/3}\,x+c^{1/3}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d+2\,b\,c\right )}{9\,c^{5/3}\,d^{7/3}}+\frac {2\,\ln \left (2\,d^{1/3}\,x-c^{1/3}+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d+2\,b\,c\right )}{9\,c^{5/3}\,d^{7/3}}-\frac {2\,\ln \left (c^{1/3}-2\,d^{1/3}\,x+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d+2\,b\,c\right )}{9\,c^{5/3}\,d^{7/3}} \] Input:
int((a + b*x^3)^2/(c + d*x^3)^2,x)
Output:
(b^2*x)/d^2 + (x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(3*c*(c*d^2 + d^3*x^3)) + (2*log(d^(1/3)*x + c^(1/3))*(a*d - b*c)*(a*d + 2*b*c))/(9*c^(5/3)*d^(7/3 )) + (2*log(3^(1/2)*c^(1/3)*1i + 2*d^(1/3)*x - c^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(a*d - b*c)*(a*d + 2*b*c))/(9*c^(5/3)*d^(7/3)) - (2*log(3^(1/2)*c^(1/ 3)*1i - 2*d^(1/3)*x + c^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(a*d - b*c)*(a*d + 2 *b*c))/(9*c^(5/3)*d^(7/3))
Time = 0.30 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx =\text {Too large to display} \] Input:
int((b*x^3+a)^2/(d*x^3+c)^2,x)
Output:
( - 2*c**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3)))* a**2*c*d**2 - 2*c**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)* sqrt(3)))*a**2*d**3*x**3 - 2*c**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)* x)/(c**(1/3)*sqrt(3)))*a*b*c**2*d - 2*c**(1/3)*sqrt(3)*atan((c**(1/3) - 2* d**(1/3)*x)/(c**(1/3)*sqrt(3)))*a*b*c*d**2*x**3 + 4*c**(1/3)*sqrt(3)*atan( (c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3)))*b**2*c**3 + 4*c**(1/3)*sqrt( 3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3)))*b**2*c**2*d*x**3 - c **(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*a**2*c*d**2 - c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*a**2*d**3*x** 3 - c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*a*b*c**2* d - c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*a*b*c*d** 2*x**3 + 2*c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*b* *2*c**3 + 2*c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*b **2*c**2*d*x**3 + 2*c**(1/3)*log(c**(1/3) + d**(1/3)*x)*a**2*c*d**2 + 2*c* *(1/3)*log(c**(1/3) + d**(1/3)*x)*a**2*d**3*x**3 + 2*c**(1/3)*log(c**(1/3) + d**(1/3)*x)*a*b*c**2*d + 2*c**(1/3)*log(c**(1/3) + d**(1/3)*x)*a*b*c*d* *2*x**3 - 4*c**(1/3)*log(c**(1/3) + d**(1/3)*x)*b**2*c**3 - 4*c**(1/3)*log (c**(1/3) + d**(1/3)*x)*b**2*c**2*d*x**3 + 3*d**(1/3)*a**2*c*d**2*x - 6*d* *(1/3)*a*b*c**2*d*x + 12*d**(1/3)*b**2*c**3*x + 9*d**(1/3)*b**2*c**2*d*x** 4)/(9*d**(1/3)*c**2*d**2*(c + d*x**3))