Integrand size = 19, antiderivative size = 98 \[ \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx=\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x}{d^3}-\frac {b^2 (b c-3 a d) x^3}{3 d^2}+\frac {b^3 x^5}{5 d}-\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}} \] Output:
b*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)*x/d^3-1/3*b^2*(-3*a*d+b*c)*x^3/d^2+1/5*b^3 *x^5/d-(-a*d+b*c)^3*arctan(d^(1/2)*x/c^(1/2))/c^(1/2)/d^(7/2)
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx=\frac {b x \left (45 a^2 d^2+15 a b d \left (-3 c+d x^2\right )+b^2 \left (15 c^2-5 c d x^2+3 d^2 x^4\right )\right )}{15 d^3}-\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}} \] Input:
Integrate[(a + b*x^2)^3/(c + d*x^2),x]
Output:
(b*x*(45*a^2*d^2 + 15*a*b*d*(-3*c + d*x^2) + b^2*(15*c^2 - 5*c*d*x^2 + 3*d ^2*x^4)))/(15*d^3) - ((b*c - a*d)^3*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]* d^(7/2))
Time = 0.25 (sec) , antiderivative size = 98, 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 = {300, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \int \left (\frac {b \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{d^3}+\frac {a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3}{d^3 \left (c+d x^2\right )}-\frac {b^2 x^2 (b c-3 a d)}{d^2}+\frac {b^3 x^4}{d}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b x \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{d^3}-\frac {(b c-a d)^3 \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}}-\frac {b^2 x^3 (b c-3 a d)}{3 d^2}+\frac {b^3 x^5}{5 d}\) |
Input:
Int[(a + b*x^2)^3/(c + d*x^2),x]
Output:
(b*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x)/d^3 - (b^2*(b*c - 3*a*d)*x^3)/(3*d ^2) + (b^3*x^5)/(5*d) - ((b*c - a*d)^3*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[ c]*d^(7/2))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {b \left (\frac {1}{5} b^{2} d^{2} x^{5}+x^{3} a b \,d^{2}-\frac {1}{3} x^{3} b^{2} c d +3 a^{2} d^{2} x -3 a b c d x +b^{2} c^{2} x \right )}{d^{3}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{d^{3} \sqrt {c d}}\) | \(116\) |
risch | \(\frac {b^{3} x^{5}}{5 d}+\frac {b^{2} x^{3} a}{d}-\frac {b^{3} x^{3} c}{3 d^{2}}+\frac {3 b \,a^{2} x}{d}-\frac {3 b^{2} a c x}{d^{2}}+\frac {b^{3} c^{2} x}{d^{3}}-\frac {\ln \left (x d +\sqrt {-c d}\right ) a^{3}}{2 \sqrt {-c d}}+\frac {3 \ln \left (x d +\sqrt {-c d}\right ) a^{2} b c}{2 d \sqrt {-c d}}-\frac {3 \ln \left (x d +\sqrt {-c d}\right ) a \,b^{2} c^{2}}{2 d^{2} \sqrt {-c d}}+\frac {\ln \left (x d +\sqrt {-c d}\right ) b^{3} c^{3}}{2 d^{3} \sqrt {-c d}}+\frac {\ln \left (-x d +\sqrt {-c d}\right ) a^{3}}{2 \sqrt {-c d}}-\frac {3 \ln \left (-x d +\sqrt {-c d}\right ) a^{2} b c}{2 d \sqrt {-c d}}+\frac {3 \ln \left (-x d +\sqrt {-c d}\right ) a \,b^{2} c^{2}}{2 d^{2} \sqrt {-c d}}-\frac {\ln \left (-x d +\sqrt {-c d}\right ) b^{3} c^{3}}{2 d^{3} \sqrt {-c d}}\) | \(284\) |
Input:
int((b*x^2+a)^3/(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
b/d^3*(1/5*b^2*d^2*x^5+x^3*a*b*d^2-1/3*x^3*b^2*c*d+3*a^2*d^2*x-3*a*b*c*d*x +b^2*c^2*x)+(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/d^3/(c*d)^(1/2)* arctan(x*d/(c*d)^(1/2))
Time = 0.09 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.96 \[ \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx=\left [\frac {6 \, b^{3} c d^{3} x^{5} - 10 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} x^{3} + 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 30 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} x}{30 \, c d^{4}}, \frac {3 \, b^{3} c d^{3} x^{5} - 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} x^{3} - 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + 15 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} x}{15 \, c d^{4}}\right ] \] Input:
integrate((b*x^2+a)^3/(d*x^2+c),x, algorithm="fricas")
Output:
[1/30*(6*b^3*c*d^3*x^5 - 10*(b^3*c^2*d^2 - 3*a*b^2*c*d^3)*x^3 + 15*(b^3*c^ 3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-c*d)*log((d*x^2 - 2*sqr t(-c*d)*x - c)/(d*x^2 + c)) + 30*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c* d^3)*x)/(c*d^4), 1/15*(3*b^3*c*d^3*x^5 - 5*(b^3*c^2*d^2 - 3*a*b^2*c*d^3)*x ^3 - 15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c*d)*arct an(sqrt(c*d)*x/c) + 15*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3)*x)/(c *d^4)]
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (92) = 184\).
Time = 0.29 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.43 \[ \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx=\frac {b^{3} x^{5}}{5 d} + x^{3} \left (\frac {a b^{2}}{d} - \frac {b^{3} c}{3 d^{2}}\right ) + x \left (\frac {3 a^{2} b}{d} - \frac {3 a b^{2} c}{d^{2}} + \frac {b^{3} c^{2}}{d^{3}}\right ) - \frac {\sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3} \log {\left (- \frac {c d^{3} \sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3} \log {\left (\frac {c d^{3} \sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} \] Input:
integrate((b*x**2+a)**3/(d*x**2+c),x)
Output:
b**3*x**5/(5*d) + x**3*(a*b**2/d - b**3*c/(3*d**2)) + x*(3*a**2*b/d - 3*a* b**2*c/d**2 + b**3*c**2/d**3) - sqrt(-1/(c*d**7))*(a*d - b*c)**3*log(-c*d* *3*sqrt(-1/(c*d**7))*(a*d - b*c)**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b** 2*c**2*d - b**3*c**3) + x)/2 + sqrt(-1/(c*d**7))*(a*d - b*c)**3*log(c*d**3 *sqrt(-1/(c*d**7))*(a*d - b*c)**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2* c**2*d - b**3*c**3) + x)/2
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{3} d^{2} x^{5} - 5 \, {\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{3} + 15 \, {\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x}{15 \, d^{3}} \] Input:
integrate((b*x^2+a)^3/(d*x^2+c),x, algorithm="maxima")
Output:
-(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(d*x/sqrt(c*d)) /(sqrt(c*d)*d^3) + 1/15*(3*b^3*d^2*x^5 - 5*(b^3*c*d - 3*a*b^2*d^2)*x^3 + 1 5*(b^3*c^2 - 3*a*b^2*c*d + 3*a^2*b*d^2)*x)/d^3
Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{3} d^{4} x^{5} - 5 \, b^{3} c d^{3} x^{3} + 15 \, a b^{2} d^{4} x^{3} + 15 \, b^{3} c^{2} d^{2} x - 45 \, a b^{2} c d^{3} x + 45 \, a^{2} b d^{4} x}{15 \, d^{5}} \] Input:
integrate((b*x^2+a)^3/(d*x^2+c),x, algorithm="giac")
Output:
-(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(d*x/sqrt(c*d)) /(sqrt(c*d)*d^3) + 1/15*(3*b^3*d^4*x^5 - 5*b^3*c*d^3*x^3 + 15*a*b^2*d^4*x^ 3 + 15*b^3*c^2*d^2*x - 45*a*b^2*c*d^3*x + 45*a^2*b*d^4*x)/d^5
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx=x^3\,\left (\frac {a\,b^2}{d}-\frac {b^3\,c}{3\,d^2}\right )+x\,\left (\frac {3\,a^2\,b}{d}-\frac {c\,\left (\frac {3\,a\,b^2}{d}-\frac {b^3\,c}{d^2}\right )}{d}\right )+\frac {b^3\,x^5}{5\,d}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {c}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{\sqrt {c}\,d^{7/2}} \] Input:
int((a + b*x^2)^3/(c + d*x^2),x)
Output:
x^3*((a*b^2)/d - (b^3*c)/(3*d^2)) + x*((3*a^2*b)/d - (c*((3*a*b^2)/d - (b^ 3*c)/d^2))/d) + (b^3*x^5)/(5*d) + (atan((d^(1/2)*x*(a*d - b*c)^3)/(c^(1/2) *(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))*(a*d - b*c)^3)/(c^( 1/2)*d^(7/2))
Time = 0.19 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x^2\right )^3}{c+d x^2} \, dx=\frac {15 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{3} d^{3}-45 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a^{2} b c \,d^{2}+45 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a \,b^{2} c^{2} d -15 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b^{3} c^{3}+45 a^{2} b c \,d^{3} x -45 a \,b^{2} c^{2} d^{2} x +15 a \,b^{2} c \,d^{3} x^{3}+15 b^{3} c^{3} d x -5 b^{3} c^{2} d^{2} x^{3}+3 b^{3} c \,d^{3} x^{5}}{15 c \,d^{4}} \] Input:
int((b*x^2+a)^3/(d*x^2+c),x)
Output:
(15*sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**3*d**3 - 45*sqrt(d)*s qrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a**2*b*c*d**2 + 45*sqrt(d)*sqrt(c)*at an((d*x)/(sqrt(d)*sqrt(c)))*a*b**2*c**2*d - 15*sqrt(d)*sqrt(c)*atan((d*x)/ (sqrt(d)*sqrt(c)))*b**3*c**3 + 45*a**2*b*c*d**3*x - 45*a*b**2*c**2*d**2*x + 15*a*b**2*c*d**3*x**3 + 15*b**3*c**3*d*x - 5*b**3*c**2*d**2*x**3 + 3*b** 3*c*d**3*x**5)/(15*c*d**4)