Integrand size = 19, antiderivative size = 142 \[ \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx=\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^3 (4 b c-a d) x^5}{5 b^2}+\frac {d^4 x^7}{7 b}+\frac {(b c-a d)^4 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{9/2}} \]
d*(-a*d+2*b*c)*(a^2*d^2-2*a*b*c*d+2*b^2*c^2)*x/b^4+1/3*d^2*(a^2*d^2-4*a*b* c*d+6*b^2*c^2)*x^3/b^3+1/5*d^3*(-a*d+4*b*c)*x^5/b^2+1/7*d^4*x^7/b+(-a*d+b* c)^4*arctan(x*b^(1/2)/a^(1/2))/b^(9/2)/a^(1/2)
Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx=\frac {d x \left (-105 a^3 d^3+35 a^2 b d^2 \left (12 c+d x^2\right )-7 a b^2 d \left (90 c^2+20 c d x^2+3 d^2 x^4\right )+3 b^3 \left (140 c^3+70 c^2 d x^2+28 c d^2 x^4+5 d^3 x^6\right )\right )}{105 b^4}+\frac {(b c-a d)^4 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{9/2}} \]
(d*x*(-105*a^3*d^3 + 35*a^2*b*d^2*(12*c + d*x^2) - 7*a*b^2*d*(90*c^2 + 20* c*d*x^2 + 3*d^2*x^4) + 3*b^3*(140*c^3 + 70*c^2*d*x^2 + 28*c*d^2*x^4 + 5*d^ 3*x^6)))/(105*b^4) + ((b*c - a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]* b^(9/2))
Time = 0.30 (sec) , antiderivative size = 142, 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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \int \left (\frac {d (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac {d^2 x^2 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{b^3}+\frac {a^4 d^4-4 a^3 b c d^3+6 a^2 b^2 c^2 d^2-4 a b^3 c^3 d+b^4 c^4}{b^4 \left (a+b x^2\right )}+\frac {d^3 x^4 (4 b c-a d)}{b^2}+\frac {d^4 x^6}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac {d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{3 b^3}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^4}{\sqrt {a} b^{9/2}}+\frac {d^3 x^5 (4 b c-a d)}{5 b^2}+\frac {d^4 x^7}{7 b}\) |
(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^3)/(3*b^3) + (d^3*(4*b*c - a*d)*x^5)/(5*b^2) + (d^4*x^7)/(7*b) + ((b*c - a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^( 9/2))
3.1.20.3.1 Defintions of rubi rules used
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 = 2.33 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.38
method | result | size |
default | \(-\frac {d \left (-\frac {b^{3} d^{3} x^{7}}{7}+\frac {\left (\left (a d -2 b c \right ) b^{2} d^{2}-2 b^{3} c \,d^{2}\right ) x^{5}}{5}+\frac {\left (2 \left (a d -2 b c \right ) b^{2} c d -b d \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right )\right ) x^{3}}{3}+\left (a d -2 b c \right ) \left (a^{2} d^{2}-2 a b c d +2 b^{2} c^{2}\right ) x \right )}{b^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\) | \(196\) |
risch | \(\frac {d^{4} x^{7}}{7 b}-\frac {d^{4} x^{5} a}{5 b^{2}}+\frac {4 d^{3} x^{5} c}{5 b}-\frac {4 d^{3} a c \,x^{3}}{3 b^{2}}+\frac {2 d^{2} c^{2} x^{3}}{b}+\frac {d^{4} a^{2} x^{3}}{3 b^{3}}-\frac {d^{4} a^{3} x}{b^{4}}+\frac {4 d^{3} a^{2} c x}{b^{3}}-\frac {6 d^{2} a \,c^{2} x}{b^{2}}+\frac {4 d \,c^{3} x}{b}-\frac {\ln \left (b x +\sqrt {-a b}\right ) a^{4} d^{4}}{2 b^{4} \sqrt {-a b}}+\frac {2 \ln \left (b x +\sqrt {-a b}\right ) a^{3} c \,d^{3}}{b^{3} \sqrt {-a b}}-\frac {3 \ln \left (b x +\sqrt {-a b}\right ) a^{2} c^{2} d^{2}}{b^{2} \sqrt {-a b}}+\frac {2 \ln \left (b x +\sqrt {-a b}\right ) a \,c^{3} d}{b \sqrt {-a b}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c^{4}}{2 \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) a^{4} d^{4}}{2 b^{4} \sqrt {-a b}}-\frac {2 \ln \left (-b x +\sqrt {-a b}\right ) a^{3} c \,d^{3}}{b^{3} \sqrt {-a b}}+\frac {3 \ln \left (-b x +\sqrt {-a b}\right ) a^{2} c^{2} d^{2}}{b^{2} \sqrt {-a b}}-\frac {2 \ln \left (-b x +\sqrt {-a b}\right ) a \,c^{3} d}{b \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c^{4}}{2 \sqrt {-a b}}\) | \(405\) |
-d/b^4*(-1/7*b^3*d^3*x^7+1/5*((a*d-2*b*c)*b^2*d^2-2*b^3*c*d^2)*x^5+1/3*(2* (a*d-2*b*c)*b^2*c*d-b*d*(a^2*d^2-2*a*b*c*d+2*b^2*c^2))*x^3+(a*d-2*b*c)*(a^ 2*d^2-2*a*b*c*d+2*b^2*c^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)/b^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))
Time = 0.26 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.01 \[ \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx=\left [\frac {30 \, a b^{4} d^{4} x^{7} + 42 \, {\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 70 \, {\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} - 105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 210 \, {\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{210 \, a b^{5}}, \frac {15 \, a b^{4} d^{4} x^{7} + 21 \, {\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 35 \, {\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} + 105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 105 \, {\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{105 \, a b^{5}}\right ] \]
[1/210*(30*a*b^4*d^4*x^7 + 42*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^5 + 70*(6*a* b^4*c^2*d^2 - 4*a^2*b^3*c*d^3 + a^3*b^2*d^4)*x^3 - 105*(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)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 210*(4*a*b^4*c^3*d - 6*a^2*b^3*c^2*d ^2 + 4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)/(a*b^5), 1/105*(15*a*b^4*d^4*x^7 + 21 *(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^5 + 35*(6*a*b^4*c^2*d^2 - 4*a^2*b^3*c*d^3 + a^3*b^2*d^4)*x^3 + 105*(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)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + 105*(4*a*b^4*c^3 *d - 6*a^2*b^3*c^2*d^2 + 4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)/(a*b^5)]
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (136) = 272\).
Time = 0.48 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.30 \[ \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx=x^{5} \left (- \frac {a d^{4}}{5 b^{2}} + \frac {4 c d^{3}}{5 b}\right ) + x^{3} \left (\frac {a^{2} d^{4}}{3 b^{3}} - \frac {4 a c d^{3}}{3 b^{2}} + \frac {2 c^{2} d^{2}}{b}\right ) + x \left (- \frac {a^{3} d^{4}}{b^{4}} + \frac {4 a^{2} c d^{3}}{b^{3}} - \frac {6 a c^{2} d^{2}}{b^{2}} + \frac {4 c^{3} d}{b}\right ) - \frac {\sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4} \log {\left (- \frac {a b^{4} \sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4} \log {\left (\frac {a b^{4} \sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac {d^{4} x^{7}}{7 b} \]
x**5*(-a*d**4/(5*b**2) + 4*c*d**3/(5*b)) + x**3*(a**2*d**4/(3*b**3) - 4*a* c*d**3/(3*b**2) + 2*c**2*d**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) - sqrt(-1/(a*b**9))*(a*d - b*c)**4*log (-a*b**4*sqrt(-1/(a*b**9))*(a*d - b*c)**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)/2 + sqrt(-1/(a*b* *9))*(a*d - b*c)**4*log(a*b**4*sqrt(-1/(a*b**9))*(a*d - b*c)**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)/2 + d**4*x**7/(7*b)
Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.32 \[ \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx=\frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{3} d^{4} x^{7} + 21 \, {\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{5} + 35 \, {\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} + 105 \, {\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x}{105 \, b^{4}} \]
(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)*ar ctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/105*(15*b^3*d^4*x^7 + 21*(4*b^3*c* d^3 - a*b^2*d^4)*x^5 + 35*(6*b^3*c^2*d^2 - 4*a*b^2*c*d^3 + a^2*b*d^4)*x^3 + 105*(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
Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.39 \[ \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx=\frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{6} d^{4} x^{7} + 84 \, b^{6} c d^{3} x^{5} - 21 \, a b^{5} d^{4} x^{5} + 210 \, b^{6} c^{2} d^{2} x^{3} - 140 \, a b^{5} c d^{3} x^{3} + 35 \, a^{2} b^{4} d^{4} x^{3} + 420 \, b^{6} c^{3} d x - 630 \, a b^{5} c^{2} d^{2} x + 420 \, a^{2} b^{4} c d^{3} x - 105 \, a^{3} b^{3} d^{4} x}{105 \, b^{7}} \]
(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)*ar ctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/105*(15*b^6*d^4*x^7 + 84*b^6*c*d^3 *x^5 - 21*a*b^5*d^4*x^5 + 210*b^6*c^2*d^2*x^3 - 140*a*b^5*c*d^3*x^3 + 35*a ^2*b^4*d^4*x^3 + 420*b^6*c^3*d*x - 630*a*b^5*c^2*d^2*x + 420*a^2*b^4*c*d^3 *x - 105*a^3*b^3*d^4*x)/b^7
Time = 4.75 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.52 \[ \int \frac {\left (c+d x^2\right )^4}{a+b x^2} \, dx=x\,\left (\frac {4\,c^3\,d}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{b}+\frac {6\,c^2\,d^2}{b}\right )}{b}\right )-x^5\,\left (\frac {a\,d^4}{5\,b^2}-\frac {4\,c\,d^3}{5\,b}\right )+x^3\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{3\,b}+\frac {2\,c^2\,d^2}{b}\right )+\frac {d^4\,x^7}{7\,b}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^4}{\sqrt {a}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )\,{\left (a\,d-b\,c\right )}^4}{\sqrt {a}\,b^{9/2}} \]
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^5*((a*d^4)/(5*b^2) - (4*c*d^3)/(5*b)) + x^3*((a*((a*d^4)/b^2 - (4*c* d^3)/b))/(3*b) + (2*c^2*d^2)/b) + (d^4*x^7)/(7*b) + (atan((b^(1/2)*x*(a*d - b*c)^4)/(a^(1/2)*(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)))*(a*d - b*c)^4)/(a^(1/2)*b^(9/2))