Integrand size = 30, antiderivative size = 368 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 (b e-a f)^3 x}{3 a b^5 \left (a+b x^2\right )^{3/2}}+\frac {(b c-a d) (b e-a f)^2 \left (2 b^2 c e-13 a^2 d f+a b (4 d e+7 c f)\right ) x}{3 a^2 b^5 \sqrt {a+b x^2}}+\frac {f \left (41 a^2 d^2 f^2-22 a b d f (3 d e+2 c f)+8 b^2 \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{16 b^5}+\frac {d f^2 (18 b d e+12 b c f-17 a d f) x^3 \sqrt {a+b x^2}}{24 b^4}+\frac {d^2 f^3 x^5 \sqrt {a+b x^2}}{6 b^3}-\frac {\left (105 a^3 d^2 f^3-70 a^2 b d f^2 (3 d e+2 c f)+40 a b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-16 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{11/2}} \] Output:
1/3*(-a*d+b*c)^2*(-a*f+b*e)^3*x/a/b^5/(b*x^2+a)^(3/2)+1/3*(-a*d+b*c)*(-a*f +b*e)^2*(2*b^2*c*e-13*a^2*d*f+a*b*(7*c*f+4*d*e))*x/a^2/b^5/(b*x^2+a)^(1/2) +1/16*f*(41*a^2*d^2*f^2-22*a*b*d*f*(2*c*f+3*d*e)+8*b^2*(c^2*f^2+6*c*d*e*f+ 3*d^2*e^2))*x*(b*x^2+a)^(1/2)/b^5+1/24*d*f^2*(-17*a*d*f+12*b*c*f+18*b*d*e) *x^3*(b*x^2+a)^(1/2)/b^4+1/6*d^2*f^3*x^5*(b*x^2+a)^(1/2)/b^3-1/16*(105*a^3 *d^2*f^3-70*a^2*b*d*f^2*(2*c*f+3*d*e)+40*a*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2* e^2)-16*b^3*e*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*arctanh(b^(1/2)*x/(b*x^2+a)^( 1/2))/b^(11/2)
Time = 1.63 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.25 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {b} x \left (315 a^6 d^2 f^3+32 b^6 c^2 e^3 x^2+210 a^5 b d f^2 \left (-3 d e-2 c f+2 d f x^2\right )+16 a b^5 c e^2 \left (2 d e x^2+3 c \left (e+f x^2\right )\right )+a^4 b^2 f \left (120 c^2 f^2+80 c d f \left (9 e-7 f x^2\right )+d^2 \left (360 e^2-840 e f x^2+63 f^2 x^4\right )\right )+4 a^2 b^4 x^2 \left (6 c^2 f^2 \left (-8 e+f x^2\right )+6 c d f \left (-16 e^2+6 e f x^2+f^2 x^4\right )+d^2 \left (-16 e^3+18 e^2 f x^2+9 e f^2 x^4+2 f^3 x^6\right )\right )-2 a^3 b^3 \left (8 c^2 f^2 \left (9 e-10 f x^2\right )+6 c d f \left (24 e^2-80 e f x^2+7 f^2 x^4\right )+3 d^2 \left (8 e^3-80 e^2 f x^2+21 e f^2 x^4+3 f^3 x^6\right )\right )\right )}{a^2 \left (a+b x^2\right )^{3/2}}-3 \left (-105 a^3 d^2 f^3+70 a^2 b d f^2 (3 d e+2 c f)-40 a b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+16 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{11/2}} \] Input:
Integrate[((c + d*x^2)^2*(e + f*x^2)^3)/(a + b*x^2)^(5/2),x]
Output:
((Sqrt[b]*x*(315*a^6*d^2*f^3 + 32*b^6*c^2*e^3*x^2 + 210*a^5*b*d*f^2*(-3*d* e - 2*c*f + 2*d*f*x^2) + 16*a*b^5*c*e^2*(2*d*e*x^2 + 3*c*(e + f*x^2)) + a^ 4*b^2*f*(120*c^2*f^2 + 80*c*d*f*(9*e - 7*f*x^2) + d^2*(360*e^2 - 840*e*f*x ^2 + 63*f^2*x^4)) + 4*a^2*b^4*x^2*(6*c^2*f^2*(-8*e + f*x^2) + 6*c*d*f*(-16 *e^2 + 6*e*f*x^2 + f^2*x^4) + d^2*(-16*e^3 + 18*e^2*f*x^2 + 9*e*f^2*x^4 + 2*f^3*x^6)) - 2*a^3*b^3*(8*c^2*f^2*(9*e - 10*f*x^2) + 6*c*d*f*(24*e^2 - 80 *e*f*x^2 + 7*f^2*x^4) + 3*d^2*(8*e^3 - 80*e^2*f*x^2 + 21*e*f^2*x^4 + 3*f^3 *x^6))))/(a^2*(a + b*x^2)^(3/2)) - 3*(-105*a^3*d^2*f^3 + 70*a^2*b*d*f^2*(3 *d*e + 2*c*f) - 40*a*b^2*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + 16*b^3*e*(d ^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(48* b^(11/2))
Leaf count is larger than twice the leaf count of optimal. \(750\) vs. \(2(368)=736\).
Time = 0.83 (sec) , antiderivative size = 750, normalized size of antiderivative = 2.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {433, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {f x^6 \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{\left (a+b x^2\right )^{5/2}}+\frac {e x^4 \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{\left (a+b x^2\right )^{5/2}}+\frac {c^2 e^3}{\left (a+b x^2\right )^{5/2}}+\frac {c e^2 x^2 (3 c f+2 d e)}{\left (a+b x^2\right )^{5/2}}+\frac {d f^2 x^8 (2 c f+3 d e)}{\left (a+b x^2\right )^{5/2}}+\frac {d^2 f^3 x^{10}}{\left (a+b x^2\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {105 a^3 d^2 f^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{11/2}}+\frac {35 a^2 d f^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 c f+3 d e)}{8 b^{9/2}}+\frac {105 a^2 d^2 f^3 x \sqrt {a+b x^2}}{16 b^5}+\frac {2 c^2 e^3 x}{3 a^2 \sqrt {a+b x^2}}-\frac {5 a f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{2 b^{7/2}}+\frac {e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{b^{5/2}}-\frac {35 a d f^2 x \sqrt {a+b x^2} (2 c f+3 d e)}{8 b^4}-\frac {35 a d^2 f^3 x^3 \sqrt {a+b x^2}}{8 b^4}+\frac {5 f x \sqrt {a+b x^2} \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{2 b^3}+\frac {35 d f^2 x^3 \sqrt {a+b x^2} (2 c f+3 d e)}{12 b^3}+\frac {7 d^2 f^3 x^5 \sqrt {a+b x^2}}{2 b^3}-\frac {e x \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{b^2 \sqrt {a+b x^2}}-\frac {5 f x^3 \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{3 b^2 \sqrt {a+b x^2}}-\frac {7 d f^2 x^5 (2 c f+3 d e)}{3 b^2 \sqrt {a+b x^2}}-\frac {3 d^2 f^3 x^7}{b^2 \sqrt {a+b x^2}}-\frac {f x^5 \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{3 b \left (a+b x^2\right )^{3/2}}-\frac {e x^3 \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{3 b \left (a+b x^2\right )^{3/2}}+\frac {c^2 e^3 x}{3 a \left (a+b x^2\right )^{3/2}}+\frac {c e^2 x^3 (3 c f+2 d e)}{3 a \left (a+b x^2\right )^{3/2}}-\frac {d f^2 x^7 (2 c f+3 d e)}{3 b \left (a+b x^2\right )^{3/2}}-\frac {d^2 f^3 x^9}{3 b \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[((c + d*x^2)^2*(e + f*x^2)^3)/(a + b*x^2)^(5/2),x]
Output:
(c^2*e^3*x)/(3*a*(a + b*x^2)^(3/2)) + (c*e^2*(2*d*e + 3*c*f)*x^3)/(3*a*(a + b*x^2)^(3/2)) - (e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)*x^3)/(3*b*(a + b*x^ 2)^(3/2)) - (f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^5)/(3*b*(a + b*x^2)^(3/ 2)) - (d*f^2*(3*d*e + 2*c*f)*x^7)/(3*b*(a + b*x^2)^(3/2)) - (d^2*f^3*x^9)/ (3*b*(a + b*x^2)^(3/2)) + (2*c^2*e^3*x)/(3*a^2*Sqrt[a + b*x^2]) - (e*(d^2* e^2 + 6*c*d*e*f + 3*c^2*f^2)*x)/(b^2*Sqrt[a + b*x^2]) - (5*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^3)/(3*b^2*Sqrt[a + b*x^2]) - (7*d*f^2*(3*d*e + 2*c* f)*x^5)/(3*b^2*Sqrt[a + b*x^2]) - (3*d^2*f^3*x^7)/(b^2*Sqrt[a + b*x^2]) + (105*a^2*d^2*f^3*x*Sqrt[a + b*x^2])/(16*b^5) - (35*a*d*f^2*(3*d*e + 2*c*f) *x*Sqrt[a + b*x^2])/(8*b^4) + (5*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x*Sqr t[a + b*x^2])/(2*b^3) - (35*a*d^2*f^3*x^3*Sqrt[a + b*x^2])/(8*b^4) + (35*d *f^2*(3*d*e + 2*c*f)*x^3*Sqrt[a + b*x^2])/(12*b^3) + (7*d^2*f^3*x^5*Sqrt[a + b*x^2])/(2*b^3) - (105*a^3*d^2*f^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] )/(16*b^(11/2)) + (35*a^2*d*f^2*(3*d*e + 2*c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2)) - (5*a*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*ArcTanh [(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(7/2)) + (e*(d^2*e^2 + 6*c*d*e*f + 3*c ^2*f^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(5/2)
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) ^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
Time = 1.40 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.20
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {105 \left (a^{3} d^{2} f^{3}-\frac {4 \left (c f +\frac {3 d e}{2}\right ) d b \,f^{2} a^{2}}{3}+\frac {8 a \,b^{2} f \left (c^{2} f^{2}+6 c d e f +3 d^{2} e^{2}\right )}{21}-\frac {16 \left (c^{2} f^{2}+2 c d e f +\frac {1}{3} d^{2} e^{2}\right ) b^{3} e}{35}\right ) a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{16}+\frac {2 \left (-\frac {9 a^{3} \left (\left (\frac {1}{8} d^{2} x^{6}+\frac {7}{12} c d \,x^{4}-\frac {10}{9} c^{2} x^{2}\right ) f^{3}+e \left (\frac {7}{8} d^{2} x^{4}-\frac {20}{3} c d \,x^{2}+c^{2}\right ) f^{2}+2 d \left (-\frac {5 x^{2} d}{3}+c \right ) e^{2} f +\frac {d^{2} e^{3}}{3}\right ) b^{\frac {7}{2}}}{2}+\frac {15 \left (\left (\frac {21}{40} d^{2} x^{4}-\frac {14}{3} c d \,x^{2}+c^{2}\right ) f^{2}+6 \left (-\frac {7 x^{2} d}{6}+c \right ) d e f +3 d^{2} e^{2}\right ) f \,a^{4} b^{\frac {5}{2}}}{4}-\frac {105 d \left (\left (-x^{2} d +c \right ) f +\frac {3 d e}{2}\right ) f^{2} a^{5} b^{\frac {3}{2}}}{8}+\frac {315 a^{6} d^{2} f^{3} \sqrt {b}}{32}+\left (-6 \left (-\frac {\left (\frac {1}{3} d^{2} x^{4}+c d \,x^{2}+c^{2}\right ) x^{2} f^{3}}{8}+e \left (-\frac {3}{16} d^{2} x^{4}-\frac {3}{4} c d \,x^{2}+c^{2}\right ) f^{2}+2 \left (-\frac {3 x^{2} d}{16}+c \right ) d \,e^{2} f +\frac {d^{2} e^{3}}{3}\right ) x^{2} a^{2}+\frac {3 c b \left (c f \,x^{2}+e \left (\frac {2 x^{2} d}{3}+c \right )\right ) e^{2} a}{2}+b^{2} c^{2} e^{3} x^{2}\right ) b^{\frac {9}{2}}\right ) x}{3}}{b^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(443\) |
| default | \(c^{2} e^{3} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+f^{2} d \left (2 c f +3 d e \right ) \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )+c \,e^{2} \left (3 c f +2 d e \right ) \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )+f \left (c^{2} f^{2}+6 c d e f +3 d^{2} e^{2}\right ) \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+e \left (3 c^{2} f^{2}+6 c d e f +d^{2} e^{2}\right ) \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+d^{2} f^{3} \left (\frac {x^{9}}{6 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {3 a \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )}{2 b}\right )\) | \(552\) |
| risch | \(\text {Expression too large to display}\) | \(1404\) |
Input:
int((d*x^2+c)^2*(f*x^2+e)^3/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3/(b*x^2+a)^(3/2)/b^(11/2)*(-315/32*(a^3*d^2*f^3-4/3*(c*f+3/2*d*e)*d*b*f ^2*a^2+8/21*a*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)-16/35*(c^2*f^2+2*c*d*e*f +1/3*d^2*e^2)*b^3*e)*a^2*(b*x^2+a)^(3/2)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2) )+(-9/2*a^3*((1/8*d^2*x^6+7/12*c*d*x^4-10/9*c^2*x^2)*f^3+e*(7/8*d^2*x^4-20 /3*c*d*x^2+c^2)*f^2+2*d*(-5/3*x^2*d+c)*e^2*f+1/3*d^2*e^3)*b^(7/2)+15/4*((2 1/40*d^2*x^4-14/3*c*d*x^2+c^2)*f^2+6*(-7/6*x^2*d+c)*d*e*f+3*d^2*e^2)*f*a^4 *b^(5/2)-105/8*d*((-d*x^2+c)*f+3/2*d*e)*f^2*a^5*b^(3/2)+315/32*a^6*d^2*f^3 *b^(1/2)+(-6*(-1/8*(1/3*d^2*x^4+c*d*x^2+c^2)*x^2*f^3+e*(-3/16*d^2*x^4-3/4* c*d*x^2+c^2)*f^2+2*(-3/16*x^2*d+c)*d*e^2*f+1/3*d^2*e^3)*x^2*a^2+3/2*c*b*(c *f*x^2+e*(2/3*x^2*d+c))*e^2*a+b^2*c^2*e^3*x^2)*b^(9/2))*x)/a^2
Leaf count of result is larger than twice the leaf count of optimal. 862 vs. \(2 (340) = 680\).
Time = 0.94 (sec) , antiderivative size = 1732, normalized size of antiderivative = 4.71 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^2*(f*x^2+e)^3/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
[-1/96*(3*(16*a^4*b^3*d^2*e^3 + (16*a^2*b^5*d^2*e^3 + 24*(4*a^2*b^5*c*d - 5*a^3*b^4*d^2)*e^2*f + 6*(8*a^2*b^5*c^2 - 40*a^3*b^4*c*d + 35*a^4*b^3*d^2) *e*f^2 - 5*(8*a^3*b^4*c^2 - 28*a^4*b^3*c*d + 21*a^5*b^2*d^2)*f^3)*x^4 + 24 *(4*a^4*b^3*c*d - 5*a^5*b^2*d^2)*e^2*f + 6*(8*a^4*b^3*c^2 - 40*a^5*b^2*c*d + 35*a^6*b*d^2)*e*f^2 - 5*(8*a^5*b^2*c^2 - 28*a^6*b*c*d + 21*a^7*d^2)*f^3 + 2*(16*a^3*b^4*d^2*e^3 + 24*(4*a^3*b^4*c*d - 5*a^4*b^3*d^2)*e^2*f + 6*(8 *a^3*b^4*c^2 - 40*a^4*b^3*c*d + 35*a^5*b^2*d^2)*e*f^2 - 5*(8*a^4*b^3*c^2 - 28*a^5*b^2*c*d + 21*a^6*b*d^2)*f^3)*x^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b* x^2 + a)*sqrt(b)*x - a) - 2*(8*a^2*b^5*d^2*f^3*x^9 + 6*(6*a^2*b^5*d^2*e*f^ 2 + (4*a^2*b^5*c*d - 3*a^3*b^4*d^2)*f^3)*x^7 + 3*(24*a^2*b^5*d^2*e^2*f + 6 *(8*a^2*b^5*c*d - 7*a^3*b^4*d^2)*e*f^2 + (8*a^2*b^5*c^2 - 28*a^3*b^4*c*d + 21*a^4*b^3*d^2)*f^3)*x^5 + 4*(8*(b^7*c^2 + a*b^6*c*d - 2*a^2*b^5*d^2)*e^3 + 12*(a*b^6*c^2 - 8*a^2*b^5*c*d + 10*a^3*b^4*d^2)*e^2*f - 6*(8*a^2*b^5*c^ 2 - 40*a^3*b^4*c*d + 35*a^4*b^3*d^2)*e*f^2 + 5*(8*a^3*b^4*c^2 - 28*a^4*b^3 *c*d + 21*a^5*b^2*d^2)*f^3)*x^3 + 3*(16*(a*b^6*c^2 - a^3*b^4*d^2)*e^3 - 24 *(4*a^3*b^4*c*d - 5*a^4*b^3*d^2)*e^2*f - 6*(8*a^3*b^4*c^2 - 40*a^4*b^3*c*d + 35*a^5*b^2*d^2)*e*f^2 + 5*(8*a^4*b^3*c^2 - 28*a^5*b^2*c*d + 21*a^6*b*d^ 2)*f^3)*x)*sqrt(b*x^2 + a))/(a^2*b^8*x^4 + 2*a^3*b^7*x^2 + a^4*b^6), -1/48 *(3*(16*a^4*b^3*d^2*e^3 + (16*a^2*b^5*d^2*e^3 + 24*(4*a^2*b^5*c*d - 5*a^3* b^4*d^2)*e^2*f + 6*(8*a^2*b^5*c^2 - 40*a^3*b^4*c*d + 35*a^4*b^3*d^2)*e*...
\[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{2} \left (e + f x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x**2+c)**2*(f*x**2+e)**3/(b*x**2+a)**(5/2),x)
Output:
Integral((c + d*x**2)**2*(e + f*x**2)**3/(a + b*x**2)**(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (340) = 680\).
Time = 0.05 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.18 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x^2+c)^2*(f*x^2+e)^3/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
1/6*d^2*f^3*x^9/((b*x^2 + a)^(3/2)*b) - 3/8*a*d^2*f^3*x^7/((b*x^2 + a)^(3/ 2)*b^2) + 21/16*a^2*d^2*f^3*x^5/((b*x^2 + a)^(3/2)*b^3) + 35/16*a^3*d^2*f^ 3*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^3 + 1/4* (3*d^2*e*f^2 + 2*c*d*f^3)*x^7/((b*x^2 + a)^(3/2)*b) + 2/3*c^2*e^3*x/(sqrt( b*x^2 + a)*a^2) + 1/3*c^2*e^3*x/((b*x^2 + a)^(3/2)*a) + 35/16*a^3*d^2*f^3* x/(sqrt(b*x^2 + a)*b^5) - 7/8*(3*d^2*e*f^2 + 2*c*d*f^3)*a*x^5/((b*x^2 + a) ^(3/2)*b^2) + 1/2*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*x^5/((b*x^2 + a)^( 3/2)*b) - 105/16*a^3*d^2*f^3*arcsinh(b*x/sqrt(a*b))/b^(11/2) - 1/3*(d^2*e^ 3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^ 2 + a)^(3/2)*b^2)) - 35/24*(3*d^2*e*f^2 + 2*c*d*f^3)*a^2*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 + 5/6*(3*d^2*e^2*f + 6*c* d*e*f^2 + c^2*f^3)*a*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/ 2)*b^2))/b - 35/24*(3*d^2*e*f^2 + 2*c*d*f^3)*a^2*x/(sqrt(b*x^2 + a)*b^4) + 5/6*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*a*x/(sqrt(b*x^2 + a)*b^3) - 1/3 *(d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*x/(sqrt(b*x^2 + a)*b^2) - 1/3*(2*c* d*e^3 + 3*c^2*e^2*f)*x/((b*x^2 + a)^(3/2)*b) + 1/3*(2*c*d*e^3 + 3*c^2*e^2* f)*x/(sqrt(b*x^2 + a)*a*b) + 35/8*(3*d^2*e*f^2 + 2*c*d*f^3)*a^2*arcsinh(b* x/sqrt(a*b))/b^(9/2) - 5/2*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*a*arcsinh (b*x/sqrt(a*b))/b^(7/2) + (d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*arcsinh(b* x/sqrt(a*b))/b^(5/2)
Time = 0.15 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.74 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left ({\left (2 \, {\left (\frac {4 \, d^{2} f^{3} x^{2}}{b} + \frac {3 \, {\left (6 \, a^{2} b^{8} d^{2} e f^{2} + 4 \, a^{2} b^{8} c d f^{3} - 3 \, a^{3} b^{7} d^{2} f^{3}\right )}}{a^{2} b^{9}}\right )} x^{2} + \frac {3 \, {\left (24 \, a^{2} b^{8} d^{2} e^{2} f + 48 \, a^{2} b^{8} c d e f^{2} - 42 \, a^{3} b^{7} d^{2} e f^{2} + 8 \, a^{2} b^{8} c^{2} f^{3} - 28 \, a^{3} b^{7} c d f^{3} + 21 \, a^{4} b^{6} d^{2} f^{3}\right )}}{a^{2} b^{9}}\right )} x^{2} + \frac {4 \, {\left (8 \, b^{10} c^{2} e^{3} + 8 \, a b^{9} c d e^{3} - 16 \, a^{2} b^{8} d^{2} e^{3} + 12 \, a b^{9} c^{2} e^{2} f - 96 \, a^{2} b^{8} c d e^{2} f + 120 \, a^{3} b^{7} d^{2} e^{2} f - 48 \, a^{2} b^{8} c^{2} e f^{2} + 240 \, a^{3} b^{7} c d e f^{2} - 210 \, a^{4} b^{6} d^{2} e f^{2} + 40 \, a^{3} b^{7} c^{2} f^{3} - 140 \, a^{4} b^{6} c d f^{3} + 105 \, a^{5} b^{5} d^{2} f^{3}\right )}}{a^{2} b^{9}}\right )} x^{2} + \frac {3 \, {\left (16 \, a b^{9} c^{2} e^{3} - 16 \, a^{3} b^{7} d^{2} e^{3} - 96 \, a^{3} b^{7} c d e^{2} f + 120 \, a^{4} b^{6} d^{2} e^{2} f - 48 \, a^{3} b^{7} c^{2} e f^{2} + 240 \, a^{4} b^{6} c d e f^{2} - 210 \, a^{5} b^{5} d^{2} e f^{2} + 40 \, a^{4} b^{6} c^{2} f^{3} - 140 \, a^{5} b^{5} c d f^{3} + 105 \, a^{6} b^{4} d^{2} f^{3}\right )}}{a^{2} b^{9}}\right )} x}{48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (16 \, b^{3} d^{2} e^{3} + 96 \, b^{3} c d e^{2} f - 120 \, a b^{2} d^{2} e^{2} f + 48 \, b^{3} c^{2} e f^{2} - 240 \, a b^{2} c d e f^{2} + 210 \, a^{2} b d^{2} e f^{2} - 40 \, a b^{2} c^{2} f^{3} + 140 \, a^{2} b c d f^{3} - 105 \, a^{3} d^{2} f^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {11}{2}}} \] Input:
integrate((d*x^2+c)^2*(f*x^2+e)^3/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
1/48*(((2*(4*d^2*f^3*x^2/b + 3*(6*a^2*b^8*d^2*e*f^2 + 4*a^2*b^8*c*d*f^3 - 3*a^3*b^7*d^2*f^3)/(a^2*b^9))*x^2 + 3*(24*a^2*b^8*d^2*e^2*f + 48*a^2*b^8*c *d*e*f^2 - 42*a^3*b^7*d^2*e*f^2 + 8*a^2*b^8*c^2*f^3 - 28*a^3*b^7*c*d*f^3 + 21*a^4*b^6*d^2*f^3)/(a^2*b^9))*x^2 + 4*(8*b^10*c^2*e^3 + 8*a*b^9*c*d*e^3 - 16*a^2*b^8*d^2*e^3 + 12*a*b^9*c^2*e^2*f - 96*a^2*b^8*c*d*e^2*f + 120*a^3 *b^7*d^2*e^2*f - 48*a^2*b^8*c^2*e*f^2 + 240*a^3*b^7*c*d*e*f^2 - 210*a^4*b^ 6*d^2*e*f^2 + 40*a^3*b^7*c^2*f^3 - 140*a^4*b^6*c*d*f^3 + 105*a^5*b^5*d^2*f ^3)/(a^2*b^9))*x^2 + 3*(16*a*b^9*c^2*e^3 - 16*a^3*b^7*d^2*e^3 - 96*a^3*b^7 *c*d*e^2*f + 120*a^4*b^6*d^2*e^2*f - 48*a^3*b^7*c^2*e*f^2 + 240*a^4*b^6*c* d*e*f^2 - 210*a^5*b^5*d^2*e*f^2 + 40*a^4*b^6*c^2*f^3 - 140*a^5*b^5*c*d*f^3 + 105*a^6*b^4*d^2*f^3)/(a^2*b^9))*x/(b*x^2 + a)^(3/2) - 1/16*(16*b^3*d^2* e^3 + 96*b^3*c*d*e^2*f - 120*a*b^2*d^2*e^2*f + 48*b^3*c^2*e*f^2 - 240*a*b^ 2*c*d*e*f^2 + 210*a^2*b*d^2*e*f^2 - 40*a*b^2*c^2*f^3 + 140*a^2*b*c*d*f^3 - 105*a^3*d^2*f^3)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)
Timed out. \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^2\,{\left (f\,x^2+e\right )}^3}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((c + d*x^2)^2*(e + f*x^2)^3)/(a + b*x^2)^(5/2),x)
Output:
int(((c + d*x^2)^2*(e + f*x^2)^3)/(a + b*x^2)^(5/2), x)
\[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (d \,x^{2}+c \right )^{2} \left (f \,x^{2}+e \right )^{3}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:
int((d*x^2+c)^2*(f*x^2+e)^3/(b*x^2+a)^(5/2),x)
Output:
int((d*x^2+c)^2*(f*x^2+e)^3/(b*x^2+a)^(5/2),x)