Integrand size = 30, antiderivative size = 368 \[ \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^3 (b e-a f)^2 x}{3 a b^5 \left (a+b x^2\right )^{3/2}}+\frac {(b c-a d)^2 (b e-a f) \left (2 b^2 c e-13 a^2 d f+a b (7 d e+4 c f)\right ) x}{3 a^2 b^5 \sqrt {a+b x^2}}+\frac {d \left (41 a^2 d^2 f^2-22 a b d f (2 d e+3 c f)+8 b^2 \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{16 b^5}+\frac {d^2 f (12 b d e+18 b c f-17 a d f) x^3 \sqrt {a+b x^2}}{24 b^4}+\frac {d^3 f^2 x^5 \sqrt {a+b x^2}}{6 b^3}-\frac {\left (105 a^3 d^3 f^2-70 a^2 b d^2 f (2 d e+3 c f)-16 b^3 c \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+40 a b^2 d \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{11/2}} \] Output:
1/3*(-a*d+b*c)^3*(-a*f+b*e)^2*x/a/b^5/(b*x^2+a)^(3/2)+1/3*(-a*d+b*c)^2*(-a *f+b*e)*(2*b^2*c*e-13*a^2*d*f+a*b*(4*c*f+7*d*e))*x/a^2/b^5/(b*x^2+a)^(1/2) +1/16*d*(41*a^2*d^2*f^2-22*a*b*d*f*(3*c*f+2*d*e)+8*b^2*(3*c^2*f^2+6*c*d*e* f+d^2*e^2))*x*(b*x^2+a)^(1/2)/b^5+1/24*d^2*f*(-17*a*d*f+18*b*c*f+12*b*d*e) *x^3*(b*x^2+a)^(1/2)/b^4+1/6*d^3*f^2*x^5*(b*x^2+a)^(1/2)/b^3-1/16*(105*a^3 *d^3*f^2-70*a^2*b*d^2*f*(3*c*f+2*d*e)-16*b^3*c*(c^2*f^2+6*c*d*e*f+3*d^2*e^ 2)+40*a*b^2*d*(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 = 1.66 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.25 \[ \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {b} x \left (315 a^6 d^3 f^2+32 b^6 c^3 e^2 x^2+16 a b^5 c^2 e \left (3 c e+3 d e x^2+2 c f x^2\right )+210 a^5 b d^2 f \left (-2 d e-3 c f+2 d f x^2\right )+4 a^2 b^4 x^2 \left (-16 c^3 f^2+6 c^2 d f \left (-16 e+3 f x^2\right )+2 d^3 x^2 \left (3 e^2+3 e f x^2+f^2 x^4\right )+3 c d^2 \left (-16 e^2+12 e f x^2+3 f^2 x^4\right )\right )-2 a^3 b^3 \left (24 c^3 f^2+48 c^2 d f \left (3 e-5 f x^2\right )+d^3 x^2 \left (-80 e^2+42 e f x^2+9 f^2 x^4\right )+3 c d^2 \left (24 e^2-160 e f x^2+21 f^2 x^4\right )\right )+a^4 b^2 d \left (360 c^2 f^2+120 c d f \left (6 e-7 f x^2\right )+d^2 \left (120 e^2-560 e f x^2+63 f^2 x^4\right )\right )\right )}{a^2 \left (a+b x^2\right )^{3/2}}-3 \left (-105 a^3 d^3 f^2+70 a^2 b d^2 f (2 d e+3 c f)+16 b^3 c \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-40 a b^2 d \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{11/2}} \] Input:
Integrate[((c + d*x^2)^3*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x]
Output:
((Sqrt[b]*x*(315*a^6*d^3*f^2 + 32*b^6*c^3*e^2*x^2 + 16*a*b^5*c^2*e*(3*c*e + 3*d*e*x^2 + 2*c*f*x^2) + 210*a^5*b*d^2*f*(-2*d*e - 3*c*f + 2*d*f*x^2) + 4*a^2*b^4*x^2*(-16*c^3*f^2 + 6*c^2*d*f*(-16*e + 3*f*x^2) + 2*d^3*x^2*(3*e^ 2 + 3*e*f*x^2 + f^2*x^4) + 3*c*d^2*(-16*e^2 + 12*e*f*x^2 + 3*f^2*x^4)) - 2 *a^3*b^3*(24*c^3*f^2 + 48*c^2*d*f*(3*e - 5*f*x^2) + d^3*x^2*(-80*e^2 + 42* e*f*x^2 + 9*f^2*x^4) + 3*c*d^2*(24*e^2 - 160*e*f*x^2 + 21*f^2*x^4)) + a^4* b^2*d*(360*c^2*f^2 + 120*c*d*f*(6*e - 7*f*x^2) + d^2*(120*e^2 - 560*e*f*x^ 2 + 63*f^2*x^4))))/(a^2*(a + b*x^2)^(3/2)) - 3*(-105*a^3*d^3*f^2 + 70*a^2* b*d^2*f*(2*d*e + 3*c*f) + 16*b^3*c*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) - 40* a*b^2*d*(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.
\(\displaystyle \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 433 |
\(\displaystyle \int \left (\frac {c^3 e^2}{\left (a+b x^2\right )^{5/2}}+\frac {d x^6 \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{\left (a+b x^2\right )^{5/2}}+\frac {c x^4 \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{\left (a+b x^2\right )^{5/2}}+\frac {c^2 e x^2 (2 c f+3 d e)}{\left (a+b x^2\right )^{5/2}}+\frac {d^2 f x^8 (3 c f+2 d e)}{\left (a+b x^2\right )^{5/2}}+\frac {d^3 f^2 x^{10}}{\left (a+b x^2\right )^{5/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {105 a^3 d^3 f^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{11/2}}+\frac {35 a^2 d^2 f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (3 c f+2 d e)}{8 b^{9/2}}+\frac {105 a^2 d^3 f^2 x \sqrt {a+b x^2}}{16 b^5}+\frac {2 c^3 e^2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{b^{5/2}}-\frac {5 a d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{2 b^{7/2}}-\frac {35 a d^2 f x \sqrt {a+b x^2} (3 c f+2 d e)}{8 b^4}-\frac {35 a d^3 f^2 x^3 \sqrt {a+b x^2}}{8 b^4}+\frac {5 d x \sqrt {a+b x^2} \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{2 b^3}+\frac {35 d^2 f x^3 \sqrt {a+b x^2} (3 c f+2 d e)}{12 b^3}+\frac {7 d^3 f^2 x^5 \sqrt {a+b x^2}}{2 b^3}-\frac {c x \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{b^2 \sqrt {a+b x^2}}-\frac {5 d x^3 \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{3 b^2 \sqrt {a+b x^2}}-\frac {7 d^2 f x^5 (3 c f+2 d e)}{3 b^2 \sqrt {a+b x^2}}-\frac {3 d^3 f^2 x^7}{b^2 \sqrt {a+b x^2}}+\frac {c^3 e^2 x}{3 a \left (a+b x^2\right )^{3/2}}-\frac {d x^5 \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )}{3 b \left (a+b x^2\right )^{3/2}}-\frac {c x^3 \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )}{3 b \left (a+b x^2\right )^{3/2}}+\frac {c^2 e x^3 (2 c f+3 d e)}{3 a \left (a+b x^2\right )^{3/2}}-\frac {d^2 f x^7 (3 c f+2 d e)}{3 b \left (a+b x^2\right )^{3/2}}-\frac {d^3 f^2 x^9}{3 b \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[((c + d*x^2)^3*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x]
Output:
(c^3*e^2*x)/(3*a*(a + b*x^2)^(3/2)) + (c^2*e*(3*d*e + 2*c*f)*x^3)/(3*a*(a + b*x^2)^(3/2)) - (c*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^3)/(3*b*(a + b*x^ 2)^(3/2)) - (d*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)*x^5)/(3*b*(a + b*x^2)^(3/ 2)) - (d^2*f*(2*d*e + 3*c*f)*x^7)/(3*b*(a + b*x^2)^(3/2)) - (d^3*f^2*x^9)/ (3*b*(a + b*x^2)^(3/2)) + (2*c^3*e^2*x)/(3*a^2*Sqrt[a + b*x^2]) - (c*(3*d^ 2*e^2 + 6*c*d*e*f + c^2*f^2)*x)/(b^2*Sqrt[a + b*x^2]) - (5*d*(d^2*e^2 + 6* c*d*e*f + 3*c^2*f^2)*x^3)/(3*b^2*Sqrt[a + b*x^2]) - (7*d^2*f*(2*d*e + 3*c* f)*x^5)/(3*b^2*Sqrt[a + b*x^2]) - (3*d^3*f^2*x^7)/(b^2*Sqrt[a + b*x^2]) + (105*a^2*d^3*f^2*x*Sqrt[a + b*x^2])/(16*b^5) - (35*a*d^2*f*(2*d*e + 3*c*f) *x*Sqrt[a + b*x^2])/(8*b^4) + (5*d*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)*x*Sqr t[a + b*x^2])/(2*b^3) - (35*a*d^3*f^2*x^3*Sqrt[a + b*x^2])/(8*b^4) + (35*d ^2*f*(2*d*e + 3*c*f)*x^3*Sqrt[a + b*x^2])/(12*b^3) + (7*d^3*f^2*x^5*Sqrt[a + b*x^2])/(2*b^3) - (105*a^3*d^3*f^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] )/(16*b^(11/2)) + (35*a^2*d^2*f*(2*d*e + 3*c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2)) + (c*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*ArcTanh[(Sq rt[b]*x)/Sqrt[a + b*x^2]])/b^(5/2) - (5*a*d*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f ^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(7/2))
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 = 1.34 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.21
method | result | size |
pseudoelliptic | \(\frac {-\frac {105 \left (a^{3} d^{3} f^{2}-2 d^{2} \left (c f +\frac {2 d e}{3}\right ) b f \,a^{2}+\frac {8 d \left (c^{2} f^{2}+2 c d e f +\frac {1}{3} d^{2} e^{2}\right ) b^{2} a}{7}-\frac {16 b^{3} c \left (c^{2} f^{2}+6 c d e f +3 d^{2} e^{2}\right )}{105}\right ) a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{16}+\frac {2 \left (-\frac {3 a^{3} \left (\left (\frac {3}{8} f^{2} x^{6}+\frac {7}{4} e f \,x^{4}-\frac {10}{3} e^{2} x^{2}\right ) d^{3}+3 c \left (\frac {7}{8} f^{2} x^{4}-\frac {20}{3} e f \,x^{2}+e^{2}\right ) d^{2}+6 c^{2} \left (-\frac {5 f \,x^{2}}{3}+e \right ) f d +f^{2} c^{3}\right ) b^{\frac {7}{2}}}{2}+\frac {45 \left (\left (\frac {7}{40} f^{2} x^{4}-\frac {14}{9} e f \,x^{2}+\frac {1}{3} e^{2}\right ) d^{2}+2 c \left (-\frac {7 f \,x^{2}}{6}+e \right ) f d +c^{2} f^{2}\right ) d \,a^{4} b^{\frac {5}{2}}}{4}-\frac {315 d^{2} \left (\frac {2 \left (-f \,x^{2}+e \right ) d}{3}+c f \right ) f \,a^{5} b^{\frac {3}{2}}}{16}+\frac {315 a^{6} d^{3} f^{2} \sqrt {b}}{32}+\left (-2 \left (-\frac {3 \left (\frac {1}{3} f^{2} x^{4}+e f \,x^{2}+e^{2}\right ) x^{2} d^{3}}{8}+3 c \left (-\frac {3}{16} f^{2} x^{4}-\frac {3}{4} e f \,x^{2}+e^{2}\right ) d^{2}+6 c^{2} \left (-\frac {3 f \,x^{2}}{16}+e \right ) f d +f^{2} c^{3}\right ) x^{2} a^{2}+\frac {3 c^{2} b \left (d e \,x^{2}+c \left (\frac {2 f \,x^{2}}{3}+e \right )\right ) e a}{2}+b^{2} c^{3} e^{2} x^{2}\right ) b^{\frac {9}{2}}\right ) x}{3}}{b^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(444\) |
default | \(e^{2} c^{3} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d^{2} f \left (3 c f +2 d e \right ) \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )+c^{2} e \left (2 c f +3 d e \right ) \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )+d \left (3 c^{2} f^{2}+6 c d e f +d^{2} e^{2}\right ) \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+c \left (c^{2} f^{2}+6 c d e f +3 d^{2} e^{2}\right ) \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+f^{2} d^{3} \left (\frac {x^{9}}{6 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {3 a \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )}{2 b}\right )\) | \(552\) |
risch | \(\text {Expression too large to display}\) | \(1404\) |
Input:
int((d*x^2+c)^3*(f*x^2+e)^2/(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^3*f^2-2*d^2*(c*f+2/3*d*e)*b*f *a^2+8/7*d*(c^2*f^2+2*c*d*e*f+1/3*d^2*e^2)*b^2*a-16/105*b^3*c*(c^2*f^2+6*c *d*e*f+3*d^2*e^2))*a^2*(b*x^2+a)^(3/2)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+ (-3/2*a^3*((3/8*f^2*x^6+7/4*e*f*x^4-10/3*e^2*x^2)*d^3+3*c*(7/8*f^2*x^4-20/ 3*e*f*x^2+e^2)*d^2+6*c^2*(-5/3*f*x^2+e)*f*d+f^2*c^3)*b^(7/2)+45/4*((7/40*f ^2*x^4-14/9*e*f*x^2+1/3*e^2)*d^2+2*c*(-7/6*f*x^2+e)*f*d+c^2*f^2)*d*a^4*b^( 5/2)-315/16*d^2*(2/3*(-f*x^2+e)*d+c*f)*f*a^5*b^(3/2)+315/32*a^6*d^3*f^2*b^ (1/2)+(-2*(-3/8*(1/3*f^2*x^4+e*f*x^2+e^2)*x^2*d^3+3*c*(-3/16*f^2*x^4-3/4*e *f*x^2+e^2)*d^2+6*c^2*(-3/16*f*x^2+e)*f*d+f^2*c^3)*x^2*a^2+3/2*c^2*b*(d*e* x^2+c*(2/3*f*x^2+e))*e*a+b^2*c^3*e^2*x^2)*b^(9/2))*x)/a^2
Leaf count of result is larger than twice the leaf count of optimal. 873 vs. \(2 (340) = 680\).
Time = 1.19 (sec) , antiderivative size = 1754, normalized size of antiderivative = 4.77 \[ \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^3*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
[-1/96*(3*((8*(6*a^2*b^5*c*d^2 - 5*a^3*b^4*d^3)*e^2 + 4*(24*a^2*b^5*c^2*d - 60*a^3*b^4*c*d^2 + 35*a^4*b^3*d^3)*e*f + (16*a^2*b^5*c^3 - 120*a^3*b^4*c ^2*d + 210*a^4*b^3*c*d^2 - 105*a^5*b^2*d^3)*f^2)*x^4 + 8*(6*a^4*b^3*c*d^2 - 5*a^5*b^2*d^3)*e^2 + 4*(24*a^4*b^3*c^2*d - 60*a^5*b^2*c*d^2 + 35*a^6*b*d ^3)*e*f + (16*a^4*b^3*c^3 - 120*a^5*b^2*c^2*d + 210*a^6*b*c*d^2 - 105*a^7* d^3)*f^2 + 2*(8*(6*a^3*b^4*c*d^2 - 5*a^4*b^3*d^3)*e^2 + 4*(24*a^3*b^4*c^2* d - 60*a^4*b^3*c*d^2 + 35*a^5*b^2*d^3)*e*f + (16*a^3*b^4*c^3 - 120*a^4*b^3 *c^2*d + 210*a^5*b^2*c*d^2 - 105*a^6*b*d^3)*f^2)*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^3*f^2*x^9 + 6*(4*a^2* b^5*d^3*e*f + 3*(2*a^2*b^5*c*d^2 - a^3*b^4*d^3)*f^2)*x^7 + 3*(8*a^2*b^5*d^ 3*e^2 + 4*(12*a^2*b^5*c*d^2 - 7*a^3*b^4*d^3)*e*f + 3*(8*a^2*b^5*c^2*d - 14 *a^3*b^4*c*d^2 + 7*a^4*b^3*d^3)*f^2)*x^5 + 4*(4*(2*b^7*c^3 + 3*a*b^6*c^2*d - 12*a^2*b^5*c*d^2 + 10*a^3*b^4*d^3)*e^2 + 4*(2*a*b^6*c^3 - 24*a^2*b^5*c^ 2*d + 60*a^3*b^4*c*d^2 - 35*a^4*b^3*d^3)*e*f - (16*a^2*b^5*c^3 - 120*a^3*b ^4*c^2*d + 210*a^4*b^3*c*d^2 - 105*a^5*b^2*d^3)*f^2)*x^3 + 3*(8*(2*a*b^6*c ^3 - 6*a^3*b^4*c*d^2 + 5*a^4*b^3*d^3)*e^2 - 4*(24*a^3*b^4*c^2*d - 60*a^4*b ^3*c*d^2 + 35*a^5*b^2*d^3)*e*f - (16*a^3*b^4*c^3 - 120*a^4*b^3*c^2*d + 210 *a^5*b^2*c*d^2 - 105*a^6*b*d^3)*f^2)*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*((8*(6*a^2*b^5*c*d^2 - 5*a^3*b^4*d^3)*e^2 + 4*(24*a^2*b^5*c^2*d - 60*a^3*b^4*c*d^2 + 35*a^4*b^3*d^3)*e*f + (16*a...
\[ \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{3} \left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x**2+c)**3*(f*x**2+e)**2/(b*x**2+a)**(5/2),x)
Output:
Integral((c + d*x**2)**3*(e + f*x**2)**2/(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 = 0.07 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.18 \[ \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x^2+c)^3*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
1/6*d^3*f^2*x^9/((b*x^2 + a)^(3/2)*b) - 3/8*a*d^3*f^2*x^7/((b*x^2 + a)^(3/ 2)*b^2) + 21/16*a^2*d^3*f^2*x^5/((b*x^2 + a)^(3/2)*b^3) + 35/16*a^3*d^3*f^ 2*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* (2*d^3*e*f + 3*c*d^2*f^2)*x^7/((b*x^2 + a)^(3/2)*b) + 2/3*c^3*e^2*x/(sqrt( b*x^2 + a)*a^2) + 1/3*c^3*e^2*x/((b*x^2 + a)^(3/2)*a) + 35/16*a^3*d^3*f^2* x/(sqrt(b*x^2 + a)*b^5) - 7/8*(2*d^3*e*f + 3*c*d^2*f^2)*a*x^5/((b*x^2 + a) ^(3/2)*b^2) + 1/2*(d^3*e^2 + 6*c*d^2*e*f + 3*c^2*d*f^2)*x^5/((b*x^2 + a)^( 3/2)*b) - 105/16*a^3*d^3*f^2*arcsinh(b*x/sqrt(a*b))/b^(11/2) - 1/3*(3*c*d^ 2*e^2 + 6*c^2*d*e*f + c^3*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*(2*d^3*e*f + 3*c*d^2*f^2)*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*(d^3*e^2 + 6*c*d^2* e*f + 3*c^2*d*f^2)*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*(2*d^3*e*f + 3*c*d^2*f^2)*a^2*x/(sqrt(b*x^2 + a)*b^4) + 5/6*(d^3*e^2 + 6*c*d^2*e*f + 3*c^2*d*f^2)*a*x/(sqrt(b*x^2 + a)*b^3) - 1/3 *(3*c*d^2*e^2 + 6*c^2*d*e*f + c^3*f^2)*x/(sqrt(b*x^2 + a)*b^2) - 1/3*(3*c^ 2*d*e^2 + 2*c^3*e*f)*x/((b*x^2 + a)^(3/2)*b) + 1/3*(3*c^2*d*e^2 + 2*c^3*e* f)*x/(sqrt(b*x^2 + a)*a*b) + 35/8*(2*d^3*e*f + 3*c*d^2*f^2)*a^2*arcsinh(b* x/sqrt(a*b))/b^(9/2) - 5/2*(d^3*e^2 + 6*c*d^2*e*f + 3*c^2*d*f^2)*a*arcsinh (b*x/sqrt(a*b))/b^(7/2) + (3*c*d^2*e^2 + 6*c^2*d*e*f + c^3*f^2)*arcsinh(b* x/sqrt(a*b))/b^(5/2)
Time = 0.16 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.74 \[ \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left ({\left (2 \, {\left (\frac {4 \, d^{3} f^{2} x^{2}}{b} + \frac {3 \, {\left (4 \, a^{2} b^{8} d^{3} e f + 6 \, a^{2} b^{8} c d^{2} f^{2} - 3 \, a^{3} b^{7} d^{3} f^{2}\right )}}{a^{2} b^{9}}\right )} x^{2} + \frac {3 \, {\left (8 \, a^{2} b^{8} d^{3} e^{2} + 48 \, a^{2} b^{8} c d^{2} e f - 28 \, a^{3} b^{7} d^{3} e f + 24 \, a^{2} b^{8} c^{2} d f^{2} - 42 \, a^{3} b^{7} c d^{2} f^{2} + 21 \, a^{4} b^{6} d^{3} f^{2}\right )}}{a^{2} b^{9}}\right )} x^{2} + \frac {4 \, {\left (8 \, b^{10} c^{3} e^{2} + 12 \, a b^{9} c^{2} d e^{2} - 48 \, a^{2} b^{8} c d^{2} e^{2} + 40 \, a^{3} b^{7} d^{3} e^{2} + 8 \, a b^{9} c^{3} e f - 96 \, a^{2} b^{8} c^{2} d e f + 240 \, a^{3} b^{7} c d^{2} e f - 140 \, a^{4} b^{6} d^{3} e f - 16 \, a^{2} b^{8} c^{3} f^{2} + 120 \, a^{3} b^{7} c^{2} d f^{2} - 210 \, a^{4} b^{6} c d^{2} f^{2} + 105 \, a^{5} b^{5} d^{3} f^{2}\right )}}{a^{2} b^{9}}\right )} x^{2} + \frac {3 \, {\left (16 \, a b^{9} c^{3} e^{2} - 48 \, a^{3} b^{7} c d^{2} e^{2} + 40 \, a^{4} b^{6} d^{3} e^{2} - 96 \, a^{3} b^{7} c^{2} d e f + 240 \, a^{4} b^{6} c d^{2} e f - 140 \, a^{5} b^{5} d^{3} e f - 16 \, a^{3} b^{7} c^{3} f^{2} + 120 \, a^{4} b^{6} c^{2} d f^{2} - 210 \, a^{5} b^{5} c d^{2} f^{2} + 105 \, a^{6} b^{4} d^{3} f^{2}\right )}}{a^{2} b^{9}}\right )} x}{48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (48 \, b^{3} c d^{2} e^{2} - 40 \, a b^{2} d^{3} e^{2} + 96 \, b^{3} c^{2} d e f - 240 \, a b^{2} c d^{2} e f + 140 \, a^{2} b d^{3} e f + 16 \, b^{3} c^{3} f^{2} - 120 \, a b^{2} c^{2} d f^{2} + 210 \, a^{2} b c d^{2} f^{2} - 105 \, a^{3} d^{3} f^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {11}{2}}} \] Input:
integrate((d*x^2+c)^3*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
1/48*(((2*(4*d^3*f^2*x^2/b + 3*(4*a^2*b^8*d^3*e*f + 6*a^2*b^8*c*d^2*f^2 - 3*a^3*b^7*d^3*f^2)/(a^2*b^9))*x^2 + 3*(8*a^2*b^8*d^3*e^2 + 48*a^2*b^8*c*d^ 2*e*f - 28*a^3*b^7*d^3*e*f + 24*a^2*b^8*c^2*d*f^2 - 42*a^3*b^7*c*d^2*f^2 + 21*a^4*b^6*d^3*f^2)/(a^2*b^9))*x^2 + 4*(8*b^10*c^3*e^2 + 12*a*b^9*c^2*d*e ^2 - 48*a^2*b^8*c*d^2*e^2 + 40*a^3*b^7*d^3*e^2 + 8*a*b^9*c^3*e*f - 96*a^2* b^8*c^2*d*e*f + 240*a^3*b^7*c*d^2*e*f - 140*a^4*b^6*d^3*e*f - 16*a^2*b^8*c ^3*f^2 + 120*a^3*b^7*c^2*d*f^2 - 210*a^4*b^6*c*d^2*f^2 + 105*a^5*b^5*d^3*f ^2)/(a^2*b^9))*x^2 + 3*(16*a*b^9*c^3*e^2 - 48*a^3*b^7*c*d^2*e^2 + 40*a^4*b ^6*d^3*e^2 - 96*a^3*b^7*c^2*d*e*f + 240*a^4*b^6*c*d^2*e*f - 140*a^5*b^5*d^ 3*e*f - 16*a^3*b^7*c^3*f^2 + 120*a^4*b^6*c^2*d*f^2 - 210*a^5*b^5*c*d^2*f^2 + 105*a^6*b^4*d^3*f^2)/(a^2*b^9))*x/(b*x^2 + a)^(3/2) - 1/16*(48*b^3*c*d^ 2*e^2 - 40*a*b^2*d^3*e^2 + 96*b^3*c^2*d*e*f - 240*a*b^2*c*d^2*e*f + 140*a^ 2*b*d^3*e*f + 16*b^3*c^3*f^2 - 120*a*b^2*c^2*d*f^2 + 210*a^2*b*c*d^2*f^2 - 105*a^3*d^3*f^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)
Timed out. \[ \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3\,{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((c + d*x^2)^3*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x)
Output:
int(((c + d*x^2)^3*(e + f*x^2)^2)/(a + b*x^2)^(5/2), x)
\[ \int \frac {\left (c+d x^2\right )^3 \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (d \,x^{2}+c \right )^{3} \left (f \,x^{2}+e \right )^{2}}{\left (b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:
int((d*x^2+c)^3*(f*x^2+e)^2/(b*x^2+a)^(5/2),x)
Output:
int((d*x^2+c)^3*(f*x^2+e)^2/(b*x^2+a)^(5/2),x)