Integrand size = 32, antiderivative size = 649 \[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {\left (192 a^3 d^3 f^2-12 a^2 b d^2 f (28 d e+33 c f)-b^3 c \left (245 d^2 e^2+322 c d e f+15 c^2 f^2\right )+a b^2 d \left (140 d^2 e^2+658 c d e f+219 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{105 b^4 d \sqrt {a+b x^2}}+\frac {\left (48 a^2 d^2 f^2-3 a b d f (28 d e+31 c f)+b^2 \left (35 d^2 e^2+154 c d e f+45 c^2 f^2\right )\right ) x^3 \sqrt {c+d x^2}}{105 b^3 \sqrt {a+b x^2}}+\frac {d f (14 b d e+15 b c f-8 a d f) x^5 \sqrt {c+d x^2}}{35 b^2 \sqrt {a+b x^2}}+\frac {d^2 f^2 x^7 \sqrt {c+d x^2}}{7 b \sqrt {a+b x^2}}+\frac {\left (105 b^4 c^2 d e^2+384 a^4 d^3 f^2-24 a^3 b d^2 f (28 d e+31 c f)-a b^3 c \left (455 d^2 e^2+532 c d e f+15 c^2 f^2\right )+a^2 b^2 d \left (280 d^2 e^2+1232 c d e f+369 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 \sqrt {a} b^{9/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {2 \sqrt {a} \left (96 a^3 d^2 f^2-105 b^3 c e (d e+c f)-6 a^2 b d f (28 d e+29 c f)+a b^2 \left (70 d^2 e^2+287 c d e f+75 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{9/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/105*(192*a^3*d^3*f^2-12*a^2*b*d^2*f*(33*c*f+28*d*e)-b^3*c*(15*c^2*f^2+3 22*c*d*e*f+245*d^2*e^2)+a*b^2*d*(219*c^2*f^2+658*c*d*e*f+140*d^2*e^2))*x*( d*x^2+c)^(1/2)/b^4/d/(b*x^2+a)^(1/2)+1/105*(48*a^2*d^2*f^2-3*a*b*d*f*(31*c *f+28*d*e)+b^2*(45*c^2*f^2+154*c*d*e*f+35*d^2*e^2))*x^3*(d*x^2+c)^(1/2)/b^ 3/(b*x^2+a)^(1/2)+1/35*d*f*(-8*a*d*f+15*b*c*f+14*b*d*e)*x^5*(d*x^2+c)^(1/2 )/b^2/(b*x^2+a)^(1/2)+1/7*d^2*f^2*x^7*(d*x^2+c)^(1/2)/b/(b*x^2+a)^(1/2)+1/ 105*(105*b^4*c^2*d*e^2+384*a^4*d^3*f^2-24*a^3*b*d^2*f*(31*c*f+28*d*e)-a*b^ 3*c*(15*c^2*f^2+532*c*d*e*f+455*d^2*e^2)+a^2*b^2*d*(369*c^2*f^2+1232*c*d*e *f+280*d^2*e^2))*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^( 1/2),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(9/2)/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/( b*x^2+a))^(1/2)-2/105*a^(1/2)*(96*a^3*d^2*f^2-105*b^3*c*e*(c*f+d*e)-6*a^2* b*d*f*(29*c*f+28*d*e)+a*b^2*(75*c^2*f^2+287*c*d*e*f+70*d^2*e^2))*(d*x^2+c) ^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(9/2 )/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 9.93 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.82 \[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (c+d x^2\right ) \left (105 b^4 c^2 e^2+192 a^4 d^2 f^2+12 a^3 b d f \left (-28 d e-29 c f+4 d f x^2\right )+a^2 b^2 \left (150 c^2 f^2+c d f \left (574 e-93 f x^2\right )+4 d^2 \left (35 e^2-21 e f x^2-6 f^2 x^4\right )\right )+a b^3 \left (15 c^2 f \left (-14 e+3 f x^2\right )+d^2 x^2 \left (35 e^2+42 e f x^2+15 f^2 x^4\right )+c d \left (-210 e^2+154 e f x^2+45 f^2 x^4\right )\right )\right )+i c \left (105 b^4 c^2 d e^2+384 a^4 d^3 f^2-24 a^3 b d^2 f (28 d e+31 c f)-a b^3 c \left (455 d^2 e^2+532 c d e f+15 c^2 f^2\right )+a^2 b^2 d \left (280 d^2 e^2+1232 c d e f+369 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (-105 b^3 c d e^2+192 a^3 d^2 f^2-12 a^2 b d f (28 d e+17 c f)+a b^2 \left (140 d^2 e^2+322 c d e f+15 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{105 b^5 d \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((c + d*x^2)^(5/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*(Sqrt[b/a]*d*x*(c + d*x^2)*(105*b^4*c^2*e^2 + 192*a^4*d^2*f^2 + 12*a^3*b*d*f*(-28*d*e - 29*c*f + 4*d*f*x^2) + a^2*b^2*(150*c^2*f^2 + c*d* f*(574*e - 93*f*x^2) + 4*d^2*(35*e^2 - 21*e*f*x^2 - 6*f^2*x^4)) + a*b^3*(1 5*c^2*f*(-14*e + 3*f*x^2) + d^2*x^2*(35*e^2 + 42*e*f*x^2 + 15*f^2*x^4) + c *d*(-210*e^2 + 154*e*f*x^2 + 45*f^2*x^4))) + I*c*(105*b^4*c^2*d*e^2 + 384* a^4*d^3*f^2 - 24*a^3*b*d^2*f*(28*d*e + 31*c*f) - a*b^3*c*(455*d^2*e^2 + 53 2*c*d*e*f + 15*c^2*f^2) + a^2*b^2*d*(280*d^2*e^2 + 1232*c*d*e*f + 369*c^2* f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a ]*x], (a*d)/(b*c)] - I*c*(-(b*c) + a*d)*(-105*b^3*c*d*e^2 + 192*a^3*d^2*f^ 2 - 12*a^2*b*d*f*(28*d*e + 17*c*f) + a*b^2*(140*d^2*e^2 + 322*c*d*e*f + 15 *c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqr t[b/a]*x], (a*d)/(b*c)]))/(105*b^5*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 1.77 (sec) , antiderivative size = 1252, normalized size of antiderivative = 1.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{3/2}}+\frac {2 e f x^2 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{3/2}}+\frac {f^2 x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {f^2 \left (d x^2+c\right )^{5/2} x^3}{b \sqrt {b x^2+a}}+\frac {8 d f^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} x^3}{7 b^2}+\frac {3 d (15 b c-16 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 b^3}-\frac {2 e f \left (d x^2+c\right )^{5/2} x}{b \sqrt {b x^2+a}}+\frac {12 d e f \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} x}{5 b^2}+\frac {(b c-a d) e^2 \left (d x^2+c\right )^{3/2} x}{a b \sqrt {b x^2+a}}-\frac {d (3 b c-4 a d) e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 a b^2}+\frac {2 \left (25 b^2 c^2-58 a b d c+32 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{35 b^4}+\frac {2 d (23 b c-24 a d) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b^3}-\frac {d \left (3 b^2 c^2-13 a b d c+8 a^2 d^2\right ) e^2 \sqrt {b x^2+a} x}{3 a b^3 \sqrt {d x^2+c}}+\frac {\left (5 b^3 c^3-123 a b^2 d c^2+248 a^2 b d^2 c-128 a^3 d^3\right ) f^2 \sqrt {b x^2+a} x}{35 b^5 \sqrt {d x^2+c}}+\frac {4 d \left (19 b^2 c^2-44 a b d c+24 a^2 d^2\right ) e f \sqrt {b x^2+a} x}{15 b^4 \sqrt {d x^2+c}}+\frac {\sqrt {c} \sqrt {d} \left (3 b^2 c^2-13 a b d c+8 a^2 d^2\right ) e^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a b^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} \left (5 b^3 c^3-123 a b^2 d c^2+248 a^2 b d^2 c-128 a^3 d^3\right ) f^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{35 b^5 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {4 \sqrt {c} \sqrt {d} \left (19 b^2 c^2-44 a b d c+24 a^2 d^2\right ) e f \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 c^{3/2} \sqrt {d} (3 b c-2 a d) e^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a b^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 c^{3/2} \left (25 b^2 c^2-58 a b d c+32 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{35 b^4 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 c^{3/2} \left (15 b^2 c^2-41 a b d c+24 a^2 d^2\right ) e f \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a b^3 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[((c + d*x^2)^(5/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2),x]
Output:
-1/3*(d*(3*b^2*c^2 - 13*a*b*c*d + 8*a^2*d^2)*e^2*x*Sqrt[a + b*x^2])/(a*b^3 *Sqrt[c + d*x^2]) + (4*d*(19*b^2*c^2 - 44*a*b*c*d + 24*a^2*d^2)*e*f*x*Sqrt [a + b*x^2])/(15*b^4*Sqrt[c + d*x^2]) + ((5*b^3*c^3 - 123*a*b^2*c^2*d + 24 8*a^2*b*c*d^2 - 128*a^3*d^3)*f^2*x*Sqrt[a + b*x^2])/(35*b^5*Sqrt[c + d*x^2 ]) - (d*(3*b*c - 4*a*d)*e^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*a*b^2) + (2*d*(23*b*c - 24*a*d)*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b^3) + (2*(25*b^2*c^2 - 58*a*b*c*d + 32*a^2*d^2)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d *x^2])/(35*b^4) + (3*d*(15*b*c - 16*a*d)*f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*b^3) + ((b*c - a*d)*e^2*x*(c + d*x^2)^(3/2))/(a*b*Sqrt[a + b*x ^2]) + (12*d*e*f*x*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(5*b^2) + (8*d*f^2*x ^3*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(7*b^2) - (2*e*f*x*(c + d*x^2)^(5/2) )/(b*Sqrt[a + b*x^2]) - (f^2*x^3*(c + d*x^2)^(5/2))/(b*Sqrt[a + b*x^2]) + (Sqrt[c]*Sqrt[d]*(3*b^2*c^2 - 13*a*b*c*d + 8*a^2*d^2)*e^2*Sqrt[a + b*x^2]* EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b^3*Sqrt[(c* (a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*Sqrt[c]*Sqrt[d]*(19*b^ 2*c^2 - 44*a*b*c*d + 24*a^2*d^2)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqr t[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^4*Sqrt[(c*(a + b*x^2))/(a*(c + d *x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(5*b^3*c^3 - 123*a*b^2*c^2*d + 248*a^2 *b*c*d^2 - 128*a^3*d^3)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/S qrt[c]], 1 - (b*c)/(a*d)])/(35*b^5*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c +...
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 = 21.45 (sec) , antiderivative size = 1088, normalized size of antiderivative = 1.68
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1088\) |
| elliptic | \(\text {Expression too large to display}\) | \(1638\) |
| default | \(\text {Expression too large to display}\) | \(1983\) |
Input:
int((d*x^2+c)^(5/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/105/b^4*x*(15*b^2*d^2*f^2*x^4-39*a*b*d^2*f^2*x^2+45*b^2*c*d*f^2*x^2+42*b ^2*d^2*e*f*x^2+87*a^2*d^2*f^2-138*a*b*c*d*f^2-126*a*b*d^2*e*f+45*b^2*c^2*f ^2+154*b^2*c*d*e*f+35*b^2*d^2*e^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)-1/105/b ^4*(-(279*a^3*d^3*f^2-534*a^2*b*c*d^2*f^2-462*a^2*b*d^3*e*f+264*a*b^2*c^2* d*f^2+812*a*b^2*c*d^2*e*f+175*a*b^2*d^3*e^2-15*b^3*c^3*f^2-322*b^3*c^2*d*e *f-245*b^3*c*d^2*e^2)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b* c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))+105*(a^ 5*d^3*f^2-3*a^4*b*c*d^2*f^2-2*a^4*b*d^3*e*f+3*a^3*b^2*c^2*d*f^2+6*a^3*b^2* c*d^2*e*f+a^3*b^2*d^3*e^2-a^2*b^3*c^3*f^2-6*a^2*b^3*c^2*d*e*f-3*a^2*b^3*c* d^2*e^2+2*a*b^4*c^3*e*f+3*a*b^4*c^2*d*e^2-b^5*c^3*e^2)/b*(-(b*d*x^2+b*c)/a /(a*d-b*c)*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(1/a+b*c/(a*d-b*c)/a)/(-b/a)^ (1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1 /2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-b/(a*d-b*c)/a*c/(-b /a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c )^(1/2)*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(- b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))-(105*a^4*d^3*f^2-402*a^3*b*c*d^2*f^ 2-210*a^3*b*d^3*e*f+453*a^2*b^2*c^2*d*f^2+756*a^2*b^2*c*d^2*e*f+105*a^2*b^ 2*d^3*e^2-150*a*b^3*c^3*f^2-784*a*b^3*c^2*d*e*f-350*a*b^3*c*d^2*e^2+210*b^ 4*c^3*e*f+315*b^4*c^2*d*e^2)/b/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/...
Leaf count of result is larger than twice the leaf count of optimal. 1226 vs. \(2 (608) = 1216\).
Time = 0.12 (sec) , antiderivative size = 1226, normalized size of antiderivative = 1.89 \[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((d*x^2+c)^(5/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="fricas ")
Output:
1/105*(((35*(3*b^5*c^3*d - 13*a*b^4*c^2*d^2 + 8*a^2*b^3*c*d^3)*e^2 - 28*(1 9*a*b^4*c^3*d - 44*a^2*b^3*c^2*d^2 + 24*a^3*b^2*c*d^3)*e*f - 3*(5*a*b^4*c^ 4 - 123*a^2*b^3*c^3*d + 248*a^3*b^2*c^2*d^2 - 128*a^4*b*c*d^3)*f^2)*x^3 + (35*(3*a*b^4*c^3*d - 13*a^2*b^3*c^2*d^2 + 8*a^3*b^2*c*d^3)*e^2 - 28*(19*a^ 2*b^3*c^3*d - 44*a^3*b^2*c^2*d^2 + 24*a^4*b*c*d^3)*e*f - 3*(5*a^2*b^3*c^4 - 123*a^3*b^2*c^3*d + 248*a^4*b*c^2*d^2 - 128*a^5*c*d^3)*f^2)*x)*sqrt(b*d) *sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((35*(3*b^5*c^3* d - 13*a*b^4*c^2*d^2 + 4*a^2*b^3*d^4 + 2*(4*a^2*b^3 - 3*a*b^4)*c*d^3)*e^2 - 14*(38*a*b^4*c^3*d + 24*a^3*b^2*d^4 - (88*a^2*b^3 - 15*a*b^4)*c^2*d^2 + (48*a^3*b^2 - 41*a^2*b^3)*c*d^3)*e*f - 3*(5*a*b^4*c^4 - 123*a^2*b^3*c^3*d - 64*a^4*b*d^4 + 2*(124*a^3*b^2 - 25*a^2*b^3)*c^2*d^2 - 4*(32*a^4*b - 29*a ^3*b^2)*c*d^3)*f^2)*x^3 + (35*(3*a*b^4*c^3*d - 13*a^2*b^3*c^2*d^2 + 4*a^3* b^2*d^4 + 2*(4*a^3*b^2 - 3*a^2*b^3)*c*d^3)*e^2 - 14*(38*a^2*b^3*c^3*d + 24 *a^4*b*d^4 - (88*a^3*b^2 - 15*a^2*b^3)*c^2*d^2 + (48*a^4*b - 41*a^3*b^2)*c *d^3)*e*f - 3*(5*a^2*b^3*c^4 - 123*a^3*b^2*c^3*d - 64*a^5*d^4 + 2*(124*a^4 *b - 25*a^3*b^2)*c^2*d^2 - 4*(32*a^5 - 29*a^4*b)*c*d^3)*f^2)*x)*sqrt(b*d)* sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (15*a*b^4*d^4*f^2 *x^8 + 3*(14*a*b^4*d^4*e*f + (15*a*b^4*c*d^3 - 8*a^2*b^3*d^4)*f^2)*x^6 + ( 35*a*b^4*d^4*e^2 + 14*(11*a*b^4*c*d^3 - 6*a^2*b^3*d^4)*e*f + 3*(15*a*b^4*c ^2*d^2 - 31*a^2*b^3*c*d^3 + 16*a^3*b^2*d^4)*f^2)*x^4 - 35*(3*a*b^4*c^2*...
\[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x**2+c)**(5/2)*(f*x**2+e)**2/(b*x**2+a)**(3/2),x)
Output:
Integral((c + d*x**2)**(5/2)*(e + f*x**2)**2/(a + b*x**2)**(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(5/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="maxima ")
Output:
integrate((d*x^2 + c)^(5/2)*(f*x^2 + e)^2/(b*x^2 + a)^(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(5/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(5/2)*(f*x^2 + e)^2/(b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(((c + d*x^2)^(5/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2),x)
Output:
int(((c + d*x^2)^(5/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((d*x^2+c)^(5/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x)
Output:
( - 144*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*c*d**2*f**2*x + 96*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*f**2*x**3 + 279*sqrt(c + d*x**2)*sqrt( a + b*x**2)*a**2*b*c**2*d*f**2*x + 252*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a **2*b*c*d**2*e*f*x - 186*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c*d**2*f **2*x**3 - 168*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3*e*f*x**3 - 48 *sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3*f**2*x**5 - 135*sqrt(c + d* x**2)*sqrt(a + b*x**2)*a*b**2*c**3*f**2*x - 462*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*e*f*x + 90*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2* c**2*d*f**2*x**3 - 105*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e** 2*x + 308*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e*f*x**3 + 90*sq rt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*f**2*x**5 + 70*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*a*b**2*d**3*e**2*x**3 + 84*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*e*f*x**5 + 30*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2 *d**3*f**2*x**7 + 210*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*e*f*x + 315*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**2*d*e**2*x - 384*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2 *a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**5*d**4*f**2 + 936*int((sqrt (c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**4*b*c*d**3*f**2 + 672*int ((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b...