Integrand size = 32, antiderivative size = 1122 \[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Output:
1/3465*(40*a^5*d^5*f^2-5*a^4*b*d^4*f*(9*c*f+44*d*e)+5*a^3*b^2*d^3*(-14*c^2 *f^2+88*c*d*e*f+99*d^2*e^2)+8*b^5*c^3*(16*c^2*f^2-44*c*d*e*f+33*d^2*e^2)+a ^2*b^3*c*d^2*(403*c^2*f^2-1452*c*d*e*f+1914*d^2*e^2)-a*b^4*c^2*d*(408*c^2* f^2-1232*c*d*e*f+1089*d^2*e^2))*x*(d*x^2+c)^(1/2)/b^2/d^5/(b*x^2+a)^(1/2)- 1/3465*(20*a^4*d^4*f^2-10*a^3*b*d^3*f*(2*c*f+11*d*e)-15*a^2*b^2*d^2*(-12*c ^2*f^2+44*c*d*e*f+99*d^2*e^2)+4*b^4*c^2*(16*c^2*f^2-44*c*d*e*f+33*d^2*e^2) -2*a*b^3*c*d*(98*c^2*f^2-297*c*d*e*f+264*d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^ 2+c)^(1/2)/b^2/d^4+1/3465*(15*a^3*d^3*f^2+10*a^2*b*d^2*f*(13*c*f+165*d*e)+ 5*a*b^2*d*(-29*c^2*f^2+88*c*d*e*f+297*d^2*e^2)+3*b^3*c*(16*c^2*f^2-44*c*d* e*f+33*d^2*e^2))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d^3+1/693*(113*a^2* d^2*f^2+2*a*b*d*f*(12*c*f+209*d*e)+b^2*(-8*c^2*f^2+22*c*d*e*f+99*d^2*e^2)) *x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^2+1/99*b*f*(23*a*d*f+b*c*f+22*b*d*e )*x^7*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d+1/11*b^2*f^2*x^9*(b*x^2+a)^(1/2)*( d*x^2+c)^(1/2)-1/3465*a^(1/2)*(40*a^5*d^5*f^2-5*a^4*b*d^4*f*(9*c*f+44*d*e) +5*a^3*b^2*d^3*(-14*c^2*f^2+88*c*d*e*f+99*d^2*e^2)+8*b^5*c^3*(16*c^2*f^2-4 4*c*d*e*f+33*d^2*e^2)+a^2*b^3*c*d^2*(403*c^2*f^2-1452*c*d*e*f+1914*d^2*e^2 )-a*b^4*c^2*d*(408*c^2*f^2-1232*c*d*e*f+1089*d^2*e^2))*(d*x^2+c)^(1/2)*Ell ipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(5/2)/d^5/ (b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+2/3465*a^(3/2)*(10*a^4*d^4 *f^2-5*a^3*b*d^3*f*(2*c*f+11*d*e)+30*a^2*b^2*d^2*(3*c^2*f^2-11*c*d*e*f+...
Result contains complex when optimal does not.
Time = 9.68 (sec) , antiderivative size = 784, normalized size of antiderivative = 0.70 \[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-20 a^4 d^4 f^2+5 a^3 b d^3 f \left (22 d e+4 c f+3 d f x^2\right )+b^4 \left (-64 c^4 f^2+16 c^3 d f \left (11 e+3 f x^2\right )-4 c^2 d^2 \left (33 e^2+33 e f x^2+10 f^2 x^4\right )+c d^3 x^2 \left (99 e^2+110 e f x^2+35 f^2 x^4\right )+5 d^4 x^4 \left (99 e^2+154 e f x^2+63 f^2 x^4\right )\right )+5 a^2 b^2 d^2 \left (-36 c^2 f^2+2 c d f \left (66 e+13 f x^2\right )+d^2 \left (297 e^2+330 e f x^2+113 f^2 x^4\right )\right )+a b^3 d \left (196 c^3 f^2-c^2 d f \left (594 e+145 f x^2\right )+8 c d^2 \left (66 e^2+55 e f x^2+15 f^2 x^4\right )+5 d^3 x^2 \left (297 e^2+418 e f x^2+161 f^2 x^4\right )\right )\right )-i c \left (40 a^5 d^5 f^2-5 a^4 b d^4 f (44 d e+9 c f)+a b^4 c^2 d \left (-1089 d^2 e^2+1232 c d e f-408 c^2 f^2\right )+5 a^3 b^2 d^3 \left (99 d^2 e^2+88 c d e f-14 c^2 f^2\right )+8 b^5 c^3 \left (33 d^2 e^2-44 c d e f+16 c^2 f^2\right )+a^2 b^3 c d^2 \left (1914 d^2 e^2-1452 c d e f+403 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (b c-a d) \left (-20 a^4 d^4 f^2+5 a^3 b d^3 f (22 d e+c f)+a b^3 c d \left (-957 d^2 e^2+1056 c d e f-344 c^2 f^2\right )+8 b^4 c^2 \left (33 d^2 e^2-44 c d e f+16 c^2 f^2\right )+15 a^2 b^2 d^2 \left (99 d^2 e^2-66 c d e f+17 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3465 a^2 \left (\frac {b}{a}\right )^{5/2} d^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^(5/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2,x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(-20*a^4*d^4*f^2 + 5*a^3*b*d^3*f*(2 2*d*e + 4*c*f + 3*d*f*x^2) + b^4*(-64*c^4*f^2 + 16*c^3*d*f*(11*e + 3*f*x^2 ) - 4*c^2*d^2*(33*e^2 + 33*e*f*x^2 + 10*f^2*x^4) + c*d^3*x^2*(99*e^2 + 110 *e*f*x^2 + 35*f^2*x^4) + 5*d^4*x^4*(99*e^2 + 154*e*f*x^2 + 63*f^2*x^4)) + 5*a^2*b^2*d^2*(-36*c^2*f^2 + 2*c*d*f*(66*e + 13*f*x^2) + d^2*(297*e^2 + 33 0*e*f*x^2 + 113*f^2*x^4)) + a*b^3*d*(196*c^3*f^2 - c^2*d*f*(594*e + 145*f* x^2) + 8*c*d^2*(66*e^2 + 55*e*f*x^2 + 15*f^2*x^4) + 5*d^3*x^2*(297*e^2 + 4 18*e*f*x^2 + 161*f^2*x^4))) - I*c*(40*a^5*d^5*f^2 - 5*a^4*b*d^4*f*(44*d*e + 9*c*f) + a*b^4*c^2*d*(-1089*d^2*e^2 + 1232*c*d*e*f - 408*c^2*f^2) + 5*a^ 3*b^2*d^3*(99*d^2*e^2 + 88*c*d*e*f - 14*c^2*f^2) + 8*b^5*c^3*(33*d^2*e^2 - 44*c*d*e*f + 16*c^2*f^2) + a^2*b^3*c*d^2*(1914*d^2*e^2 - 1452*c*d*e*f + 4 03*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[S qrt[b/a]*x], (a*d)/(b*c)] + I*c*(b*c - a*d)*(-20*a^4*d^4*f^2 + 5*a^3*b*d^3 *f*(22*d*e + c*f) + a*b^3*c*d*(-957*d^2*e^2 + 1056*c*d*e*f - 344*c^2*f^2) + 8*b^4*c^2*(33*d^2*e^2 - 44*c*d*e*f + 16*c^2*f^2) + 15*a^2*b^2*d^2*(99*d^ 2*e^2 - 66*c*d*e*f + 17*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]* EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3465*a^2*(b/a)^(5/2)*d^5* Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 2.56 (sec) , antiderivative size = 1706, normalized size of antiderivative = 1.52, 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.
| \(\displaystyle \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (e^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}+2 e f x^2 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}+f^2 x^4 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{11} f^2 \left (b x^2+a\right )^{5/2} \sqrt {d x^2+c} x^5+\frac {(b c+5 a d) f^2 \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{99 d}-\frac {\left (8 b^2 c^2-17 a b d c-15 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{693 d^2}+\frac {2}{9} e f \left (b x^2+a\right )^{5/2} \sqrt {d x^2+c} x^3+\frac {2 (b c+5 a d) e f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3}{63 d}+\frac {\left (48 b^3 c^3-145 a b^2 d c^2+130 a^2 b d^2 c+15 a^3 d^3\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{3465 b d^3}-\frac {2 \left (2 b^2 c^2-5 a b d c-5 a^2 d^2\right ) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{105 d^2}+\frac {b e^2 \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2} x}{7 d}-\frac {2 (2 b c-5 a d) e^2 \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x}{35 d}-\frac {\left (4 b^2 c^2-13 a b d c-15 a^2 d^2\right ) e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{105 d^2}-\frac {4 \left (16 b^4 c^4-49 a b^3 d c^3+45 a^2 b^2 d^2 c^2-5 a^3 b d^3 c+5 a^4 d^4\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3465 b^2 d^4}+\frac {2 \left (8 b^3 c^3-27 a b^2 d c^2+30 a^2 b d^2 c+5 a^3 d^3\right ) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{315 b d^3}+\frac {\left (8 b^3 c^3-33 a b^2 d c^2+58 a^2 b d^2 c+15 a^3 d^3\right ) e^2 \sqrt {b x^2+a} x}{105 b d^2 \sqrt {d x^2+c}}+\frac {\left (128 b^5 c^5-408 a b^4 d c^4+403 a^2 b^3 d^2 c^3-70 a^3 b^2 d^3 c^2-45 a^4 b d^4 c+40 a^5 d^5\right ) f^2 \sqrt {b x^2+a} x}{3465 b^3 d^4 \sqrt {d x^2+c}}-\frac {4 \left (8 b^4 c^4-28 a b^3 d c^3+33 a^2 b^2 d^2 c^2-10 a^3 b d^3 c+5 a^4 d^4\right ) e f \sqrt {b x^2+a} x}{315 b^2 d^3 \sqrt {d x^2+c}}-\frac {\sqrt {c} \left (8 b^3 c^3-33 a b^2 d c^2+58 a^2 b d^2 c+15 a^3 d^3\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 )}{105 b d^{5/2} \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 (128 b^5 c^5-408 a b^4 d c^4+403 a^2 b^3 d^2 c^3-70 a^3 b^2 d^3 c^2-45 a^4 b d^4 c+40 a^5 d^5\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 )}{3465 b^3 d^{9/2} \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} \left (8 b^4 c^4-28 a b^3 d c^3+33 a^2 b^2 d^2 c^2-10 a^3 b d^3 c+5 a^4 d^4\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 )}{315 b^2 d^{7/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {4 c^{3/2} \left (b^2 c^2-4 a b d c+15 a^2 d^2\right ) 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 )}{105 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {4 c^{3/2} \left (16 b^4 c^4-49 a b^3 d c^3+45 a^2 b^2 d^2 c^2-5 a^3 b d^3 c+5 a^4 d^4\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 )}{3465 b^2 d^{9/2} \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 (8 b^3 c^3-27 a b^2 d c^2+30 a^2 b d^2 c+5 a^3 d^3\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 )}{315 b d^{7/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[(a + b*x^2)^(5/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2,x]
Output:
((8*b^3*c^3 - 33*a*b^2*c^2*d + 58*a^2*b*c*d^2 + 15*a^3*d^3)*e^2*x*Sqrt[a + b*x^2])/(105*b*d^2*Sqrt[c + d*x^2]) - (4*(8*b^4*c^4 - 28*a*b^3*c^3*d + 33 *a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 + 5*a^4*d^4)*e*f*x*Sqrt[a + b*x^2])/(315 *b^2*d^3*Sqrt[c + d*x^2]) + ((128*b^5*c^5 - 408*a*b^4*c^4*d + 403*a^2*b^3* c^3*d^2 - 70*a^3*b^2*c^2*d^3 - 45*a^4*b*c*d^4 + 40*a^5*d^5)*f^2*x*Sqrt[a + b*x^2])/(3465*b^3*d^4*Sqrt[c + d*x^2]) - ((4*b^2*c^2 - 13*a*b*c*d - 15*a^ 2*d^2)*e^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(105*d^2) + (2*(8*b^3*c^3 - 27*a*b^2*c^2*d + 30*a^2*b*c*d^2 + 5*a^3*d^3)*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(315*b*d^3) - (4*(16*b^4*c^4 - 49*a*b^3*c^3*d + 45*a^2*b^2*c^2*d ^2 - 5*a^3*b*c*d^3 + 5*a^4*d^4)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(34 65*b^2*d^4) - (2*(2*b^2*c^2 - 5*a*b*c*d - 5*a^2*d^2)*e*f*x^3*Sqrt[a + b*x^ 2]*Sqrt[c + d*x^2])/(105*d^2) + ((48*b^3*c^3 - 145*a*b^2*c^2*d + 130*a^2*b *c*d^2 + 15*a^3*d^3)*f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3465*b*d^3) - ((8*b^2*c^2 - 17*a*b*c*d - 15*a^2*d^2)*f^2*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(693*d^2) - (2*(2*b*c - 5*a*d)*e^2*x*(a + b*x^2)^(3/2)*Sqrt[c + d *x^2])/(35*d) + (2*(b*c + 5*a*d)*e*f*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2] )/(63*d) + ((b*c + 5*a*d)*f^2*x^5*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(99*d ) + (2*e*f*x^3*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/9 + (f^2*x^5*(a + b*x^2) ^(5/2)*Sqrt[c + d*x^2])/11 + (b*e^2*x*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) /(7*d) - (Sqrt[c]*(8*b^3*c^3 - 33*a*b^2*c^2*d + 58*a^2*b*c*d^2 + 15*a^3...
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 = 13.90 (sec) , antiderivative size = 2046, normalized size of antiderivative = 1.82
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2046\) |
| elliptic | \(\text {Expression too large to display}\) | \(2349\) |
| default | \(\text {Expression too large to display}\) | \(3341\) |
Input:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
-1/3465/b^2*x*(-315*b^4*d^4*f^2*x^8-805*a*b^3*d^4*f^2*x^6-35*b^4*c*d^3*f^2 *x^6-770*b^4*d^4*e*f*x^6-565*a^2*b^2*d^4*f^2*x^4-120*a*b^3*c*d^3*f^2*x^4-2 090*a*b^3*d^4*e*f*x^4+40*b^4*c^2*d^2*f^2*x^4-110*b^4*c*d^3*e*f*x^4-495*b^4 *d^4*e^2*x^4-15*a^3*b*d^4*f^2*x^2-130*a^2*b^2*c*d^3*f^2*x^2-1650*a^2*b^2*d ^4*e*f*x^2+145*a*b^3*c^2*d^2*f^2*x^2-440*a*b^3*c*d^3*e*f*x^2-1485*a*b^3*d^ 4*e^2*x^2-48*b^4*c^3*d*f^2*x^2+132*b^4*c^2*d^2*e*f*x^2-99*b^4*c*d^3*e^2*x^ 2+20*a^4*d^4*f^2-20*a^3*b*c*d^3*f^2-110*a^3*b*d^4*e*f+180*a^2*b^2*c^2*d^2* f^2-660*a^2*b^2*c*d^3*e*f-1485*a^2*b^2*d^4*e^2-196*a*b^3*c^3*d*f^2+594*a*b ^3*c^2*d^2*e*f-528*a*b^3*c*d^3*e^2+64*b^4*c^4*f^2-176*b^4*c^3*d*e*f+132*b^ 4*c^2*d^2*e^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^4+1/3465/b^2/d^4*(-(40*a^ 5*d^5*f^2-45*a^4*b*c*d^4*f^2-220*a^4*b*d^5*e*f-70*a^3*b^2*c^2*d^3*f^2+440* a^3*b^2*c*d^4*e*f+495*a^3*b^2*d^5*e^2+403*a^2*b^3*c^3*d^2*f^2-1452*a^2*b^3 *c^2*d^3*e*f+1914*a^2*b^3*c*d^4*e^2-408*a*b^4*c^4*d*f^2+1232*a*b^4*c^3*d^2 *e*f-1089*a*b^4*c^2*d^3*e^2+128*b^5*c^5*f^2-352*b^5*c^4*d*e*f+264*b^5*c^3* 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)))+64*a*b^4*c^5*f^2/(-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))+20*a^5*c*d^4*f^ 2/(-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...
Time = 0.11 (sec) , antiderivative size = 1200, normalized size of antiderivative = 1.07 \[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="fricas ")
Output:
-1/3465*((33*(8*b^5*c^4*d^2 - 33*a*b^4*c^3*d^3 + 58*a^2*b^3*c^2*d^4 + 15*a ^3*b^2*c*d^5)*e^2 - 44*(8*b^5*c^5*d - 28*a*b^4*c^4*d^2 + 33*a^2*b^3*c^3*d^ 3 - 10*a^3*b^2*c^2*d^4 + 5*a^4*b*c*d^5)*e*f + (128*b^5*c^6 - 408*a*b^4*c^5 *d + 403*a^2*b^3*c^4*d^2 - 70*a^3*b^2*c^3*d^3 - 45*a^4*b*c^2*d^4 + 40*a^5* c*d^5)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b *c)) - (33*(8*b^5*c^4*d^2 - 33*a*b^4*c^3*d^3 + 60*a^3*b^2*d^6 + 2*(29*a^2* b^3 + 2*a*b^4)*c^2*d^4 + (15*a^3*b^2 - 16*a^2*b^3)*c*d^5)*e^2 - 22*(16*b^5 *c^5*d - 56*a*b^4*c^4*d^2 + 5*a^4*b*d^6 + 2*(33*a^2*b^3 + 4*a*b^4)*c^3*d^3 - (20*a^3*b^2 + 27*a^2*b^3)*c^2*d^4 + 10*(a^4*b + 3*a^3*b^2)*c*d^5)*e*f + (128*b^5*c^6 - 408*a*b^4*c^5*d + 20*a^5*d^6 + (403*a^2*b^3 + 64*a*b^4)*c^ 4*d^2 - 14*(5*a^3*b^2 + 14*a^2*b^3)*c^3*d^3 - 45*(a^4*b - 4*a^3*b^2)*c^2*d ^4 + 20*(2*a^5 - a^4*b)*c*d^5)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcs in(sqrt(-c/d)/x), a*d/(b*c)) - (315*b^5*d^6*f^2*x^10 + 35*(22*b^5*d^6*e*f + (b^5*c*d^5 + 23*a*b^4*d^6)*f^2)*x^8 + 5*(99*b^5*d^6*e^2 + 22*(b^5*c*d^5 + 19*a*b^4*d^6)*e*f - (8*b^5*c^2*d^4 - 24*a*b^4*c*d^5 - 113*a^2*b^3*d^6)*f ^2)*x^6 + (99*(b^5*c*d^5 + 15*a*b^4*d^6)*e^2 - 22*(6*b^5*c^2*d^4 - 20*a*b^ 4*c*d^5 - 75*a^2*b^3*d^6)*e*f + (48*b^5*c^3*d^3 - 145*a*b^4*c^2*d^4 + 130* a^2*b^3*c*d^5 + 15*a^3*b^2*d^6)*f^2)*x^4 + 33*(8*b^5*c^3*d^3 - 33*a*b^4*c^ 2*d^4 + 58*a^2*b^3*c*d^5 + 15*a^3*b^2*d^6)*e^2 - 44*(8*b^5*c^4*d^2 - 28*a* b^4*c^3*d^3 + 33*a^2*b^3*c^2*d^4 - 10*a^3*b^2*c*d^5 + 5*a^4*b*d^6)*e*f ...
\[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int \left (a + b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}\, dx \] Input:
integrate((b*x**2+a)**(5/2)*(d*x**2+c)**(1/2)*(f*x**2+e)**2,x)
Output:
Integral((a + b*x**2)**(5/2)*sqrt(c + d*x**2)*(e + f*x**2)**2, x)
\[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2, x)
\[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2, x)
Timed out. \[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int {\left (b\,x^2+a\right )}^{5/2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2 \,d x \] Input:
int((a + b*x^2)^(5/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2,x)
Output:
int((a + b*x^2)^(5/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2, x)
\[ \int \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x)
Output:
( - 20*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**4*d**4*f**2*x + 20*sqrt(c + d* x**2)*sqrt(a + b*x**2)*a**3*b*c*d**3*f**2*x + 110*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b*d**4*e*f*x + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*b* d**4*f**2*x**3 - 180*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c**2*d**2 *f**2*x + 660*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c*d**3*e*f*x + 1 30*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*c*d**3*f**2*x**3 + 1485*sqr t(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*d**4*e**2*x + 1650*sqrt(c + d*x** 2)*sqrt(a + b*x**2)*a**2*b**2*d**4*e*f*x**3 + 565*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b**2*d**4*f**2*x**5 + 196*sqrt(c + d*x**2)*sqrt(a + b*x**2) *a*b**3*c**3*d*f**2*x - 594*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c**2* d**2*e*f*x - 145*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c**2*d**2*f**2*x **3 + 528*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c*d**3*e**2*x + 440*sqr t(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*c*d**3*e*f*x**3 + 120*sqrt(c + d*x** 2)*sqrt(a + b*x**2)*a*b**3*c*d**3*f**2*x**5 + 1485*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*d**4*e**2*x**3 + 2090*sqrt(c + d*x**2)*sqrt(a + b*x**2)* a*b**3*d**4*e*f*x**5 + 805*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**3*d**4*f **2*x**7 - 64*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**4*c**4*f**2*x + 176*sqr t(c + d*x**2)*sqrt(a + b*x**2)*b**4*c**3*d*e*f*x + 48*sqrt(c + d*x**2)*sqr t(a + b*x**2)*b**4*c**3*d*f**2*x**3 - 132*sqrt(c + d*x**2)*sqrt(a + b*x**2 )*b**4*c**2*d**2*e**2*x - 132*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**4*c*...