Integrand size = 32, antiderivative size = 847 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\frac {\left (8 a^4 d^4 f^2-a^3 b d^3 f (36 d e+7 c f)+9 a^2 b^2 d^2 \left (7 d^2 e^2+6 c d e f-c^2 f^2\right )-2 b^4 c^2 \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )+a b^3 c d \left (147 d^2 e^2-114 c d e f+32 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{315 b^2 d^4 \sqrt {a+b x^2}}-\frac {\left (4 a^3 d^3 f^2-3 a^2 b d^2 f (6 d e+c f)-3 a b^2 d \left (42 d^2 e^2+18 c d e f-5 c^2 f^2\right )-b^3 c \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b^2 d^3}+\frac {\left (3 a^2 d^2 f^2+a b d f (144 d e+11 c f)+3 b^2 \left (21 d^2 e^2+6 c d e f-2 c^2 f^2\right )\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^2}+\frac {f (18 b d e+b c f+10 a d f) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d}+\frac {1}{9} b f^2 x^7 \sqrt {a+b x^2} \sqrt {c+d x^2}-\frac {\sqrt {a} \left (8 a^4 d^4 f^2-a^3 b d^3 f (36 d e+7 c f)+9 a^2 b^2 d^2 \left (7 d^2 e^2+6 c d e f-c^2 f^2\right )-2 b^4 c^2 \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )+a b^3 c d \left (147 d^2 e^2-114 c d e f+32 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{5/2} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (4 a^3 d^3 f^2-3 a^2 b d^2 f (6 d e+c f)+3 a b^2 d \left (63 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 c \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{5/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/315*(8*a^4*d^4*f^2-a^3*b*d^3*f*(7*c*f+36*d*e)+9*a^2*b^2*d^2*(-c^2*f^2+6* c*d*e*f+7*d^2*e^2)-2*b^4*c^2*(8*c^2*f^2-24*c*d*e*f+21*d^2*e^2)+a*b^3*c*d*( 32*c^2*f^2-114*c*d*e*f+147*d^2*e^2))*x*(d*x^2+c)^(1/2)/b^2/d^4/(b*x^2+a)^( 1/2)-1/315*(4*a^3*d^3*f^2-3*a^2*b*d^2*f*(c*f+6*d*e)-3*a*b^2*d*(-5*c^2*f^2+ 18*c*d*e*f+42*d^2*e^2)-b^3*c*(8*c^2*f^2-24*c*d*e*f+21*d^2*e^2))*x*(b*x^2+a )^(1/2)*(d*x^2+c)^(1/2)/b^2/d^3+1/315*(3*a^2*d^2*f^2+a*b*d*f*(11*c*f+144*d *e)+3*b^2*(-2*c^2*f^2+6*c*d*e*f+21*d^2*e^2))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c) ^(1/2)/b/d^2+1/63*f*(10*a*d*f+b*c*f+18*b*d*e)*x^5*(b*x^2+a)^(1/2)*(d*x^2+c )^(1/2)/d+1/9*b*f^2*x^7*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)-1/315*a^(1/2)*(8*a ^4*d^4*f^2-a^3*b*d^3*f*(7*c*f+36*d*e)+9*a^2*b^2*d^2*(-c^2*f^2+6*c*d*e*f+7* d^2*e^2)-2*b^4*c^2*(8*c^2*f^2-24*c*d*e*f+21*d^2*e^2)+a*b^3*c*d*(32*c^2*f^2 -114*c*d*e*f+147*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))/b^(5/2)/d^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c) /c/(b*x^2+a))^(1/2)+1/315*a^(3/2)*(4*a^3*d^3*f^2-3*a^2*b*d^2*f*(c*f+6*d*e) +3*a*b^2*d*(5*c^2*f^2-18*c*d*e*f+63*d^2*e^2)-b^3*c*(8*c^2*f^2-24*c*d*e*f+2 1*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^(5/2)/d^3/(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 = 7.90 (sec) , antiderivative size = 575, normalized size of antiderivative = 0.68 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-4 a^3 d^3 f^2+3 a^2 b d^2 f \left (6 d e+c f+d f x^2\right )+a b^2 d \left (-15 c^2 f^2+c d f \left (54 e+11 f x^2\right )+2 d^2 \left (63 e^2+72 e f x^2+25 f^2 x^4\right )\right )+b^3 \left (8 c^3 f^2-6 c^2 d f \left (4 e+f x^2\right )+c d^2 \left (21 e^2+18 e f x^2+5 f^2 x^4\right )+d^3 x^2 \left (63 e^2+90 e f x^2+35 f^2 x^4\right )\right )\right )-i c \left (8 a^4 d^4 f^2-a^3 b d^3 f (36 d e+7 c f)+9 a^2 b^2 d^2 \left (7 d^2 e^2+6 c d e f-c^2 f^2\right )-2 b^4 c^2 \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )+a b^3 c d \left (147 d^2 e^2-114 c d e f+32 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i c (-b c+a d) \left (-9 a^2 b d^3 e f+2 a^3 d^3 f^2-3 a b^2 d \left (21 d^2 e^2-15 c d e f+4 c^2 f^2\right )+b^3 c \left (21 d^2 e^2-24 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 a^2 \left (\frac {b}{a}\right )^{5/2} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^(3/2)*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)*(-4*a^3*d^3*f^2 + 3*a^2*b*d^2*f*(6* d*e + c*f + d*f*x^2) + a*b^2*d*(-15*c^2*f^2 + c*d*f*(54*e + 11*f*x^2) + 2* d^2*(63*e^2 + 72*e*f*x^2 + 25*f^2*x^4)) + b^3*(8*c^3*f^2 - 6*c^2*d*f*(4*e + f*x^2) + c*d^2*(21*e^2 + 18*e*f*x^2 + 5*f^2*x^4) + d^3*x^2*(63*e^2 + 90* e*f*x^2 + 35*f^2*x^4))) - I*c*(8*a^4*d^4*f^2 - a^3*b*d^3*f*(36*d*e + 7*c*f ) + 9*a^2*b^2*d^2*(7*d^2*e^2 + 6*c*d*e*f - c^2*f^2) - 2*b^4*c^2*(21*d^2*e^ 2 - 24*c*d*e*f + 8*c^2*f^2) + a*b^3*c*d*(147*d^2*e^2 - 114*c*d*e*f + 32*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)] + (2*I)*c*(-(b*c) + a*d)*(-9*a^2*b*d^3*e*f + 2*a^3*d^ 3*f^2 - 3*a*b^2*d*(21*d^2*e^2 - 15*c*d*e*f + 4*c^2*f^2) + b^3*c*(21*d^2*e^ 2 - 24*c*d*e*f + 8*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellip ticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*a^2*(b/a)^(5/2)*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 2.00 (sec) , antiderivative size = 1342, normalized size of antiderivative = 1.58, 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.
| \(\displaystyle \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (e^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}+2 e f x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}+f^2 x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} f^2 \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5+\frac {(b c+3 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{63 d}+\frac {2}{7} e f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3+\frac {\left (\frac {3 d a^2}{b}+11 c a-\frac {6 b c^2}{d}\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{315 d}+\frac {2 (b c+3 a d) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 d}+\frac {b e^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} x}{5 d}-\frac {2 (b c-3 a d) e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 d}+\frac {\left (8 b^3 c^3-15 a b^2 d c^2+3 a^2 b d^2 c-4 a^3 d^3\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{315 b^2 d^3}+\frac {2 \left (\frac {3 d a^2}{b}+9 c a-\frac {4 b c^2}{d}\right ) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{105 d}+\frac {\left (\frac {3 d a^2}{b}+7 c a-\frac {2 b c^2}{d}\right ) e^2 \sqrt {b x^2+a} x}{15 \sqrt {d x^2+c}}-\frac {\left (16 b^4 c^4-32 a b^3 d c^3+9 a^2 b^2 d^2 c^2+7 a^3 b d^3 c-8 a^4 d^4\right ) f^2 \sqrt {b x^2+a} x}{315 b^3 d^3 \sqrt {d x^2+c}}+\frac {2 (b c-2 a d) \left (8 b^2 c^2-3 a b d c+3 a^2 d^2\right ) e f \sqrt {b x^2+a} x}{105 b^2 d^2 \sqrt {d x^2+c}}+\frac {\sqrt {c} \left (2 b^2 c^2-7 a b d c-3 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 )}{15 b d^{3/2} \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 (16 b^4 c^4-32 a b^3 d c^3+9 a^2 b^2 d^2 c^2+7 a^3 b d^3 c-8 a^4 d^4\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 )}{315 b^3 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 {2 \sqrt {c} (b c-2 a d) \left (8 b^2 c^2-3 a b d c+3 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 )}{105 b^2 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 {c^{3/2} (b c-9 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 )}{15 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} \left (8 b^3 c^3-15 a b^2 d c^2+3 a^2 b d^2 c-4 a^3 d^3\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 )}{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 {2 c^{3/2} \left (4 b^2 c^2-9 a b d c-3 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 )}{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}}\) |
Input:
Int[(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2,x]
Output:
((7*a*c - (2*b*c^2)/d + (3*a^2*d)/b)*e^2*x*Sqrt[a + b*x^2])/(15*Sqrt[c + d *x^2]) + (2*(b*c - 2*a*d)*(8*b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*e*f*x*Sqrt[a + b*x^2])/(105*b^2*d^2*Sqrt[c + d*x^2]) - ((16*b^4*c^4 - 32*a*b^3*c^3*d + 9*a^2*b^2*c^2*d^2 + 7*a^3*b*c*d^3 - 8*a^4*d^4)*f^2*x*Sqrt[a + b*x^2])/(31 5*b^3*d^3*Sqrt[c + d*x^2]) - (2*(b*c - 3*a*d)*e^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d) + (2*(9*a*c - (4*b*c^2)/d + (3*a^2*d)/b)*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(105*d) + ((8*b^3*c^3 - 15*a*b^2*c^2*d + 3*a^2*b*c *d^2 - 4*a^3*d^3)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(315*b^2*d^3) + ( 2*(b*c + 3*a*d)*e*f*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*d) + ((11*a*c - (6*b*c^2)/d + (3*a^2*d)/b)*f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(31 5*d) + ((b*c + 3*a*d)*f^2*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(63*d) + (2 *e*f*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/7 + (f^2*x^5*(a + b*x^2)^(3/2) *Sqrt[c + d*x^2])/9 + (b*e^2*x*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(5*d) + (Sqrt[c]*(2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*e^2*Sqrt[a + b*x^2]*EllipticE [ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*Sqrt[c]*(b*c - 2*a*d)*(8*b ^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt [d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b^2*d^(5/2)*Sqrt[(c*(a + b*x^2))/( a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*(16*b^4*c^4 - 32*a*b^3*c^3*d + 9*a^2*b^2*c^2*d^2 + 7*a^3*b*c*d^3 - 8*a^4*d^4)*f^2*Sqrt[a + b*x^2]*Ell...
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 = 11.47 (sec) , antiderivative size = 1300, normalized size of antiderivative = 1.53
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1300\) |
| risch | \(\text {Expression too large to display}\) | \(1494\) |
| default | \(\text {Expression too large to display}\) | \(2488\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/9*b*f^2*x^7 *(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/7*(2*a*f^2*d*b+b^2*c*f^2+2*b^2*d*e* f-1/9*b*f^2*(8*a*d+8*b*c))/b/d*x^5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5 *(a^2*d*f^2+11/9*a*b*c*f^2+4*a*b*d*e*f+2*b^2*c*e*f+b^2*d*e^2-1/7*(2*a*f^2* d*b+b^2*c*f^2+2*b^2*d*e*f-1/9*b*f^2*(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d* x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(a^2*c*f^2+2*a^2*e*f*d+4*a*c*e *f*b+2*a*b*e^2*d+b^2*e^2*c-5/7*(2*a*f^2*d*b+b^2*c*f^2+2*b^2*d*e*f-1/9*b*f^ 2*(8*a*d+8*b*c))/b/d*a*c-1/5*(a^2*d*f^2+11/9*a*b*c*f^2+4*a*b*d*e*f+2*b^2*c *e*f+b^2*d*e^2-1/7*(2*a*f^2*d*b+b^2*c*f^2+2*b^2*d*e*f-1/9*b*f^2*(8*a*d+8*b *c))/b/d*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+ a*c)^(1/2)+(a^2*c*e^2-1/3*(a^2*c*f^2+2*a^2*e*f*d+4*a*c*e*f*b+2*a*b*e^2*d+b ^2*e^2*c-5/7*(2*a*f^2*d*b+b^2*c*f^2+2*b^2*d*e*f-1/9*b*f^2*(8*a*d+8*b*c))/b /d*a*c-1/5*(a^2*d*f^2+11/9*a*b*c*f^2+4*a*b*d*e*f+2*b^2*c*e*f+b^2*d*e^2-1/7 *(2*a*f^2*d*b+b^2*c*f^2+2*b^2*d*e*f-1/9*b*f^2*(8*a*d+8*b*c))/b/d*(6*a*d+6* b*c))/b/d*(4*a*d+4*b*c))/b/d*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))-(2*a^2*c*e*f+a^2*e^2*d+2*a*c*e^2*b-3/5*(a^2*d*f^2+11 /9*a*b*c*f^2+4*a*b*d*e*f+2*b^2*c*e*f+b^2*d*e^2-1/7*(2*a*f^2*d*b+b^2*c*f^2+ 2*b^2*d*e*f-1/9*b*f^2*(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*a*c-1/3*(a^2*c *f^2+2*a^2*e*f*d+4*a*c*e*f*b+2*a*b*e^2*d+b^2*e^2*c-5/7*(2*a*f^2*d*b+b^2...
Time = 0.11 (sec) , antiderivative size = 880, normalized size of antiderivative = 1.04 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="fricas ")
Output:
1/315*((21*(2*b^4*c^3*d^2 - 7*a*b^3*c^2*d^3 - 3*a^2*b^2*c*d^4)*e^2 - 6*(8* b^4*c^4*d - 19*a*b^3*c^3*d^2 + 9*a^2*b^2*c^2*d^3 - 6*a^3*b*c*d^4)*e*f + (1 6*b^4*c^5 - 32*a*b^3*c^4*d + 9*a^2*b^2*c^3*d^2 + 7*a^3*b*c^2*d^3 - 8*a^4*c *d^4)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b* c)) - (21*(2*b^4*c^3*d^2 - 7*a*b^3*c^2*d^3 - 9*a^2*b^2*d^5 - (3*a^2*b^2 - a*b^3)*c*d^4)*e^2 - 6*(8*b^4*c^4*d - 19*a*b^3*c^3*d^2 - 3*a^3*b*d^5 + (9*a ^2*b^2 + 4*a*b^3)*c^2*d^3 - 3*(2*a^3*b + 3*a^2*b^2)*c*d^4)*e*f + (16*b^4*c ^5 - 32*a*b^3*c^4*d - 4*a^4*d^5 + (9*a^2*b^2 + 8*a*b^3)*c^3*d^2 + (7*a^3*b - 15*a^2*b^2)*c^2*d^3 - (8*a^4 - 3*a^3*b)*c*d^4)*f^2)*sqrt(b*d)*x*sqrt(-c /d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (35*b^4*d^5*f^2*x^8 + 5* (18*b^4*d^5*e*f + (b^4*c*d^4 + 10*a*b^3*d^5)*f^2)*x^6 + (63*b^4*d^5*e^2 + 18*(b^4*c*d^4 + 8*a*b^3*d^5)*e*f - (6*b^4*c^2*d^3 - 11*a*b^3*c*d^4 - 3*a^2 *b^2*d^5)*f^2)*x^4 - 21*(2*b^4*c^2*d^3 - 7*a*b^3*c*d^4 - 3*a^2*b^2*d^5)*e^ 2 + 6*(8*b^4*c^3*d^2 - 19*a*b^3*c^2*d^3 + 9*a^2*b^2*c*d^4 - 6*a^3*b*d^5)*e *f - (16*b^4*c^4*d - 32*a*b^3*c^3*d^2 + 9*a^2*b^2*c^2*d^3 + 7*a^3*b*c*d^4 - 8*a^4*d^5)*f^2 + (21*(b^4*c*d^4 + 6*a*b^3*d^5)*e^2 - 6*(4*b^4*c^2*d^3 - 9*a*b^3*c*d^4 - 3*a^2*b^2*d^5)*e*f + (8*b^4*c^3*d^2 - 15*a*b^3*c^2*d^3 + 3 *a^2*b^2*c*d^4 - 4*a^3*b*d^5)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/( b^3*d^5*x)
\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int \left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)**(1/2)*(f*x**2+e)**2,x)
Output:
Integral((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)**2, x)
\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2, x)
\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2, x)
Timed out. \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int {\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2 \,d x \] Input:
int((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2,x)
Output:
int((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2, x)
\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*f**2*x + 3*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*a**2*b*c*d**2*f**2*x + 18*sqrt(c + d*x**2)*sqrt(a + b *x**2)*a**2*b*d**3*e*f*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3 *f**2*x**3 - 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*f**2*x + 5 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e*f*x + 11*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*a*b**2*c*d**2*f**2*x**3 + 126*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*e**2*x + 144*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b* *2*d**3*e*f*x**3 + 50*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*f**2*x **5 + 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*f**2*x - 24*sqrt(c + d *x**2)*sqrt(a + b*x**2)*b**3*c**2*d*e*f*x - 6*sqrt(c + d*x**2)*sqrt(a + b* x**2)*b**3*c**2*d*f**2*x**3 + 21*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c* d**2*e**2*x + 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*f*x**3 + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*f**2*x**5 + 63*sqrt(c + d* x**2)*sqrt(a + b*x**2)*b**3*d**3*e**2*x**3 + 90*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*d**3*e*f*x**5 + 35*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*d** 3*f**2*x**7 + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x* *2 + b*c*x**2 + b*d*x**4),x)*a**4*d**4*f**2 - 7*int((sqrt(c + d*x**2)*sqrt (a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*b*c*d**3 *f**2 - 36*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*b*d**4*e*f - 9*int((sqrt(c + d*x**2)*sqrt(...