\(\int \frac {(a+b x^2)^{5/2} (e+f x^2)^2}{\sqrt {c+d x^2}} \, dx\) [69]

Optimal result

Integrand size = 32, antiderivative size = 874 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {\left (10 a^4 d^4 f^2-5 a^3 b d^3 f (18 d e-5 c f)-8 b^4 c^2 \left (21 d^2 e^2-36 c d e f+16 c^2 f^2\right )-3 a^2 b^2 d^2 \left (161 d^2 e^2-206 c d e f+81 c^2 f^2\right )+a b^3 c d \left (483 d^2 e^2-768 c d e f+328 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{315 b d^5 \sqrt {a+b x^2}}+\frac {\left (5 a^3 d^3 f^2+15 a^2 b d^2 f (18 d e-7 c f)-4 b^3 c \left (21 d^2 e^2-36 c d e f+16 c^2 f^2\right )+3 a b^2 d \left (77 d^2 e^2-122 c d e f+52 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (75 a^2 d^2 f^2+5 a b d f (54 d e-23 c f)+3 b^2 \left (21 d^2 e^2-36 c d e f+16 c^2 f^2\right )\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}+\frac {b f (18 b d e-8 b c f+19 a d f) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b^2 f^2 x^7 \sqrt {a+b x^2} \sqrt {c+d x^2}}{9 d}+\frac {\sqrt {a} \left (10 a^4 d^4 f^2-5 a^3 b d^3 f (18 d e-5 c f)-8 b^4 c^2 \left (21 d^2 e^2-36 c d e f+16 c^2 f^2\right )-3 a^2 b^2 d^2 \left (161 d^2 e^2-206 c d e f+81 c^2 f^2\right )+a b^3 c d \left (483 d^2 e^2-768 c d e f+328 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{3/2} d^5 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (5 a^3 c d^3 f^2-15 a^2 b d^2 \left (21 d^2 e^2-18 c d e f+7 c^2 f^2\right )-4 b^3 c^2 \left (21 d^2 e^2-36 c d e f+16 c^2 f^2\right )+3 a b^2 c d \left (77 d^2 e^2-122 c d e f+52 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{3/2} c d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/315*(10*a^4*d^4*f^2-5*a^3*b*d^3*f*(-5*c*f+18*d*e)-8*b^4*c^2*(16*c^2*f^2 
-36*c*d*e*f+21*d^2*e^2)-3*a^2*b^2*d^2*(81*c^2*f^2-206*c*d*e*f+161*d^2*e^2) 
+a*b^3*c*d*(328*c^2*f^2-768*c*d*e*f+483*d^2*e^2))*x*(d*x^2+c)^(1/2)/b/d^5/ 
(b*x^2+a)^(1/2)+1/315*(5*a^3*d^3*f^2+15*a^2*b*d^2*f*(-7*c*f+18*d*e)-4*b^3* 
c*(16*c^2*f^2-36*c*d*e*f+21*d^2*e^2)+3*a*b^2*d*(52*c^2*f^2-122*c*d*e*f+77* 
d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d^4+1/315*(75*a^2*d^2*f^2+5* 
a*b*d*f*(-23*c*f+54*d*e)+3*b^2*(16*c^2*f^2-36*c*d*e*f+21*d^2*e^2))*x^3*(b* 
x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3+1/63*b*f*(19*a*d*f-8*b*c*f+18*b*d*e)*x^5* 
(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^2+1/9*b^2*f^2*x^7*(b*x^2+a)^(1/2)*(d*x^2 
+c)^(1/2)/d+1/315*a^(1/2)*(10*a^4*d^4*f^2-5*a^3*b*d^3*f*(-5*c*f+18*d*e)-8* 
b^4*c^2*(16*c^2*f^2-36*c*d*e*f+21*d^2*e^2)-3*a^2*b^2*d^2*(81*c^2*f^2-206*c 
*d*e*f+161*d^2*e^2)+a*b^3*c*d*(328*c^2*f^2-768*c*d*e*f+483*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^(3/2)/d^5/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/315*a^(3/2 
)*(5*a^3*c*d^3*f^2-15*a^2*b*d^2*(7*c^2*f^2-18*c*d*e*f+21*d^2*e^2)-4*b^3*c^ 
2*(16*c^2*f^2-36*c*d*e*f+21*d^2*e^2)+3*a*b^2*c*d*(52*c^2*f^2-122*c*d*e*f+7 
7*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^(3/2)/c/d^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/ 
2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.75 (sec) , antiderivative size = 603, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (5 a^3 d^3 f^2+15 a^2 b d^2 f \left (18 d e-7 c f+5 d f x^2\right )+b^3 \left (-64 c^3 f^2+48 c^2 d f \left (3 e+f x^2\right )-4 c d^2 \left (21 e^2+27 e f x^2+10 f^2 x^4\right )+d^3 x^2 \left (63 e^2+90 e f x^2+35 f^2 x^4\right )\right )+a b^2 d \left (156 c^2 f^2-c d f \left (366 e+115 f x^2\right )+d^2 \left (231 e^2+270 e f x^2+95 f^2 x^4\right )\right )\right )+i c \left (10 a^4 d^4 f^2+5 a^3 b d^3 f (-18 d e+5 c f)-8 b^4 c^2 \left (21 d^2 e^2-36 c d e f+16 c^2 f^2\right )-3 a^2 b^2 d^2 \left (161 d^2 e^2-206 c d e f+81 c^2 f^2\right )+a b^3 c d \left (483 d^2 e^2-768 c d e f+328 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (b c-a d) \left (5 a^3 c d^3 f^2+45 a^2 b d^2 \left (7 d^2 e^2-8 c d e f+3 c^2 f^2\right )+8 b^3 c^2 \left (21 d^2 e^2-36 c d e f+16 c^2 f^2\right )-3 a b^2 c d \left (133 d^2 e^2-208 c d e f+88 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 b \sqrt {\frac {b}{a}} d^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[((a + b*x^2)^(5/2)*(e + f*x^2)^2)/Sqrt[c + d*x^2],x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(5*a^3*d^3*f^2 + 15*a^2*b*d^2*f*(18 
*d*e - 7*c*f + 5*d*f*x^2) + b^3*(-64*c^3*f^2 + 48*c^2*d*f*(3*e + f*x^2) - 
4*c*d^2*(21*e^2 + 27*e*f*x^2 + 10*f^2*x^4) + d^3*x^2*(63*e^2 + 90*e*f*x^2 
+ 35*f^2*x^4)) + a*b^2*d*(156*c^2*f^2 - c*d*f*(366*e + 115*f*x^2) + d^2*(2 
31*e^2 + 270*e*f*x^2 + 95*f^2*x^4))) + I*c*(10*a^4*d^4*f^2 + 5*a^3*b*d^3*f 
*(-18*d*e + 5*c*f) - 8*b^4*c^2*(21*d^2*e^2 - 36*c*d*e*f + 16*c^2*f^2) - 3* 
a^2*b^2*d^2*(161*d^2*e^2 - 206*c*d*e*f + 81*c^2*f^2) + a*b^3*c*d*(483*d^2* 
e^2 - 768*c*d*e*f + 328*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*(b*c - a*d)*(5*a^3*c*d^ 
3*f^2 + 45*a^2*b*d^2*(7*d^2*e^2 - 8*c*d*e*f + 3*c^2*f^2) + 8*b^3*c^2*(21*d 
^2*e^2 - 36*c*d*e*f + 16*c^2*f^2) - 3*a*b^2*c*d*(133*d^2*e^2 - 208*c*d*e*f 
 + 88*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSin 
h[Sqrt[b/a]*x], (a*d)/(b*c)])/(315*b*Sqrt[b/a]*d^5*Sqrt[a + b*x^2]*Sqrt[c 
+ d*x^2])
 

Rubi [A] (verified)

Time = 1.99 (sec) , antiderivative size = 1379, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 433

\(\displaystyle \int \left (\frac {e^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}}+\frac {2 e f x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}}+\frac {f^2 x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b f^2 \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^5}{9 d}-\frac {4 b (2 b c-3 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{63 d^2}+\frac {2 b e f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3}{7 d}+\frac {\left (48 b^2 c^2-115 a b d c+75 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{315 d^3}-\frac {4 b (3 b c-5 a d) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 d^2}+\frac {b e^2 \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x}{5 d}-\frac {4 b (b c-2 a d) e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 d^2}-\frac {\left (64 b^3 c^3-156 a b^2 d c^2+105 a^2 b d^2 c-5 a^3 d^3\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{315 b d^4}+\frac {2 \left (24 b^2 c^2-61 a b d c+45 a^2 d^2\right ) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{105 d^3}+\frac {\left (8 b^2 c^2-23 a b d c+23 a^2 d^2\right ) e^2 \sqrt {b x^2+a} x}{15 d^2 \sqrt {d x^2+c}}+\frac {\left (128 b^4 c^4-328 a b^3 d c^3+243 a^2 b^2 d^2 c^2-25 a^3 b d^3 c-10 a^4 d^4\right ) f^2 \sqrt {b x^2+a} x}{315 b^2 d^4 \sqrt {d x^2+c}}-\frac {2 \left (48 b^3 c^3-128 a b^2 d c^2+103 a^2 b d^2 c-15 a^3 d^3\right ) e f \sqrt {b x^2+a} x}{105 b d^3 \sqrt {d x^2+c}}-\frac {\sqrt {c} \left (8 b^2 c^2-23 a b d c+23 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 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^4 c^4-328 a b^3 d c^3+243 a^2 b^2 d^2 c^2-25 a^3 b d^3 c-10 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^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 \sqrt {c} \left (48 b^3 c^3-128 a b^2 d c^2+103 a^2 b d^2 c-15 a^3 d^3\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 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 {\sqrt {c} \left (4 b^2 c^2-11 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 )}{15 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} \left (64 b^3 c^3-156 a b^2 d c^2+105 a^2 b d^2 c-5 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 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 (24 b^2 c^2-61 a b d c+45 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 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)*(e + f*x^2)^2)/Sqrt[c + d*x^2],x]
 

Output:

((8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*e^2*x*Sqrt[a + b*x^2])/(15*d^2*Sqrt 
[c + d*x^2]) - (2*(48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3 
*d^3)*e*f*x*Sqrt[a + b*x^2])/(105*b*d^3*Sqrt[c + d*x^2]) + ((128*b^4*c^4 - 
 328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 25*a^3*b*c*d^3 - 10*a^4*d^4)*f^2* 
x*Sqrt[a + b*x^2])/(315*b^2*d^4*Sqrt[c + d*x^2]) - (4*b*(b*c - 2*a*d)*e^2* 
x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d^2) + (2*(24*b^2*c^2 - 61*a*b*c*d 
+ 45*a^2*d^2)*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(105*d^3) - ((64*b^3* 
c^3 - 156*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*f^2*x*Sqrt[a + b*x^2] 
*Sqrt[c + d*x^2])/(315*b*d^4) - (4*b*(3*b*c - 5*a*d)*e*f*x^3*Sqrt[a + b*x^ 
2]*Sqrt[c + d*x^2])/(35*d^2) + ((48*b^2*c^2 - 115*a*b*c*d + 75*a^2*d^2)*f^ 
2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(315*d^3) - (4*b*(2*b*c - 3*a*d)*f^ 
2*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(63*d^2) + (b*e^2*x*(a + b*x^2)^(3/ 
2)*Sqrt[c + d*x^2])/(5*d) + (2*b*e*f*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2] 
)/(7*d) + (b*f^2*x^5*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(9*d) - (Sqrt[c]*( 
8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[ 
(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*d^(5/2)*Sqrt[(c*(a + b*x^2))/( 
a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*(48*b^3*c^3 - 128*a*b^2*c^2* 
d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sq 
rt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b*d^(7/2)*Sqrt[(c*(a + b*x^2))/( 
a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(128*b^4*c^4 - 328*a*b^3*c^...
 

Defintions of rubi rules used

rule 433

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 13.84 (sec) , antiderivative size = 1214, normalized size of antiderivative = 1.39

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1214\)
risch	\(\text {Expression too large to display}\)	\(1591\)
default	\(\text {Expression too large to display}\)	\(2551\)

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),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^2/d*f^2 
*x^7*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/7*(3*f^2*a*b^2+2*e*f*b^3-1/9*b^ 
2/d*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*(3* 
a^2*b*f^2+6*a*b^2*e*f+e^2*b^3-7/9*a*b^2*c/d*f^2-1/7*(3*f^2*a*b^2+2*e*f*b^3 
-1/9*b^2/d*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^3*f^2+6*a^2*b*e*f+3*a*b^2*e^2-5/7*(3*f^2*a*b^2+2 
*e*f*b^3-1/9*b^2/d*f^2*(8*a*d+8*b*c))/b/d*a*c-1/5*(3*a^2*b*f^2+6*a*b^2*e*f 
+e^2*b^3-7/9*a*b^2*c/d*f^2-1/7*(3*f^2*a*b^2+2*e*f*b^3-1/9*b^2/d*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^3*e^2-1/3*(a^3*f^2+6*a^2*b*e*f+3*a*b^2*e^2-5/7*(3*f^2*a* 
b^2+2*e*f*b^3-1/9*b^2/d*f^2*(8*a*d+8*b*c))/b/d*a*c-1/5*(3*a^2*b*f^2+6*a*b^ 
2*e*f+e^2*b^3-7/9*a*b^2*c/d*f^2-1/7*(3*f^2*a*b^2+2*e*f*b^3-1/9*b^2/d*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)*El 
lipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(2*a^3*e*f+3*a^2*b*e^2-3/ 
5*(3*a^2*b*f^2+6*a*b^2*e*f+e^2*b^3-7/9*a*b^2*c/d*f^2-1/7*(3*f^2*a*b^2+2*e* 
f*b^3-1/9*b^2/d*f^2*(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/b/d*a*c-1/3*(a^3*f^2 
+6*a^2*b*e*f+3*a*b^2*e^2-5/7*(3*f^2*a*b^2+2*e*f*b^3-1/9*b^2/d*f^2*(8*a*d+8 
*b*c))/b/d*a*c-1/5*(3*a^2*b*f^2+6*a*b^2*e*f+e^2*b^3-7/9*a*b^2*c/d*f^2-1/7* 
(3*f^2*a*b^2+2*e*f*b^3-1/9*b^2/d*f^2*(8*a*d+8*b*c))/b/d*(6*a*d+6*b*c))/...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 941, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="fricas 
")
 

Output:

-1/315*((21*(8*b^4*c^4*d^2 - 23*a*b^3*c^3*d^3 + 23*a^2*b^2*c^2*d^4)*e^2 - 
6*(48*b^4*c^5*d - 128*a*b^3*c^4*d^2 + 103*a^2*b^2*c^3*d^3 - 15*a^3*b*c^2*d 
^4)*e*f + (128*b^4*c^6 - 328*a*b^3*c^5*d + 243*a^2*b^2*c^4*d^2 - 25*a^3*b* 
c^3*d^3 - 10*a^4*c^2*d^4)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sq 
rt(-c/d)/x), a*d/(b*c)) - (21*(8*b^4*c^4*d^2 - 23*a*b^3*c^3*d^3 - 11*a^2*b 
^2*c*d^5 + 15*a^3*b*d^6 + (23*a^2*b^2 + 4*a*b^3)*c^2*d^4)*e^2 - 6*(48*b^4* 
c^5*d - 128*a*b^3*c^4*d^2 + 45*a^3*b*c*d^5 + (103*a^2*b^2 + 24*a*b^3)*c^3* 
d^3 - (15*a^3*b + 61*a^2*b^2)*c^2*d^4)*e*f + (128*b^4*c^6 - 328*a*b^3*c^5* 
d - 5*a^4*c*d^5 + (243*a^2*b^2 + 64*a*b^3)*c^4*d^2 - (25*a^3*b + 156*a^2*b 
^2)*c^3*d^3 - 5*(2*a^4 - 21*a^3*b)*c^2*d^4)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*el 
liptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (35*b^4*c*d^5*f^2*x^8 + 5*(18* 
b^4*c*d^5*e*f - (8*b^4*c^2*d^4 - 19*a*b^3*c*d^5)*f^2)*x^6 + (63*b^4*c*d^5* 
e^2 - 54*(2*b^4*c^2*d^4 - 5*a*b^3*c*d^5)*e*f + (48*b^4*c^3*d^3 - 115*a*b^3 
*c^2*d^4 + 75*a^2*b^2*c*d^5)*f^2)*x^4 + 21*(8*b^4*c^3*d^3 - 23*a*b^3*c^2*d 
^4 + 23*a^2*b^2*c*d^5)*e^2 - 6*(48*b^4*c^4*d^2 - 128*a*b^3*c^3*d^3 + 103*a 
^2*b^2*c^2*d^4 - 15*a^3*b*c*d^5)*e*f + (128*b^4*c^5*d - 328*a*b^3*c^4*d^2 
+ 243*a^2*b^2*c^3*d^3 - 25*a^3*b*c^2*d^4 - 10*a^4*c*d^5)*f^2 - (21*(4*b^4* 
c^2*d^4 - 11*a*b^3*c*d^5)*e^2 - 6*(24*b^4*c^3*d^3 - 61*a*b^3*c^2*d^4 + 45* 
a^2*b^2*c*d^5)*e*f + (64*b^4*c^4*d^2 - 156*a*b^3*c^3*d^3 + 105*a^2*b^2*c^2 
*d^4 - 5*a^3*b*c*d^5)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*c...
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )^{2}}{\sqrt {c + d x^{2}}}\, dx \] Input:

integrate((b*x**2+a)**(5/2)*(f*x**2+e)**2/(d*x**2+c)**(1/2),x)
 

Output:

Integral((a + b*x**2)**(5/2)*(e + f*x**2)**2/sqrt(c + d*x**2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}}{\sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="maxima 
")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)^2/sqrt(d*x^2 + c), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}}{\sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)^2/sqrt(d*x^2 + c), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,{\left (f\,x^2+e\right )}^2}{\sqrt {d\,x^2+c}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(1/2),x)
 

Output:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x)
 

Output:

(5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*f**2*x - 105*sqrt(c + d*x** 
2)*sqrt(a + b*x**2)*a**2*b*c*d**2*f**2*x + 270*sqrt(c + d*x**2)*sqrt(a + b 
*x**2)*a**2*b*d**3*e*f*x + 75*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d** 
3*f**2*x**3 + 156*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*f**2*x - 
 366*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e*f*x - 115*sqrt(c + 
d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*f**2*x**3 + 231*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a*b**2*d**3*e**2*x + 270*sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
a*b**2*d**3*e*f*x**3 + 95*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*f* 
*2*x**5 - 64*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*f**2*x + 144*sqrt 
(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**2*d*e*f*x + 48*sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*b**3*c**2*d*f**2*x**3 - 84*sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
b**3*c*d**2*e**2*x - 108*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*f 
*x**3 - 40*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*f**2*x**5 + 63*sq 
rt(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 - 10*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 - 25*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 + 90*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 + 243*int((sqrt(c + ...