\(\int \frac {\sqrt {a+b x^2} (e+f x^2)^2}{(c+d x^2)^{7/2}} \, dx\) [60]

Optimal result

Integrand size = 32, antiderivative size = 505 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {\left (d^2 e^2-2 c d e f-4 c^2 f^2\right ) x \sqrt {a+b x^2}}{5 c d^2 \left (c+d x^2\right )^{5/2}}-\frac {f^2 x^3 \sqrt {a+b x^2}}{d \left (c+d x^2\right )^{5/2}}+\frac {\left (b c \left (3 d^2 e^2+4 c d e f+8 c^2 f^2\right )-a d \left (4 d^2 e^2+2 c d e f+9 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{15 c^2 d^2 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {\left (a^2 d^2 \left (8 d^2 e^2+4 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (3 d^2 e^2+4 c d e f+8 c^2 f^2\right )-a b c d \left (13 d^2 e^2+4 c d e f+13 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 c^{5/2} d^{5/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {2 b (d e-c f) (b c (3 d e+2 c f)-a d (2 d e+3 c f)) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 c^{3/2} d^{5/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/5*(-4*c^2*f^2-2*c*d*e*f+d^2*e^2)*x*(b*x^2+a)^(1/2)/c/d^2/(d*x^2+c)^(5/2) 
-f^2*x^3*(b*x^2+a)^(1/2)/d/(d*x^2+c)^(5/2)+1/15*(b*c*(8*c^2*f^2+4*c*d*e*f+ 
3*d^2*e^2)-a*d*(9*c^2*f^2+2*c*d*e*f+4*d^2*e^2))*x*(b*x^2+a)^(1/2)/c^2/d^2/ 
(-a*d+b*c)/(d*x^2+c)^(3/2)+1/15*(a^2*d^2*(3*c^2*f^2+4*c*d*e*f+8*d^2*e^2)+b 
^2*c^2*(8*c^2*f^2+4*c*d*e*f+3*d^2*e^2)-a*b*c*d*(13*c^2*f^2+4*c*d*e*f+13*d^ 
2*e^2))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b 
*c/a/d)^(1/2))/c^(5/2)/d^(5/2)/(-a*d+b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2 
)/(d*x^2+c)^(1/2)+2/15*b*(-c*f+d*e)*(b*c*(2*c*f+3*d*e)-a*d*(3*c*f+2*d*e))* 
(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2 
))/c^(3/2)/d^(5/2)/(-a*d+b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^ 
(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.57 (sec) , antiderivative size = 499, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (3 c^2 (b c-a d)^2 (d e-c f)^2+c (b c-a d) (d e-c f) (-2 a d (2 d e+3 c f)+b c (3 d e+7 c f)) \left (c+d x^2\right )+\left (a^2 d^2 \left (8 d^2 e^2+4 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (3 d^2 e^2+4 c d e f+8 c^2 f^2\right )-a b c d \left (13 d^2 e^2+4 c d e f+13 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )+i b c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \left (\left (a^2 d^2 \left (8 d^2 e^2+4 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (3 d^2 e^2+4 c d e f+8 c^2 f^2\right )-a b c d \left (13 d^2 e^2+4 c d e f+13 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (b c \left (3 d^2 e^2+4 c d e f+8 c^2 f^2\right )-a d \left (4 d^2 e^2+2 c d e f+9 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 \sqrt {\frac {b}{a}} c^3 d^3 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*(e + f*x^2)^2)/(c + d*x^2)^(7/2),x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2 + c*(b*c - a 
*d)*(d*e - c*f)*(-2*a*d*(2*d*e + 3*c*f) + b*c*(3*d*e + 7*c*f))*(c + d*x^2) 
 + (a^2*d^2*(8*d^2*e^2 + 4*c*d*e*f + 3*c^2*f^2) + b^2*c^2*(3*d^2*e^2 + 4*c 
*d*e*f + 8*c^2*f^2) - a*b*c*d*(13*d^2*e^2 + 4*c*d*e*f + 13*c^2*f^2))*(c + 
d*x^2)^2) + I*b*c*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*(( 
a^2*d^2*(8*d^2*e^2 + 4*c*d*e*f + 3*c^2*f^2) + b^2*c^2*(3*d^2*e^2 + 4*c*d*e 
*f + 8*c^2*f^2) - a*b*c*d*(13*d^2*e^2 + 4*c*d*e*f + 13*c^2*f^2))*EllipticE 
[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (b*c - a*d)*(b*c*(3*d^2*e^2 + 4*c* 
d*e*f + 8*c^2*f^2) - a*d*(4*d^2*e^2 + 2*c*d*e*f + 9*c^2*f^2))*EllipticF[I* 
ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*Sqrt[b/a]*c^3*d^3*(b*c - a*d)^2*S 
qrt[a + b*x^2]*(c + d*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 952, normalized size of antiderivative = 1.89, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 433

\(\displaystyle \int \left (\frac {e^2 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{7/2}}+\frac {2 e f x^2 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{7/2}}+\frac {f^2 x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{7/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {f^2 \sqrt {b x^2+a} x^3}{5 d \left (d x^2+c\right )^{5/2}}+\frac {(3 b c-4 a d) e^2 \sqrt {b x^2+a} x}{15 c^2 (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {(4 b c-3 a d) f^2 \sqrt {b x^2+a} x}{15 d^2 (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {2 (2 b c-a d) e f \sqrt {b x^2+a} x}{15 c d (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {e^2 \sqrt {b x^2+a} x}{5 c \left (d x^2+c\right )^{5/2}}-\frac {2 e f \sqrt {b x^2+a} x}{5 d \left (d x^2+c\right )^{5/2}}+\frac {\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 )}{15 c^{5/2} \sqrt {d} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\left (8 b^2 c^2-13 a b d c+3 a^2 d^2\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 )}{15 \sqrt {c} d^{5/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {4 \left (b^2 c^2-a b d c+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 c^{3/2} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 b (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 )}{15 c^{3/2} \sqrt {d} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 b \sqrt {c} (2 b c-3 a d) 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 )}{15 d^{5/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 b (b c+a d) 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 \sqrt {c} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\)

Input:

Int[(Sqrt[a + b*x^2]*(e + f*x^2)^2)/(c + d*x^2)^(7/2),x]
 

Output:

(e^2*x*Sqrt[a + b*x^2])/(5*c*(c + d*x^2)^(5/2)) - (2*e*f*x*Sqrt[a + b*x^2] 
)/(5*d*(c + d*x^2)^(5/2)) - (f^2*x^3*Sqrt[a + b*x^2])/(5*d*(c + d*x^2)^(5/ 
2)) + ((3*b*c - 4*a*d)*e^2*x*Sqrt[a + b*x^2])/(15*c^2*(b*c - a*d)*(c + d*x 
^2)^(3/2)) + (2*(2*b*c - a*d)*e*f*x*Sqrt[a + b*x^2])/(15*c*d*(b*c - a*d)*( 
c + d*x^2)^(3/2)) - ((4*b*c - 3*a*d)*f^2*x*Sqrt[a + b*x^2])/(15*d^2*(b*c - 
 a*d)*(c + d*x^2)^(3/2)) + ((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)])/(15*c^ 
(5/2)*Sqrt[d]*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + 
 d*x^2]) + (4*(b^2*c^2 - a*b*c*d + a^2*d^2)*e*f*Sqrt[a + b*x^2]*EllipticE[ 
ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*c^(3/2)*d^(3/2)*(b*c - 
a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + ((8*b^2*c^ 
2 - 13*a*b*c*d + 3*a^2*d^2)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]* 
x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*Sqrt[c]*d^(5/2)*(b*c - a*d)^2*Sqrt[(c*( 
a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*b*(3*b*c - 2*a*d)*e^2*S 
qrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(1 
5*c^(3/2)*Sqrt[d]*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt 
[c + d*x^2]) - (2*b*(b*c + a*d)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt 
[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*Sqrt[c]*d^(3/2)*(b*c - a*d)^2*Sqrt[ 
(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*b*Sqrt[c]*(2*b*c - 
3*a*d)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (...
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs. \(2(472)=944\).

Time = 8.27 (sec) , antiderivative size = 1037, normalized size of antiderivative = 2.05

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1037\)
default	\(\text {Expression too large to display}\)	\(4450\)

Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(7/2),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)*(1/5*(c^2*f^2- 
2*c*d*e*f+d^2*e^2)/c/d^5*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^3 
-1/15*(6*a*c^2*d*f^2-2*a*c*d^2*e*f-4*a*d^3*e^2-7*b*c^3*f^2+4*b*c^2*d*e*f+3 
*b*c*d^2*e^2)/c^2/d^4/(a*d-b*c)*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2 
+c/d)^2+1/15*(b*d*x^2+a*d)/c^3/d^3/(a*d-b*c)^2*x*(3*a^2*c^2*d^2*f^2+4*a^2* 
c*d^3*e*f+8*a^2*d^4*e^2-13*a*b*c^3*d*f^2-4*a*b*c^2*d^2*e*f-13*a*b*c*d^3*e^ 
2+8*b^2*c^4*f^2+4*b^2*c^3*d*e*f+3*b^2*c^2*d^2*e^2)/((x^2+c/d)*(b*d*x^2+a*d 
))^(1/2)+(b*f^2/d^3-1/15*b*(6*a*c^2*d*f^2-2*a*c*d^2*e*f-4*a*d^3*e^2-7*b*c^ 
3*f^2+4*b*c^2*d*e*f+3*b*c*d^2*e^2)/d^3/(a*d-b*c)/c^2+1/15/d^3/(a*d-b*c)*(3 
*a^2*c^2*d^2*f^2+4*a^2*c*d^3*e*f+8*a^2*d^4*e^2-13*a*b*c^3*d*f^2-4*a*b*c^2* 
d^2*e*f-13*a*b*c*d^3*e^2+8*b^2*c^4*f^2+4*b^2*c^3*d*e*f+3*b^2*c^2*d^2*e^2)/ 
c^3-1/15*a/d^2/c^3/(a*d-b*c)^2*(3*a^2*c^2*d^2*f^2+4*a^2*c*d^3*e*f+8*a^2*d^ 
4*e^2-13*a*b*c^3*d*f^2-4*a*b*c^2*d^2*e*f-13*a*b*c*d^3*e^2+8*b^2*c^4*f^2+4* 
b^2*c^3*d*e*f+3*b^2*c^2*d^2*e^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))+1/15/d^3*b*(3*a^2*c^2*d^2*f^2+4*a^2*c*d^3*e*f+8*a^2* 
d^4*e^2-13*a*b*c^3*d*f^2-4*a*b*c^2*d^2*e*f-13*a*b*c*d^3*e^2+8*b^2*c^4*f^2+ 
4*b^2*c^3*d*e*f+3*b^2*c^2*d^2*e^2)/(a*d-b*c)^2/c^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))-EllipticE(x*(-b/a)^(1/2),(-1+(a...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1697 vs. \(2 (472) = 944\).

Time = 0.15 (sec) , antiderivative size = 1697, normalized size of antiderivative = 3.36 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(7/2),x, algorithm="fricas 
")
 

Output:

-1/15*((((3*b^3*c^2*d^5 - 13*a*b^2*c*d^6 + 8*a^2*b*d^7)*e^2 + 4*(b^3*c^3*d 
^4 - a*b^2*c^2*d^5 + a^2*b*c*d^6)*e*f + (8*b^3*c^4*d^3 - 13*a*b^2*c^3*d^4 
+ 3*a^2*b*c^2*d^5)*f^2)*x^6 + 3*((3*b^3*c^3*d^4 - 13*a*b^2*c^2*d^5 + 8*a^2 
*b*c*d^6)*e^2 + 4*(b^3*c^4*d^3 - a*b^2*c^3*d^4 + a^2*b*c^2*d^5)*e*f + (8*b 
^3*c^5*d^2 - 13*a*b^2*c^4*d^3 + 3*a^2*b*c^3*d^4)*f^2)*x^4 + (3*b^3*c^5*d^2 
 - 13*a*b^2*c^4*d^3 + 8*a^2*b*c^3*d^4)*e^2 + 4*(b^3*c^6*d - a*b^2*c^5*d^2 
+ a^2*b*c^4*d^3)*e*f + (8*b^3*c^7 - 13*a*b^2*c^6*d + 3*a^2*b*c^5*d^2)*f^2 
+ 3*((3*b^3*c^4*d^3 - 13*a*b^2*c^3*d^4 + 8*a^2*b*c^2*d^5)*e^2 + 4*(b^3*c^5 
*d^2 - a*b^2*c^4*d^3 + a^2*b*c^3*d^4)*e*f + (8*b^3*c^6*d - 13*a*b^2*c^5*d^ 
2 + 3*a^2*b*c^4*d^3)*f^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sq 
rt(-b/a)), a*d/(b*c)) - (((3*b^3*c^2*d^5 - (6*a^2*b + 13*a*b^2)*c*d^6 + 4* 
(a^3 + 2*a^2*b)*d^7)*e^2 + 2*(2*b^3*c^3*d^4 + (a^2*b - 2*a*b^2)*c^2*d^5 + 
(a^3 + 2*a^2*b)*c*d^6)*e*f + (8*b^3*c^4*d^3 + (4*a^2*b - 13*a*b^2)*c^3*d^4 
 - 3*(2*a^3 - a^2*b)*c^2*d^5)*f^2)*x^6 + 3*((3*b^3*c^3*d^4 - (6*a^2*b + 13 
*a*b^2)*c^2*d^5 + 4*(a^3 + 2*a^2*b)*c*d^6)*e^2 + 2*(2*b^3*c^4*d^3 + (a^2*b 
 - 2*a*b^2)*c^3*d^4 + (a^3 + 2*a^2*b)*c^2*d^5)*e*f + (8*b^3*c^5*d^2 + (4*a 
^2*b - 13*a*b^2)*c^4*d^3 - 3*(2*a^3 - a^2*b)*c^3*d^4)*f^2)*x^4 + (3*b^3*c^ 
5*d^2 - (6*a^2*b + 13*a*b^2)*c^4*d^3 + 4*(a^3 + 2*a^2*b)*c^3*d^4)*e^2 + 2* 
(2*b^3*c^6*d + (a^2*b - 2*a*b^2)*c^5*d^2 + (a^3 + 2*a^2*b)*c^4*d^3)*e*f + 
(8*b^3*c^7 + (4*a^2*b - 13*a*b^2)*c^6*d - 3*(2*a^3 - a^2*b)*c^5*d^2)*f^...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(f*x**2+e)**2/(d*x**2+c)**(7/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(sqrt(a + b*x**2)*(e + f*x**2)**2/(c + d*x**2)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(7/2),x, algorithm="maxima 
")
 

Output:

integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2/(d*x^2 + c)^(7/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(7/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2/(d*x^2 + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^2}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(e + f*x^2)^2)/(c + d*x^2)^(7/2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(e + f*x^2)^2)/(c + d*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{7/2}} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(7/2),x)
 

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*f**2*x - 2*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a*d*e*f*x - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f**2*x* 
*3 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*f**2*x**3 - sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*b*d*e**2*x - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x** 
4)/(2*a**2*c**4*d + 8*a**2*c**3*d**2*x**2 + 12*a**2*c**2*d**3*x**4 + 8*a** 
2*c*d**4*x**6 + 2*a**2*d**5*x**8 - a*b*c**5 - 2*a*b*c**4*d*x**2 + 2*a*b*c* 
*3*d**2*x**4 + 8*a*b*c**2*d**3*x**6 + 7*a*b*c*d**4*x**8 + 2*a*b*d**5*x**10 
 - b**2*c**5*x**2 - 4*b**2*c**4*d*x**4 - 6*b**2*c**3*d**2*x**6 - 4*b**2*c* 
*2*d**3*x**8 - b**2*c*d**4*x**10),x)*a**3*c**3*d**3*f**2 - 24*int((sqrt(c 
+ d*x**2)*sqrt(a + b*x**2)*x**4)/(2*a**2*c**4*d + 8*a**2*c**3*d**2*x**2 + 
12*a**2*c**2*d**3*x**4 + 8*a**2*c*d**4*x**6 + 2*a**2*d**5*x**8 - a*b*c**5 
- 2*a*b*c**4*d*x**2 + 2*a*b*c**3*d**2*x**4 + 8*a*b*c**2*d**3*x**6 + 7*a*b* 
c*d**4*x**8 + 2*a*b*d**5*x**10 - b**2*c**5*x**2 - 4*b**2*c**4*d*x**4 - 6*b 
**2*c**3*d**2*x**6 - 4*b**2*c**2*d**3*x**8 - b**2*c*d**4*x**10),x)*a**3*c* 
*2*d**4*f**2*x**2 - 24*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(2*a** 
2*c**4*d + 8*a**2*c**3*d**2*x**2 + 12*a**2*c**2*d**3*x**4 + 8*a**2*c*d**4* 
x**6 + 2*a**2*d**5*x**8 - a*b*c**5 - 2*a*b*c**4*d*x**2 + 2*a*b*c**3*d**2*x 
**4 + 8*a*b*c**2*d**3*x**6 + 7*a*b*c*d**4*x**8 + 2*a*b*d**5*x**10 - b**2*c 
**5*x**2 - 4*b**2*c**4*d*x**4 - 6*b**2*c**3*d**2*x**6 - 4*b**2*c**2*d**3*x 
**8 - b**2*c*d**4*x**10),x)*a**3*c*d**5*f**2*x**4 - 8*int((sqrt(c + d*x...