\(\int \frac {(a+b x^2)^{3/2} (c+d x^2)}{(e+f x^2)^5} \, dx\) [297]

Optimal result

Integrand size = 28, antiderivative size = 379 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx=-\frac {(d e-c f) x \left (a+b x^2\right )^{3/2}}{8 e f \left (e+f x^2\right )^4}-\frac {(4 b e (d e+c f)-a f (d e+7 c f)) x \sqrt {a+b x^2}}{48 e^2 f^2 \left (e+f x^2\right )^3}+\frac {\left (24 a b c e f^2+8 b^2 e^2 (d e+c f)-5 a^2 f^2 (d e+7 c f)\right ) x \sqrt {a+b x^2}}{192 e^3 f^2 (b e-a f) \left (e+f x^2\right )^2}-\frac {\left (8 a b^2 e^2 f (d e-5 c f)-16 b^3 e^3 (d e+c f)-15 a^3 f^3 (d e+7 c f)+2 a^2 b e f^2 (7 d e+85 c f)\right ) x \sqrt {a+b x^2}}{384 e^4 f^2 (b e-a f)^2 \left (e+f x^2\right )}+\frac {a^2 \left (48 b^2 c e^2+5 a^2 f (d e+7 c f)-8 a b e (d e+10 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{128 e^{9/2} (b e-a f)^{5/2}} \] Output:

-1/8*(-c*f+d*e)*x*(b*x^2+a)^(3/2)/e/f/(f*x^2+e)^4-1/48*(4*b*e*(c*f+d*e)-a* 
f*(7*c*f+d*e))*x*(b*x^2+a)^(1/2)/e^2/f^2/(f*x^2+e)^3+1/192*(24*a*b*c*e*f^2 
+8*b^2*e^2*(c*f+d*e)-5*a^2*f^2*(7*c*f+d*e))*x*(b*x^2+a)^(1/2)/e^3/f^2/(-a* 
f+b*e)/(f*x^2+e)^2-1/384*(8*a*b^2*e^2*f*(-5*c*f+d*e)-16*b^3*e^3*(c*f+d*e)- 
15*a^3*f^3*(7*c*f+d*e)+2*a^2*b*e*f^2*(85*c*f+7*d*e))*x*(b*x^2+a)^(1/2)/e^4 
/f^2/(-a*f+b*e)^2/(f*x^2+e)+1/128*a^2*(48*b^2*c*e^2+5*a^2*f*(7*c*f+d*e)-8* 
a*b*e*(10*c*f+d*e))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2))/e^ 
(9/2)/(-a*f+b*e)^(5/2)
 

Mathematica [A] (verified)

Time = 12.75 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx=\frac {x \left (8 d e (b e-a f) \left (a+b x^2\right ) \left (e+f x^2\right ) \left (e \left (a+b x^2\right ) \left (4 b^2 e^2 x^2 \left (3 e+f x^2\right )+2 a b e \left (15 e^2+11 e f x^2+4 f^2 x^4\right )-a^2 f \left (33 e^2+40 e f x^2+15 f^2 x^4\right )\right )+\frac {3 a^2 (6 b e-5 a f) \left (e+f x^2\right )^3 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )+a (-d e+c f) \left (1+\frac {b x^2}{a}\right ) \left (e \left (a+b x^2\right ) \left (16 b^3 e^3 x^2 \left (6 e^2+4 e f x^2+f^2 x^4\right )+8 a b^2 e^2 \left (30 e^3+26 e^2 f x^2+19 e f^2 x^4+5 f^3 x^6\right )-2 a^2 b e f \left (264 e^3+421 e^2 f x^2+314 e f^2 x^4+85 f^3 x^6\right )+a^3 f^2 \left (279 e^3+511 e^2 f x^2+385 e f^2 x^4+105 f^3 x^6\right )\right )+\frac {3 a^2 \left (48 b^2 e^2-80 a b e f+35 a^2 f^2\right ) \left (e+f x^2\right )^4 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )\right )}{384 e^5 f (b e-a f)^2 \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^4} \] Input:

Integrate[((a + b*x^2)^(3/2)*(c + d*x^2))/(e + f*x^2)^5,x]
 

Output:

(x*(8*d*e*(b*e - a*f)*(a + b*x^2)*(e + f*x^2)*(e*(a + b*x^2)*(4*b^2*e^2*x^ 
2*(3*e + f*x^2) + 2*a*b*e*(15*e^2 + 11*e*f*x^2 + 4*f^2*x^4) - a^2*f*(33*e^ 
2 + 40*e*f*x^2 + 15*f^2*x^4)) + (3*a^2*(6*b*e - 5*a*f)*(e + f*x^2)^3*ArcTa 
nh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/Sqrt[((b*e - a*f)*x^2)/(e*(a 
+ b*x^2))]) + a*(-(d*e) + c*f)*(1 + (b*x^2)/a)*(e*(a + b*x^2)*(16*b^3*e^3* 
x^2*(6*e^2 + 4*e*f*x^2 + f^2*x^4) + 8*a*b^2*e^2*(30*e^3 + 26*e^2*f*x^2 + 1 
9*e*f^2*x^4 + 5*f^3*x^6) - 2*a^2*b*e*f*(264*e^3 + 421*e^2*f*x^2 + 314*e*f^ 
2*x^4 + 85*f^3*x^6) + a^3*f^2*(279*e^3 + 511*e^2*f*x^2 + 385*e*f^2*x^4 + 1 
05*f^3*x^6)) + (3*a^2*(48*b^2*e^2 - 80*a*b*e*f + 35*a^2*f^2)*(e + f*x^2)^4 
*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]])/Sqrt[((b*e - a*f)*x^2)/ 
(e*(a + b*x^2))])))/(384*e^5*f*(b*e - a*f)^2*(a + b*x^2)^(3/2)*(e + f*x^2) 
^4)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {401, 25, 401, 25, 402, 402, 27, 291, 221}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx\)

\(\Big \downarrow \) 401

\(\displaystyle -\frac {\int -\frac {\sqrt {b x^2+a} \left (4 b (d e+c f) x^2+a (d e+7 c f)\right )}{\left (f x^2+e\right )^4}dx}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left (4 b (d e+c f) x^2+a (d e+7 c f)\right )}{\left (f x^2+e\right )^4}dx}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {-\frac {\int -\frac {4 b (2 b e (d e+c f)+a f (d e+7 c f)) x^2+a (4 b e (d e+c f)+5 a f (d e+7 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e f}-\frac {x \sqrt {a+b x^2} (4 b e (c f+d e)-a f (7 c f+d e))}{6 e f \left (e+f x^2\right )^3}}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {4 b (2 b e (d e+c f)+a f (d e+7 c f)) x^2+a (4 b e (d e+c f)+5 a f (d e+7 c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{6 e f}-\frac {x \sqrt {a+b x^2} (4 b e (c f+d e)-a f (7 c f+d e))}{6 e f \left (e+f x^2\right )^3}}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {\int \frac {2 b \left (8 b^2 (d e+c f) e^2+24 a b c f^2 e-5 a^2 f^2 (d e+7 c f)\right ) x^2+a \left (8 b^2 (d e+c f) e^2+4 a b f (d e+25 c f) e-15 a^2 f^2 (d e+7 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (-5 a^2 f^2 (7 c f+d e)+24 a b c e f^2+8 b^2 e^2 (c f+d e)\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (4 b e (c f+d e)-a f (7 c f+d e))}{6 e f \left (e+f x^2\right )^3}}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 a^2 f^2 \left (5 f (d e+7 c f) a^2-8 b e (d e+10 c f) a+48 b^2 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (-15 a^3 f^3 (7 c f+d e)+2 a^2 b e f^2 (85 c f+7 d e)+8 a b^2 e^2 f (d e-5 c f)-16 b^3 e^3 (c f+d e)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (-5 a^2 f^2 (7 c f+d e)+24 a b c e f^2+8 b^2 e^2 (c f+d e)\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (4 b e (c f+d e)-a f (7 c f+d e))}{6 e f \left (e+f x^2\right )^3}}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 f^2 \left (5 a^2 f (7 c f+d e)-8 a b e (10 c f+d e)+48 b^2 c e^2\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (-15 a^3 f^3 (7 c f+d e)+2 a^2 b e f^2 (85 c f+7 d e)+8 a b^2 e^2 f (d e-5 c f)-16 b^3 e^3 (c f+d e)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (-5 a^2 f^2 (7 c f+d e)+24 a b c e f^2+8 b^2 e^2 (c f+d e)\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (4 b e (c f+d e)-a f (7 c f+d e))}{6 e f \left (e+f x^2\right )^3}}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 f^2 \left (5 a^2 f (7 c f+d e)-8 a b e (10 c f+d e)+48 b^2 c e^2\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} \left (-15 a^3 f^3 (7 c f+d e)+2 a^2 b e f^2 (85 c f+7 d e)+8 a b^2 e^2 f (d e-5 c f)-16 b^3 e^3 (c f+d e)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (-5 a^2 f^2 (7 c f+d e)+24 a b c e f^2+8 b^2 e^2 (c f+d e)\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (4 b e (c f+d e)-a f (7 c f+d e))}{6 e f \left (e+f x^2\right )^3}}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {x \sqrt {a+b x^2} \left (-5 a^2 f^2 (7 c f+d e)+24 a b c e f^2+8 b^2 e^2 (c f+d e)\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {3 a^2 f^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (5 a^2 f (7 c f+d e)-8 a b e (10 c f+d e)+48 b^2 c e^2\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} \left (-15 a^3 f^3 (7 c f+d e)+2 a^2 b e f^2 (85 c f+7 d e)+8 a b^2 e^2 f (d e-5 c f)-16 b^3 e^3 (c f+d e)\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}}{6 e f}-\frac {x \sqrt {a+b x^2} (4 b e (c f+d e)-a f (7 c f+d e))}{6 e f \left (e+f x^2\right )^3}}{8 e f}-\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{8 e f \left (e+f x^2\right )^4}\)

Input:

Int[((a + b*x^2)^(3/2)*(c + d*x^2))/(e + f*x^2)^5,x]
 

Output:

-1/8*((d*e - c*f)*x*(a + b*x^2)^(3/2))/(e*f*(e + f*x^2)^4) + (-1/6*((4*b*e 
*(d*e + c*f) - a*f*(d*e + 7*c*f))*x*Sqrt[a + b*x^2])/(e*f*(e + f*x^2)^3) + 
 (((24*a*b*c*e*f^2 + 8*b^2*e^2*(d*e + c*f) - 5*a^2*f^2*(d*e + 7*c*f))*x*Sq 
rt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + (-1/2*((8*a*b^2*e^2*f*(d* 
e - 5*c*f) - 16*b^3*e^3*(d*e + c*f) - 15*a^3*f^3*(d*e + 7*c*f) + 2*a^2*b*e 
*f^2*(7*d*e + 85*c*f))*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)) + (3 
*a^2*f^2*(48*b^2*c*e^2 + 5*a^2*f*(d*e + 7*c*f) - 8*a*b*e*(d*e + 10*c*f))*A 
rcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a 
*f)^(3/2)))/(4*e*(b*e - a*f)))/(6*e*f))/(8*e*f)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 401

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(-\frac {35 \left (a^{2} \left (-\frac {8 b \left (a d -6 b c \right ) e^{2}}{35}+\frac {f a \left (a d -16 b c \right ) e}{7}+a^{2} c \,f^{2}\right ) \left (f \,x^{2}+e \right )^{4} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {93 \left (\frac {8 \left (d \,a^{2}+10 \left (\frac {7 x^{2} d}{15}+c \right ) b a +4 \left (\frac {2 x^{2} d}{3}+c \right ) b^{2} x^{2}\right ) b \,e^{5}}{93}-\frac {5 \left (a^{3} d +\frac {176 b \left (\frac {79 x^{2} d}{264}+c \right ) a^{2}}{5}-\frac {208 \left (-\frac {5 x^{2} d}{26}+c \right ) b^{2} x^{2} a}{15}-\frac {64 \left (\frac {x^{2} d}{4}+c \right ) b^{3} x^{4}}{15}\right ) f \,e^{4}}{93}+f^{2} \left (\left (\frac {73 x^{2} d}{279}+c \right ) a^{3}-\frac {842 \left (\frac {26 x^{2} d}{421}+c \right ) b \,x^{2} a^{2}}{279}+\frac {152 b^{2} x^{4} \left (-\frac {x^{2} d}{19}+c \right ) a}{279}+\frac {16 b^{3} c \,x^{6}}{279}\right ) e^{3}+\frac {511 a \left (\left (\frac {55 x^{2} d}{511}+c \right ) a^{2}-\frac {628 \left (\frac {7 x^{2} d}{314}+c \right ) b \,x^{2} a}{511}+\frac {40 b^{2} c \,x^{4}}{511}\right ) x^{2} f^{3} e^{2}}{279}+\frac {385 a^{2} x^{4} \left (\left (\frac {3 x^{2} d}{77}+c \right ) a -\frac {34 x^{2} b c}{77}\right ) f^{4} e}{279}+\frac {35 a^{3} c \,f^{5} x^{6}}{93}\right ) \sqrt {\left (a f -b e \right ) e}\, \sqrt {b \,x^{2}+a}\, x}{35}\right )}{128 \sqrt {\left (a f -b e \right ) e}\, \left (f \,x^{2}+e \right )^{4} \left (a f -b e \right )^{2} e^{4}}\)	\(392\)
default	\(\text {Expression too large to display}\)	\(22590\)

Input:

int((b*x^2+a)^(3/2)*(d*x^2+c)/(f*x^2+e)^5,x,method=_RETURNVERBOSE)
 

Output:

-35/128/((a*f-b*e)*e)^(1/2)*(a^2*(-8/35*b*(a*d-6*b*c)*e^2+1/7*f*a*(a*d-16* 
b*c)*e+a^2*c*f^2)*(f*x^2+e)^4*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/ 
2))-93/35*(8/93*(d*a^2+10*(7/15*x^2*d+c)*b*a+4*(2/3*x^2*d+c)*b^2*x^2)*b*e^ 
5-5/93*(a^3*d+176/5*b*(79/264*x^2*d+c)*a^2-208/15*(-5/26*x^2*d+c)*b^2*x^2* 
a-64/15*(1/4*x^2*d+c)*b^3*x^4)*f*e^4+f^2*((73/279*x^2*d+c)*a^3-842/279*(26 
/421*x^2*d+c)*b*x^2*a^2+152/279*b^2*x^4*(-1/19*x^2*d+c)*a+16/279*b^3*c*x^6 
)*e^3+511/279*a*((55/511*x^2*d+c)*a^2-628/511*(7/314*x^2*d+c)*b*x^2*a+40/5 
11*b^2*c*x^4)*x^2*f^3*e^2+385/279*a^2*x^4*((3/77*x^2*d+c)*a-34/77*x^2*b*c) 
*f^4*e+35/93*a^3*c*f^5*x^6)*((a*f-b*e)*e)^(1/2)*(b*x^2+a)^(1/2)*x)/(f*x^2+ 
e)^4/(a*f-b*e)^2/e^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1082 vs. \(2 (351) = 702\).

Time = 11.00 (sec) , antiderivative size = 2204, normalized size of antiderivative = 5.82 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(d*x^2+c)/(f*x^2+e)^5,x, algorithm="fricas")
 

Output:

[1/1536*(3*(35*a^4*c*e^4*f^2 + (35*a^4*c*f^6 + 8*(6*a^2*b^2*c - a^3*b*d)*e 
^2*f^4 - 5*(16*a^3*b*c - a^4*d)*e*f^5)*x^8 + 8*(6*a^2*b^2*c - a^3*b*d)*e^6 
 - 5*(16*a^3*b*c - a^4*d)*e^5*f + 4*(35*a^4*c*e*f^5 + 8*(6*a^2*b^2*c - a^3 
*b*d)*e^3*f^3 - 5*(16*a^3*b*c - a^4*d)*e^2*f^4)*x^6 + 6*(35*a^4*c*e^2*f^4 
+ 8*(6*a^2*b^2*c - a^3*b*d)*e^4*f^2 - 5*(16*a^3*b*c - a^4*d)*e^3*f^3)*x^4 
+ 4*(35*a^4*c*e^3*f^3 + 8*(6*a^2*b^2*c - a^3*b*d)*e^5*f - 5*(16*a^3*b*c - 
a^4*d)*e^4*f^2)*x^2)*sqrt(b*e^2 - a*e*f)*log(((8*b^2*e^2 - 8*a*b*e*f + a^2 
*f^2)*x^4 + a^2*e^2 + 2*(4*a*b*e^2 - 3*a^2*e*f)*x^2 + 4*((2*b*e - a*f)*x^3 
 + a*e*x)*sqrt(b*e^2 - a*e*f)*sqrt(b*x^2 + a))/(f^2*x^4 + 2*e*f*x^2 + e^2) 
) + 4*((16*b^4*d*e^6*f - 105*a^4*c*e*f^6 + 8*(2*b^4*c - 3*a*b^3*d)*e^5*f^2 
 + 6*(4*a*b^3*c - a^2*b^2*d)*e^4*f^3 - (210*a^2*b^2*c - 29*a^3*b*d)*e^3*f^ 
4 + 5*(55*a^3*b*c - 3*a^4*d)*e^2*f^5)*x^7 + (64*b^4*d*e^7 - 385*a^4*c*e^2* 
f^5 + 8*(8*b^4*c - 13*a*b^3*d)*e^6*f + 4*(22*a*b^3*c - 3*a^2*b^2*d)*e^5*f^ 
2 - (780*a^2*b^2*c - 107*a^3*b*d)*e^4*f^3 + (1013*a^3*b*c - 55*a^4*d)*e^3* 
f^4)*x^5 - (511*a^4*c*e^3*f^4 - 16*(6*b^4*c + 7*a*b^3*d)*e^7 - 2*(56*a*b^3 
*c - 135*a^2*b^2*d)*e^6*f + 21*(50*a^2*b^2*c - 11*a^3*b*d)*e^5*f^2 - (1353 
*a^3*b*c - 73*a^4*d)*e^4*f^3)*x^3 - 3*(93*a^4*c*e^4*f^3 - 8*(10*a*b^3*c + 
a^2*b^2*d)*e^7 + (256*a^2*b^2*c + 13*a^3*b*d)*e^6*f - (269*a^3*b*c + 5*a^4 
*d)*e^5*f^2)*x)*sqrt(b*x^2 + a))/(b^3*e^12 - 3*a*b^2*e^11*f + 3*a^2*b*e^10 
*f^2 - a^3*e^9*f^3 + (b^3*e^8*f^4 - 3*a*b^2*e^7*f^5 + 3*a^2*b*e^6*f^6 -...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)*(d*x**2+c)/(f*x**2+e)**5,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}}{{\left (f x^{2} + e\right )}^{5}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(d*x^2+c)/(f*x^2+e)^5,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)*(d*x^2 + c)/(f*x^2 + e)^5, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3048 vs. \(2 (351) = 702\).

Time = 0.52 (sec) , antiderivative size = 3048, normalized size of antiderivative = 8.04 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(d*x^2+c)/(f*x^2+e)^5,x, algorithm="giac")
 

Output:

-1/128*(48*a^2*b^(5/2)*c*e^2 - 8*a^3*b^(3/2)*d*e^2 - 80*a^3*b^(3/2)*c*e*f 
+ 5*a^4*sqrt(b)*d*e*f + 35*a^4*sqrt(b)*c*f^2)*arctan(1/2*((sqrt(b)*x - sqr 
t(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b^2*e^6 - 2*a 
*b*e^5*f + a^2*e^4*f^2)*sqrt(-b^2*e^2 + a*b*e*f)) - 1/192*(144*(sqrt(b)*x 
- sqrt(b*x^2 + a))^14*a^2*b^(5/2)*c*e^2*f^6 - 24*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^14*a^3*b^(3/2)*d*e^2*f^6 - 240*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^3*b 
^(3/2)*c*e*f^7 + 15*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^4*sqrt(b)*d*e*f^7 + 
 105*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^4*sqrt(b)*c*f^8 - 768*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^12*b^(11/2)*d*e^6*f^2 + 1536*(sqrt(b)*x - sqrt(b*x^2 + a 
))^12*a*b^(9/2)*d*e^5*f^3 - 768*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^2*b^(7/ 
2)*d*e^4*f^4 + 2016*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^2*b^(7/2)*c*e^3*f^5 
 - 336*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(5/2)*d*e^3*f^5 - 4368*(sqrt 
(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(5/2)*c*e^2*f^6 + 378*(sqrt(b)*x - sqrt( 
b*x^2 + a))^12*a^4*b^(3/2)*d*e^2*f^6 + 3150*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
12*a^4*b^(3/2)*c*e*f^7 - 105*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^5*sqrt(b)* 
d*e*f^7 - 735*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^5*sqrt(b)*c*f^8 - 2048*(s 
qrt(b)*x - sqrt(b*x^2 + a))^10*b^(13/2)*d*e^7*f - 2048*(sqrt(b)*x - sqrt(b 
*x^2 + a))^10*b^(13/2)*c*e^6*f^2 + 4096*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a 
*b^(11/2)*d*e^6*f^2 + 4096*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(11/2)*c*e 
^5*f^3 - 2048*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(9/2)*d*e^5*f^3 + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (d\,x^2+c\right )}{{\left (f\,x^2+e\right )}^5} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(c + d*x^2))/(e + f*x^2)^5,x)
 

Output:

int(((a + b*x^2)^(3/2)*(c + d*x^2))/(e + f*x^2)^5, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )}{\left (e+f x^2\right )^5} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )}{\left (f \,x^{2}+e \right )^{5}}d x \] Input:

int((b*x^2+a)^(3/2)*(d*x^2+c)/(f*x^2+e)^5,x)
 

Output:

int((b*x^2+a)^(3/2)*(d*x^2+c)/(f*x^2+e)^5,x)