\(\int \frac {(c+d x^2) (e+f x^2)^3}{(a+b x^2)^{5/2}} \, dx\) [358]

Optimal result

Integrand size = 28, antiderivative size = 227 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(b c-a d) (b e-a f)^3 x}{3 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {(b e-a f)^2 \left (2 b^2 c e-10 a^2 d f+a b (d e+7 c f)\right ) x}{3 a^2 b^4 \sqrt {a+b x^2}}+\frac {f^2 (12 b d e+4 b c f-11 a d f) x \sqrt {a+b x^2}}{8 b^4}+\frac {d f^3 x^3 \sqrt {a+b x^2}}{4 b^3}+\frac {f \left (35 a^2 d f^2+24 b^2 e (d e+c f)-20 a b f (3 d e+c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}} \] Output:

1/3*(-a*d+b*c)*(-a*f+b*e)^3*x/a/b^4/(b*x^2+a)^(3/2)+1/3*(-a*f+b*e)^2*(2*b^ 
2*c*e-10*a^2*d*f+a*b*(7*c*f+d*e))*x/a^2/b^4/(b*x^2+a)^(1/2)+1/8*f^2*(-11*a 
*d*f+4*b*c*f+12*b*d*e)*x*(b*x^2+a)^(1/2)/b^4+1/4*d*f^3*x^3*(b*x^2+a)^(1/2) 
/b^3+1/8*f*(35*a^2*d*f^2+24*b^2*e*(c*f+d*e)-20*a*b*f*(c*f+3*d*e))*arctanh( 
b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.14 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (-105 a^5 d f^3+16 b^5 c e^3 x^2+20 a^4 b f^2 \left (9 d e+3 c f-7 d f x^2\right )+8 a b^4 e^2 \left (d e x^2+3 c \left (e+f x^2\right )\right )+6 a^2 b^3 f x^2 \left (2 c f \left (-8 e+f x^2\right )+d \left (-16 e^2+6 e f x^2+f^2 x^4\right )\right )+a^3 b^2 f \left (8 c f \left (-9 e+10 f x^2\right )-3 d \left (24 e^2-80 e f x^2+7 f^2 x^4\right )\right )\right )}{24 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac {f \left (-35 a^2 d f^2-24 b^2 e (d e+c f)+20 a b f (3 d e+c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{9/2}} \] Input:

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

Output:

(x*(-105*a^5*d*f^3 + 16*b^5*c*e^3*x^2 + 20*a^4*b*f^2*(9*d*e + 3*c*f - 7*d* 
f*x^2) + 8*a*b^4*e^2*(d*e*x^2 + 3*c*(e + f*x^2)) + 6*a^2*b^3*f*x^2*(2*c*f* 
(-8*e + f*x^2) + d*(-16*e^2 + 6*e*f*x^2 + f^2*x^4)) + a^3*b^2*f*(8*c*f*(-9 
*e + 10*f*x^2) - 3*d*(24*e^2 - 80*e*f*x^2 + 7*f^2*x^4))))/(24*a^2*b^4*(a + 
 b*x^2)^(3/2)) + (f*(-35*a^2*d*f^2 - 24*b^2*e*(d*e + c*f) + 20*a*b*f*(3*d* 
e + c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8*b^(9/2))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.37, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {401, 25, 401, 27, 403, 299, 224, 219}

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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {x \left (e+f x^2\right )^2 (b e (a d+2 b c)+a f (4 b c-7 a d))}{a b \sqrt {a+b x^2}}-\frac {\int \frac {f \left (f x^2+e\right ) \left (\left (-35 d f a^2+4 b (d e+5 c f) a+8 b^2 c e\right ) x^2+a (4 b c-7 a d) e\right )}{\sqrt {b x^2+a}}dx}{a b}}{3 a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {x \left (e+f x^2\right )^2 (b e (a d+2 b c)+a f (4 b c-7 a d))}{a b \sqrt {a+b x^2}}-\frac {f \int \frac {\left (f x^2+e\right ) \left (\left (-35 d f a^2+4 b (d e+5 c f) a+8 b^2 c e\right ) x^2+a (4 b c-7 a d) e\right )}{\sqrt {b x^2+a}}dx}{a b}}{3 a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {x \left (e+f x^2\right )^2 (b e (a d+2 b c)+a f (4 b c-7 a d))}{a b \sqrt {a+b x^2}}-\frac {f \left (\frac {\int \frac {\left (105 d f^2 a^3-10 b f (11 d e+6 c f) a^2+8 b^2 e (d e+4 c f) a+16 b^3 c e^2\right ) x^2+a e \left (35 d f a^2-32 b d e a-20 b c f a+8 b^2 c e\right )}{\sqrt {b x^2+a}}dx}{4 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (-35 a^2 d f+4 a b (5 c f+d e)+8 b^2 c e\right )}{4 b}\right )}{a b}}{3 a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {x \left (e+f x^2\right )^2 (b e (a d+2 b c)+a f (4 b c-7 a d))}{a b \sqrt {a+b x^2}}-\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (105 a^3 d f^2-10 a^2 b f (6 c f+11 d e)+8 a b^2 e (4 c f+d e)+16 b^3 c e^2\right )}{2 b}-\frac {3 a^2 \left (35 a^2 d f^2-20 a b f (c f+3 d e)+24 b^2 e (c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}}{4 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (-35 a^2 d f+4 a b (5 c f+d e)+8 b^2 c e\right )}{4 b}\right )}{a b}}{3 a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {x \left (e+f x^2\right )^2 (b e (a d+2 b c)+a f (4 b c-7 a d))}{a b \sqrt {a+b x^2}}-\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (105 a^3 d f^2-10 a^2 b f (6 c f+11 d e)+8 a b^2 e (4 c f+d e)+16 b^3 c e^2\right )}{2 b}-\frac {3 a^2 \left (35 a^2 d f^2-20 a b f (c f+3 d e)+24 b^2 e (c f+d e)\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}}{4 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (-35 a^2 d f+4 a b (5 c f+d e)+8 b^2 c e\right )}{4 b}\right )}{a b}}{3 a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {x \left (e+f x^2\right )^2 (b e (a d+2 b c)+a f (4 b c-7 a d))}{a b \sqrt {a+b x^2}}-\frac {f \left (\frac {x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (-35 a^2 d f+4 a b (5 c f+d e)+8 b^2 c e\right )}{4 b}+\frac {\frac {x \sqrt {a+b x^2} \left (105 a^3 d f^2-10 a^2 b f (6 c f+11 d e)+8 a b^2 e (4 c f+d e)+16 b^3 c e^2\right )}{2 b}-\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^2 d f^2-20 a b f (c f+3 d e)+24 b^2 e (c f+d e)\right )}{2 b^{3/2}}}{4 b}\right )}{a b}}{3 a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}}\)

Input:

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

Output:

((b*c - a*d)*x*(e + f*x^2)^3)/(3*a*b*(a + b*x^2)^(3/2)) + (((b*(2*b*c + a* 
d)*e + a*(4*b*c - 7*a*d)*f)*x*(e + f*x^2)^2)/(a*b*Sqrt[a + b*x^2]) - (f*(( 
(8*b^2*c*e - 35*a^2*d*f + 4*a*b*(d*e + 5*c*f))*x*Sqrt[a + b*x^2]*(e + f*x^ 
2))/(4*b) + (((16*b^3*c*e^2 + 105*a^3*d*f^2 + 8*a*b^2*e*(d*e + 4*c*f) - 10 
*a^2*b*f*(11*d*e + 6*c*f))*x*Sqrt[a + b*x^2])/(2*b) - (3*a^2*(35*a^2*d*f^2 
 + 24*b^2*e*(d*e + c*f) - 20*a*b*f*(3*d*e + c*f))*ArcTanh[(Sqrt[b]*x)/Sqrt 
[a + b*x^2]])/(2*b^(3/2)))/(4*b)))/(a*b))/(3*a*b)
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

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

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 403

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

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {\frac {35 \left (a^{2} d \,f^{2}-\frac {4 a b f \left (c f +3 d e \right )}{7}+\frac {24 b^{2} e \left (c f +d e \right )}{35}\right ) a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}} f \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}+\frac {2 x \left (-\frac {9 \left (\frac {\left (\frac {7}{8} d \,x^{4}-\frac {10}{3} c \,x^{2}\right ) f^{2}}{3}+e \left (-\frac {10 x^{2} d}{3}+c \right ) f +d \,e^{2}\right ) f \,a^{3} b^{\frac {5}{2}}}{2}-6 x^{2} f \left (-\frac {\left (\frac {x^{2} d}{2}+c \right ) x^{2} f^{2}}{8}+e \left (-\frac {3 x^{2} d}{8}+c \right ) f +d \,e^{2}\right ) a^{2} b^{\frac {7}{2}}+\frac {15 \left (\left (-\frac {7 x^{2} d}{3}+c \right ) f +3 d e \right ) f^{2} a^{4} b^{\frac {3}{2}}}{4}-\frac {105 a^{5} d \,f^{3} \sqrt {b}}{16}+b^{\frac {9}{2}} \left (\frac {3 \left (c f \,x^{2}+e \left (\frac {x^{2} d}{3}+c \right )\right ) a}{2}+b c e \,x^{2}\right ) e^{2}\right )}{3}}{b^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\)	\(251\)
default	\(c \,e^{3} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+f^{2} \left (c f +3 d e \right ) \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+3 e f \left (c f +d e \right ) \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+e^{2} \left (3 c f +d e \right ) \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )+d \,f^{3} \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )\)	\(377\)
risch	\(-\frac {f^{2} x \left (-2 b d f \,x^{2}+11 a d f -4 b c f -12 b d e \right ) \sqrt {b \,x^{2}+a}}{8 b^{4}}+\frac {\frac {f \left (35 a^{2} d \,f^{2}-20 a b c \,f^{2}-60 a b d e f +24 b^{2} c e f +24 b^{2} d \,e^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {2 \left (a^{4} d \,f^{3}-a^{3} b c \,f^{3}-3 a^{3} b d e \,f^{2}+3 a^{2} b^{2} c e \,f^{2}+3 a^{2} b^{2} d \,e^{2} f -3 a \,b^{3} c \,e^{2} f -a \,b^{3} d \,e^{3}+b^{4} c \,e^{3}\right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{b a}-\frac {2 \left (a^{4} d \,f^{3}-a^{3} b c \,f^{3}-3 a^{3} b d e \,f^{2}+3 a^{2} b^{2} c e \,f^{2}+3 a^{2} b^{2} d \,e^{2} f -3 a \,b^{3} c \,e^{2} f -a \,b^{3} d \,e^{3}+b^{4} c \,e^{3}\right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{b a}-\frac {2 \left (7 a^{4} d \,f^{3}-5 a^{3} b c \,f^{3}-15 a^{3} b d e \,f^{2}+9 a^{2} b^{2} c e \,f^{2}+9 a^{2} b^{2} d \,e^{2} f -3 a \,b^{3} c \,e^{2} f -a \,b^{3} d \,e^{3}-b^{4} c \,e^{3}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{a^{2} b \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {2 \left (7 a^{4} d \,f^{3}-5 a^{3} b c \,f^{3}-15 a^{3} b d e \,f^{2}+9 a^{2} b^{2} c e \,f^{2}+9 a^{2} b^{2} d \,e^{2} f -3 a \,b^{3} c \,e^{2} f -a \,b^{3} d \,e^{3}-b^{4} c \,e^{3}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{a^{2} b \left (x +\frac {\sqrt {-a b}}{b}\right )}}{8 b^{4}}\)	\(844\)

Input:

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

Output:

2/3/(b*x^2+a)^(3/2)*(105/16*(a^2*d*f^2-4/7*a*b*f*(c*f+3*d*e)+24/35*b^2*e*( 
c*f+d*e))*a^2*(b*x^2+a)^(3/2)*f*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))+x*(-9/2 
*(1/3*(7/8*d*x^4-10/3*c*x^2)*f^2+e*(-10/3*x^2*d+c)*f+d*e^2)*f*a^3*b^(5/2)- 
6*x^2*f*(-1/8*(1/2*x^2*d+c)*x^2*f^2+e*(-3/8*x^2*d+c)*f+d*e^2)*a^2*b^(7/2)+ 
15/4*((-7/3*x^2*d+c)*f+3*d*e)*f^2*a^4*b^(3/2)-105/16*a^5*d*f^3*b^(1/2)+b^( 
9/2)*(3/2*(c*f*x^2+e*(1/3*x^2*d+c))*a+b*c*e*x^2)*e^2))/b^(9/2)/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (203) = 406\).

Time = 0.40 (sec) , antiderivative size = 978, normalized size of antiderivative = 4.31 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/48*(3*(24*a^4*b^2*d*e^2*f + (24*a^2*b^4*d*e^2*f + 12*(2*a^2*b^4*c - 5*a 
^3*b^3*d)*e*f^2 - 5*(4*a^3*b^3*c - 7*a^4*b^2*d)*f^3)*x^4 + 12*(2*a^4*b^2*c 
 - 5*a^5*b*d)*e*f^2 - 5*(4*a^5*b*c - 7*a^6*d)*f^3 + 2*(24*a^3*b^3*d*e^2*f 
+ 12*(2*a^3*b^3*c - 5*a^4*b^2*d)*e*f^2 - 5*(4*a^4*b^2*c - 7*a^5*b*d)*f^3)* 
x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(6*a^2*b^ 
4*d*f^3*x^7 + 3*(12*a^2*b^4*d*e*f^2 + (4*a^2*b^4*c - 7*a^3*b^3*d)*f^3)*x^5 
 + 4*(2*(2*b^6*c + a*b^5*d)*e^3 + 6*(a*b^5*c - 4*a^2*b^4*d)*e^2*f - 12*(2* 
a^2*b^4*c - 5*a^3*b^3*d)*e*f^2 + 5*(4*a^3*b^3*c - 7*a^4*b^2*d)*f^3)*x^3 + 
3*(8*a*b^5*c*e^3 - 24*a^3*b^3*d*e^2*f - 12*(2*a^3*b^3*c - 5*a^4*b^2*d)*e*f 
^2 + 5*(4*a^4*b^2*c - 7*a^5*b*d)*f^3)*x)*sqrt(b*x^2 + a))/(a^2*b^7*x^4 + 2 
*a^3*b^6*x^2 + a^4*b^5), -1/24*(3*(24*a^4*b^2*d*e^2*f + (24*a^2*b^4*d*e^2* 
f + 12*(2*a^2*b^4*c - 5*a^3*b^3*d)*e*f^2 - 5*(4*a^3*b^3*c - 7*a^4*b^2*d)*f 
^3)*x^4 + 12*(2*a^4*b^2*c - 5*a^5*b*d)*e*f^2 - 5*(4*a^5*b*c - 7*a^6*d)*f^3 
 + 2*(24*a^3*b^3*d*e^2*f + 12*(2*a^3*b^3*c - 5*a^4*b^2*d)*e*f^2 - 5*(4*a^4 
*b^2*c - 7*a^5*b*d)*f^3)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) 
- (6*a^2*b^4*d*f^3*x^7 + 3*(12*a^2*b^4*d*e*f^2 + (4*a^2*b^4*c - 7*a^3*b^3* 
d)*f^3)*x^5 + 4*(2*(2*b^6*c + a*b^5*d)*e^3 + 6*(a*b^5*c - 4*a^2*b^4*d)*e^2 
*f - 12*(2*a^2*b^4*c - 5*a^3*b^3*d)*e*f^2 + 5*(4*a^3*b^3*c - 7*a^4*b^2*d)* 
f^3)*x^3 + 3*(8*a*b^5*c*e^3 - 24*a^3*b^3*d*e^2*f - 12*(2*a^3*b^3*c - 5*a^4 
*b^2*d)*e*f^2 + 5*(4*a^4*b^2*c - 7*a^5*b*d)*f^3)*x)*sqrt(b*x^2 + a))/(a...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (203) = 406\).

Time = 0.04 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.09 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d f^{3} x^{7}}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {7 \, a d f^{3} x^{5}}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - \frac {35 \, a^{2} d f^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{24 \, b^{2}} + \frac {{\left (3 \, d e f^{2} + c f^{3}\right )} x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - {\left (d e^{2} f + c e f^{2}\right )} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} a x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{6 \, b} + \frac {2 \, c e^{3} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c e^{3} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {35 \, a^{2} d f^{3} x}{24 \, \sqrt {b x^{2} + a} b^{4}} + \frac {35 \, a^{2} d f^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {9}{2}}} + \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} a x}{6 \, \sqrt {b x^{2} + a} b^{3}} - \frac {{\left (d e^{2} f + c e f^{2}\right )} x}{\sqrt {b x^{2} + a} b^{2}} - \frac {{\left (d e^{3} + 3 \, c e^{2} f\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {{\left (d e^{3} + 3 \, c e^{2} f\right )} x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{2}}} + \frac {3 \, {\left (d e^{2} f + c e f^{2}\right )} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} \] Input:

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

Output:

1/4*d*f^3*x^7/((b*x^2 + a)^(3/2)*b) - 7/8*a*d*f^3*x^5/((b*x^2 + a)^(3/2)*b 
^2) - 35/24*a^2*d*f^3*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3 
/2)*b^2))/b^2 + 1/2*(3*d*e*f^2 + c*f^3)*x^5/((b*x^2 + a)^(3/2)*b) - (d*e^2 
*f + c*e*f^2)*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2) 
) + 5/6*(3*d*e*f^2 + c*f^3)*a*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 
 + a)^(3/2)*b^2))/b + 2/3*c*e^3*x/(sqrt(b*x^2 + a)*a^2) + 1/3*c*e^3*x/((b* 
x^2 + a)^(3/2)*a) - 35/24*a^2*d*f^3*x/(sqrt(b*x^2 + a)*b^4) + 35/8*a^2*d*f 
^3*arcsinh(b*x/sqrt(a*b))/b^(9/2) + 5/6*(3*d*e*f^2 + c*f^3)*a*x/(sqrt(b*x^ 
2 + a)*b^3) - (d*e^2*f + c*e*f^2)*x/(sqrt(b*x^2 + a)*b^2) - 1/3*(d*e^3 + 3 
*c*e^2*f)*x/((b*x^2 + a)^(3/2)*b) + 1/3*(d*e^3 + 3*c*e^2*f)*x/(sqrt(b*x^2 
+ a)*a*b) - 5/2*(3*d*e*f^2 + c*f^3)*a*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 3*( 
d*e^2*f + c*e*f^2)*arcsinh(b*x/sqrt(a*b))/b^(5/2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.48 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (\frac {2 \, d f^{3} x^{2}}{b} + \frac {12 \, a^{2} b^{6} d e f^{2} + 4 \, a^{2} b^{6} c f^{3} - 7 \, a^{3} b^{5} d f^{3}}{a^{2} b^{7}}\right )} x^{2} + \frac {4 \, {\left (4 \, b^{8} c e^{3} + 2 \, a b^{7} d e^{3} + 6 \, a b^{7} c e^{2} f - 24 \, a^{2} b^{6} d e^{2} f - 24 \, a^{2} b^{6} c e f^{2} + 60 \, a^{3} b^{5} d e f^{2} + 20 \, a^{3} b^{5} c f^{3} - 35 \, a^{4} b^{4} d f^{3}\right )}}{a^{2} b^{7}}\right )} x^{2} + \frac {3 \, {\left (8 \, a b^{7} c e^{3} - 24 \, a^{3} b^{5} d e^{2} f - 24 \, a^{3} b^{5} c e f^{2} + 60 \, a^{4} b^{4} d e f^{2} + 20 \, a^{4} b^{4} c f^{3} - 35 \, a^{5} b^{3} d f^{3}\right )}}{a^{2} b^{7}}\right )} x}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (24 \, b^{2} d e^{2} f + 24 \, b^{2} c e f^{2} - 60 \, a b d e f^{2} - 20 \, a b c f^{3} + 35 \, a^{2} d f^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {9}{2}}} \] Input:

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

Output:

1/24*((3*(2*d*f^3*x^2/b + (12*a^2*b^6*d*e*f^2 + 4*a^2*b^6*c*f^3 - 7*a^3*b^ 
5*d*f^3)/(a^2*b^7))*x^2 + 4*(4*b^8*c*e^3 + 2*a*b^7*d*e^3 + 6*a*b^7*c*e^2*f 
 - 24*a^2*b^6*d*e^2*f - 24*a^2*b^6*c*e*f^2 + 60*a^3*b^5*d*e*f^2 + 20*a^3*b 
^5*c*f^3 - 35*a^4*b^4*d*f^3)/(a^2*b^7))*x^2 + 3*(8*a*b^7*c*e^3 - 24*a^3*b^ 
5*d*e^2*f - 24*a^3*b^5*c*e*f^2 + 60*a^4*b^4*d*e*f^2 + 20*a^4*b^4*c*f^3 - 3 
5*a^5*b^3*d*f^3)/(a^2*b^7))*x/(b*x^2 + a)^(3/2) - 1/8*(24*b^2*d*e^2*f + 24 
*b^2*c*e*f^2 - 60*a*b*d*e*f^2 - 20*a*b*c*f^3 + 35*a^2*d*f^3)*log(abs(-sqrt 
(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 10.56 (sec) , antiderivative size = 1229, normalized size of antiderivative = 5.41 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 840*sqrt(a + b*x**2)*a**5*b*d*f**3*x + 480*sqrt(a + b*x**2)*a**4*b**2* 
c*f**3*x + 1440*sqrt(a + b*x**2)*a**4*b**2*d*e*f**2*x - 1120*sqrt(a + b*x* 
*2)*a**4*b**2*d*f**3*x**3 - 576*sqrt(a + b*x**2)*a**3*b**3*c*e*f**2*x + 64 
0*sqrt(a + b*x**2)*a**3*b**3*c*f**3*x**3 - 576*sqrt(a + b*x**2)*a**3*b**3* 
d*e**2*f*x + 1920*sqrt(a + b*x**2)*a**3*b**3*d*e*f**2*x**3 - 168*sqrt(a + 
b*x**2)*a**3*b**3*d*f**3*x**5 - 768*sqrt(a + b*x**2)*a**2*b**4*c*e*f**2*x* 
*3 + 96*sqrt(a + b*x**2)*a**2*b**4*c*f**3*x**5 - 768*sqrt(a + b*x**2)*a**2 
*b**4*d*e**2*f*x**3 + 288*sqrt(a + b*x**2)*a**2*b**4*d*e*f**2*x**5 + 48*sq 
rt(a + b*x**2)*a**2*b**4*d*f**3*x**7 + 192*sqrt(a + b*x**2)*a*b**5*c*e**3* 
x + 192*sqrt(a + b*x**2)*a*b**5*c*e**2*f*x**3 + 64*sqrt(a + b*x**2)*a*b**5 
*d*e**3*x**3 + 128*sqrt(a + b*x**2)*b**6*c*e**3*x**3 + 840*sqrt(b)*log((sq 
rt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**6*d*f**3 - 480*sqrt(b)*log((sqrt(a 
 + b*x**2) + sqrt(b)*x)/sqrt(a))*a**5*b*c*f**3 - 1440*sqrt(b)*log((sqrt(a 
+ b*x**2) + sqrt(b)*x)/sqrt(a))*a**5*b*d*e*f**2 + 1680*sqrt(b)*log((sqrt(a 
 + b*x**2) + sqrt(b)*x)/sqrt(a))*a**5*b*d*f**3*x**2 + 576*sqrt(b)*log((sqr 
t(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b**2*c*e*f**2 - 960*sqrt(b)*log(( 
sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b**2*c*f**3*x**2 + 576*sqrt(b) 
*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b**2*d*e**2*f - 2880*sqr 
t(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b**2*d*e*f**2*x**2 + 
 840*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b**2*d*f*...