\(\int \frac {(c+d x^2) (e+f x^2)^3}{(a+b x^2)^{3/2}} \, dx\) [337]

Optimal result

Integrand size = 28, antiderivative size = 245 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {(b c-a d) (b e-a f)^3 x}{a b^4 \sqrt {a+b x^2}}+\frac {f \left (19 a^2 d f^2+24 b^2 e (d e+c f)-14 a b f (3 d e+c f)\right ) x \sqrt {a+b x^2}}{16 b^4}+\frac {f^2 (18 b d e+6 b c f-11 a d f) x^3 \sqrt {a+b x^2}}{24 b^3}+\frac {d f^3 x^5 \sqrt {a+b x^2}}{6 b^2}-\frac {\left (35 a^3 d f^3+72 a b^2 e f (d e+c f)-30 a^2 b f^2 (3 d e+c f)-16 b^3 e^2 (d e+3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}} \] Output:

(-a*d+b*c)*(-a*f+b*e)^3*x/a/b^4/(b*x^2+a)^(1/2)+1/16*f*(19*a^2*d*f^2+24*b^ 
2*e*(c*f+d*e)-14*a*b*f*(c*f+3*d*e))*x*(b*x^2+a)^(1/2)/b^4+1/24*f^2*(-11*a* 
d*f+6*b*c*f+18*b*d*e)*x^3*(b*x^2+a)^(1/2)/b^3+1/6*d*f^3*x^5*(b*x^2+a)^(1/2 
)/b^2-1/16*(35*a^3*d*f^3+72*a*b^2*e*f*(c*f+d*e)-30*a^2*b*f^2*(c*f+3*d*e)-1 
6*b^3*e^2*(3*c*f+d*e))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {b} x \left (48 b^4 c e^3+105 a^4 d f^3+5 a^3 b f^2 \left (-54 d e-18 c f+7 d f x^2\right )+2 a^2 b^2 f \left (3 c f \left (36 e-5 f x^2\right )+d \left (108 e^2-45 e f x^2-7 f^2 x^4\right )\right )+4 a b^3 \left (3 c f \left (-12 e^2+6 e f x^2+f^2 x^4\right )+d \left (-12 e^3+18 e^2 f x^2+9 e f^2 x^4+2 f^3 x^6\right )\right )\right )}{a \sqrt {a+b x^2}}+3 \left (35 a^3 d f^3+72 a b^2 e f (d e+c f)-30 a^2 b f^2 (3 d e+c f)-16 b^3 e^2 (d e+3 c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{9/2}} \] Input:

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

Output:

((Sqrt[b]*x*(48*b^4*c*e^3 + 105*a^4*d*f^3 + 5*a^3*b*f^2*(-54*d*e - 18*c*f 
+ 7*d*f*x^2) + 2*a^2*b^2*f*(3*c*f*(36*e - 5*f*x^2) + d*(108*e^2 - 45*e*f*x 
^2 - 7*f^2*x^4)) + 4*a*b^3*(3*c*f*(-12*e^2 + 6*e*f*x^2 + f^2*x^4) + d*(-12 
*e^3 + 18*e^2*f*x^2 + 9*e*f^2*x^4 + 2*f^3*x^6))))/(a*Sqrt[a + b*x^2]) + 3* 
(35*a^3*d*f^3 + 72*a*b^2*e*f*(d*e + c*f) - 30*a^2*b*f^2*(3*d*e + c*f) - 16 
*b^3*e^2*(d*e + 3*c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(48*b^(9/2))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.27, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {401, 25, 403, 25, 403, 25, 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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\int -\frac {\left (f x^2+e\right ) \left (f \left (35 d f a^2-2 b (17 d e+15 c f) a+24 b^2 c e\right ) x^2+a e (7 a d f-6 b (d e+c f))\right )}{\sqrt {b x^2+a}}dx}{6 b}-\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (6 b c-7 a d)}{6 b}}{a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\int \frac {\left (f x^2+e\right ) \left (f \left (35 d f a^2-2 b (17 d e+15 c f) a+24 b^2 c e\right ) x^2+a e (7 a d f-6 b (d e+c f))\right )}{\sqrt {b x^2+a}}dx}{6 b}-\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (6 b c-7 a d)}{6 b}}{a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {-\frac {\frac {\int -\frac {a e \left (24 e (d e+2 c f) b^2-2 a f (31 d e+15 c f) b+35 a^2 d f^2\right )-f \left (-105 d f^2 a^3+10 b f (20 d e+9 c f) a^2-4 b^2 e (23 d e+39 c f) a+48 b^3 c e^2\right ) x^2}{\sqrt {b x^2+a}}dx}{4 b}+\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f-30 a b c f-34 a b d e+24 b^2 c e\right )}{4 b}}{6 b}-\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (6 b c-7 a d)}{6 b}}{a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f-30 a b c f-34 a b d e+24 b^2 c e\right )}{4 b}-\frac {\int \frac {a e \left (24 e (d e+2 c f) b^2-2 a f (31 d e+15 c f) b+35 a^2 d f^2\right )-f \left (-105 d f^2 a^3+10 b f (20 d e+9 c f) a^2-4 b^2 e (23 d e+39 c f) a+48 b^3 c e^2\right ) x^2}{\sqrt {b x^2+a}}dx}{4 b}}{6 b}-\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (6 b c-7 a d)}{6 b}}{a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {-\frac {\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f-30 a b c f-34 a b d e+24 b^2 c e\right )}{4 b}-\frac {-\frac {3 a \left (35 a^3 d f^3-30 a^2 b f^2 (c f+3 d e)+72 a b^2 e f (c f+d e)-16 b^3 e^2 (3 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}-\frac {f x \sqrt {a+b x^2} \left (-105 a^3 d f^2+10 a^2 b f (9 c f+20 d e)-4 a b^2 e (39 c f+23 d e)+48 b^3 c e^2\right )}{2 b}}{4 b}}{6 b}-\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (6 b c-7 a d)}{6 b}}{a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f-30 a b c f-34 a b d e+24 b^2 c e\right )}{4 b}-\frac {-\frac {3 a \left (35 a^3 d f^3-30 a^2 b f^2 (c f+3 d e)+72 a b^2 e f (c f+d e)-16 b^3 e^2 (3 c f+d e)\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}-\frac {f x \sqrt {a+b x^2} \left (-105 a^3 d f^2+10 a^2 b f (9 c f+20 d e)-4 a b^2 e (39 c f+23 d e)+48 b^3 c e^2\right )}{2 b}}{4 b}}{6 b}-\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (6 b c-7 a d)}{6 b}}{a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f-30 a b c f-34 a b d e+24 b^2 c e\right )}{4 b}-\frac {-\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^3 d f^3-30 a^2 b f^2 (c f+3 d e)+72 a b^2 e f (c f+d e)-16 b^3 e^2 (3 c f+d e)\right )}{2 b^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (-105 a^3 d f^2+10 a^2 b f (9 c f+20 d e)-4 a b^2 e (39 c f+23 d e)+48 b^3 c e^2\right )}{2 b}}{4 b}}{6 b}-\frac {f x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (6 b c-7 a d)}{6 b}}{a b}+\frac {x \left (e+f x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}}\)

Input:

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

Output:

((b*c - a*d)*x*(e + f*x^2)^3)/(a*b*Sqrt[a + b*x^2]) + (-1/6*((6*b*c - 7*a* 
d)*f*x*Sqrt[a + b*x^2]*(e + f*x^2)^2)/b - ((f*(24*b^2*c*e - 34*a*b*d*e - 3 
0*a*b*c*f + 35*a^2*d*f)*x*Sqrt[a + b*x^2]*(e + f*x^2))/(4*b) - (-1/2*(f*(4 
8*b^3*c*e^2 - 105*a^3*d*f^2 + 10*a^2*b*f*(20*d*e + 9*c*f) - 4*a*b^2*e*(23* 
d*e + 39*c*f))*x*Sqrt[a + b*x^2])/b - (3*a*(35*a^3*d*f^3 + 72*a*b^2*e*f*(d 
*e + c*f) - 30*a^2*b*f^2*(3*d*e + c*f) - 16*b^3*e^2*(d*e + 3*c*f))*ArcTanh 
[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2)))/(4*b))/(6*b))/(a*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 = 0.73 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {-\frac {35 a \left (a^{3} d \,f^{3}-\frac {6 a^{2} b \,f^{2} \left (c f +3 d e \right )}{7}+\frac {72 a \,b^{2} e f \left (c f +d e \right )}{35}-\frac {48 \left (c f +\frac {d e}{3}\right ) b^{3} e^{2}}{35}\right ) \sqrt {b \,x^{2}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{16}+\left (\frac {35 a^{4} d \,f^{3}}{16}-\frac {15 \left (-\frac {7}{18} d f \,x^{2}+c f +3 d e \right ) b \,f^{2} a^{3}}{8}+\frac {9 b^{2} f \left (-\frac {7 d \,f^{2} x^{4}}{108}-\frac {5 f \left (c f +3 d e \right ) x^{2}}{36}+c e f +d \,e^{2}\right ) a^{2}}{2}-3 \left (-\frac {d \,f^{3} x^{6}}{18}-\frac {f^{2} \left (c f +3 d e \right ) x^{4}}{12}-\frac {e f \left (c f +d e \right ) x^{2}}{2}+e^{2} \left (c f +\frac {d e}{3}\right )\right ) b^{3} a +b^{4} c \,e^{3}\right ) x \sqrt {b}}{\sqrt {b \,x^{2}+a}\, b^{\frac {9}{2}} a}\)	\(251\)
risch	\(\frac {f x \left (8 x^{4} d \,f^{2} b^{2}-22 a b d \,f^{2} x^{2}+12 b^{2} c \,f^{2} x^{2}+36 b^{2} d e f \,x^{2}+57 a^{2} d \,f^{2}-42 a b c \,f^{2}-126 a b d e f +72 b^{2} c e f +72 b^{2} d \,e^{2}\right ) \sqrt {b \,x^{2}+a}}{48 b^{4}}-\frac {b \left (35 a^{3} d \,f^{3}-30 a^{2} b c \,f^{3}-90 a^{2} b d e \,f^{2}+72 a \,b^{2} c e \,f^{2}+72 a \,b^{2} d \,e^{2} f -48 b^{3} c \,e^{2} f -16 b^{3} d \,e^{3}\right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\frac {19 a^{3} d \,f^{3} x}{\sqrt {b \,x^{2}+a}}-\frac {16 b^{4} c \,e^{3} x}{a \sqrt {b \,x^{2}+a}}-\frac {14 a^{2} b c \,f^{3} x}{\sqrt {b \,x^{2}+a}}+\frac {24 a \,b^{2} c e \,f^{2} x}{\sqrt {b \,x^{2}+a}}+\frac {24 a \,b^{2} d \,e^{2} f x}{\sqrt {b \,x^{2}+a}}-\frac {42 a^{2} b d e \,f^{2} x}{\sqrt {b \,x^{2}+a}}}{16 b^{4}}\)	\(346\)
default	\(\frac {c \,e^{3} x}{a \sqrt {b \,x^{2}+a}}+f^{2} \left (c f +3 d e \right ) \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )+3 e f \left (c f +d e \right ) \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+e^{2} \left (3 c f +d e \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+d \,f^{3} \left (\frac {x^{7}}{6 b \sqrt {b \,x^{2}+a}}-\frac {7 a \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )}{6 b}\right )\)	\(348\)

Input:

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

Output:

1/(b*x^2+a)^(1/2)*(-35/16*a*(a^3*d*f^3-6/7*a^2*b*f^2*(c*f+3*d*e)+72/35*a*b 
^2*e*f*(c*f+d*e)-48/35*(c*f+1/3*d*e)*b^3*e^2)*(b*x^2+a)^(1/2)*arctanh((b*x 
^2+a)^(1/2)/x/b^(1/2))+(35/16*a^4*d*f^3-15/8*(-7/18*d*f*x^2+c*f+3*d*e)*b*f 
^2*a^3+9/2*b^2*f*(-7/108*d*f^2*x^4-5/36*f*(c*f+3*d*e)*x^2+c*e*f+d*e^2)*a^2 
-3*(-1/18*d*f^3*x^6-1/12*f^2*(c*f+3*d*e)*x^4-1/2*e*f*(c*f+d*e)*x^2+e^2*(c* 
f+1/3*d*e))*b^3*a+b^4*c*e^3)*x*b^(1/2))/b^(9/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 864, normalized size of antiderivative = 3.53 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/96*(3*(16*a^2*b^3*d*e^3 + 24*(2*a^2*b^3*c - 3*a^3*b^2*d)*e^2*f - 18*(4 
*a^3*b^2*c - 5*a^4*b*d)*e*f^2 + 5*(6*a^4*b*c - 7*a^5*d)*f^3 + (16*a*b^4*d* 
e^3 + 24*(2*a*b^4*c - 3*a^2*b^3*d)*e^2*f - 18*(4*a^2*b^3*c - 5*a^3*b^2*d)* 
e*f^2 + 5*(6*a^3*b^2*c - 7*a^4*b*d)*f^3)*x^2)*sqrt(b)*log(-2*b*x^2 + 2*sqr 
t(b*x^2 + a)*sqrt(b)*x - a) - 2*(8*a*b^4*d*f^3*x^7 + 2*(18*a*b^4*d*e*f^2 + 
 (6*a*b^4*c - 7*a^2*b^3*d)*f^3)*x^5 + (72*a*b^4*d*e^2*f + 18*(4*a*b^4*c - 
5*a^2*b^3*d)*e*f^2 - 5*(6*a^2*b^3*c - 7*a^3*b^2*d)*f^3)*x^3 + 3*(16*(b^5*c 
 - a*b^4*d)*e^3 - 24*(2*a*b^4*c - 3*a^2*b^3*d)*e^2*f + 18*(4*a^2*b^3*c - 5 
*a^3*b^2*d)*e*f^2 - 5*(6*a^3*b^2*c - 7*a^4*b*d)*f^3)*x)*sqrt(b*x^2 + a))/( 
a*b^6*x^2 + a^2*b^5), -1/48*(3*(16*a^2*b^3*d*e^3 + 24*(2*a^2*b^3*c - 3*a^3 
*b^2*d)*e^2*f - 18*(4*a^3*b^2*c - 5*a^4*b*d)*e*f^2 + 5*(6*a^4*b*c - 7*a^5* 
d)*f^3 + (16*a*b^4*d*e^3 + 24*(2*a*b^4*c - 3*a^2*b^3*d)*e^2*f - 18*(4*a^2* 
b^3*c - 5*a^3*b^2*d)*e*f^2 + 5*(6*a^3*b^2*c - 7*a^4*b*d)*f^3)*x^2)*sqrt(-b 
)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8*a*b^4*d*f^3*x^7 + 2*(18*a*b^4*d* 
e*f^2 + (6*a*b^4*c - 7*a^2*b^3*d)*f^3)*x^5 + (72*a*b^4*d*e^2*f + 18*(4*a*b 
^4*c - 5*a^2*b^3*d)*e*f^2 - 5*(6*a^2*b^3*c - 7*a^3*b^2*d)*f^3)*x^3 + 3*(16 
*(b^5*c - a*b^4*d)*e^3 - 24*(2*a*b^4*c - 3*a^2*b^3*d)*e^2*f + 18*(4*a^2*b^ 
3*c - 5*a^3*b^2*d)*e*f^2 - 5*(6*a^3*b^2*c - 7*a^4*b*d)*f^3)*x)*sqrt(b*x^2 
+ a))/(a*b^6*x^2 + a^2*b^5)]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.60 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d f^{3} x^{7}}{6 \, \sqrt {b x^{2} + a} b} - \frac {7 \, a d f^{3} x^{5}}{24 \, \sqrt {b x^{2} + a} b^{2}} + \frac {35 \, a^{2} d f^{3} x^{3}}{48 \, \sqrt {b x^{2} + a} b^{3}} + \frac {{\left (3 \, d e f^{2} + c f^{3}\right )} x^{5}}{4 \, \sqrt {b x^{2} + a} b} + \frac {c e^{3} x}{\sqrt {b x^{2} + a} a} + \frac {35 \, a^{3} d f^{3} x}{16 \, \sqrt {b x^{2} + a} b^{4}} - \frac {35 \, a^{3} d f^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {9}{2}}} - \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} a x^{3}}{8 \, \sqrt {b x^{2} + a} b^{2}} + \frac {3 \, {\left (d e^{2} f + c e f^{2}\right )} x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {15 \, {\left (3 \, d e f^{2} + c f^{3}\right )} a^{2} x}{8 \, \sqrt {b x^{2} + a} b^{3}} + \frac {9 \, {\left (d e^{2} f + c e f^{2}\right )} a x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {{\left (d e^{3} + 3 \, c e^{2} f\right )} x}{\sqrt {b x^{2} + a} b} + \frac {15 \, {\left (3 \, d e f^{2} + c f^{3}\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {7}{2}}} - \frac {9 \, {\left (d e^{2} f + c e f^{2}\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {{\left (d e^{3} + 3 \, c e^{2} f\right )} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} \] Input:

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

Output:

1/6*d*f^3*x^7/(sqrt(b*x^2 + a)*b) - 7/24*a*d*f^3*x^5/(sqrt(b*x^2 + a)*b^2) 
 + 35/48*a^2*d*f^3*x^3/(sqrt(b*x^2 + a)*b^3) + 1/4*(3*d*e*f^2 + c*f^3)*x^5 
/(sqrt(b*x^2 + a)*b) + c*e^3*x/(sqrt(b*x^2 + a)*a) + 35/16*a^3*d*f^3*x/(sq 
rt(b*x^2 + a)*b^4) - 35/16*a^3*d*f^3*arcsinh(b*x/sqrt(a*b))/b^(9/2) - 5/8* 
(3*d*e*f^2 + c*f^3)*a*x^3/(sqrt(b*x^2 + a)*b^2) + 3/2*(d*e^2*f + c*e*f^2)* 
x^3/(sqrt(b*x^2 + a)*b) - 15/8*(3*d*e*f^2 + c*f^3)*a^2*x/(sqrt(b*x^2 + a)* 
b^3) + 9/2*(d*e^2*f + c*e*f^2)*a*x/(sqrt(b*x^2 + a)*b^2) - (d*e^3 + 3*c*e^ 
2*f)*x/(sqrt(b*x^2 + a)*b) + 15/8*(3*d*e*f^2 + c*f^3)*a^2*arcsinh(b*x/sqrt 
(a*b))/b^(7/2) - 9/2*(d*e^2*f + c*e*f^2)*a*arcsinh(b*x/sqrt(a*b))/b^(5/2) 
+ (d*e^3 + 3*c*e^2*f)*arcsinh(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.40 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (2 \, {\left (\frac {4 \, d f^{3} x^{2}}{b} + \frac {18 \, a b^{6} d e f^{2} + 6 \, a b^{6} c f^{3} - 7 \, a^{2} b^{5} d f^{3}}{a b^{7}}\right )} x^{2} + \frac {72 \, a b^{6} d e^{2} f + 72 \, a b^{6} c e f^{2} - 90 \, a^{2} b^{5} d e f^{2} - 30 \, a^{2} b^{5} c f^{3} + 35 \, a^{3} b^{4} d f^{3}}{a b^{7}}\right )} x^{2} + \frac {3 \, {\left (16 \, b^{7} c e^{3} - 16 \, a b^{6} d e^{3} - 48 \, a b^{6} c e^{2} f + 72 \, a^{2} b^{5} d e^{2} f + 72 \, a^{2} b^{5} c e f^{2} - 90 \, a^{3} b^{4} d e f^{2} - 30 \, a^{3} b^{4} c f^{3} + 35 \, a^{4} b^{3} d f^{3}\right )}}{a b^{7}}\right )} x}{48 \, \sqrt {b x^{2} + a}} - \frac {{\left (16 \, b^{3} d e^{3} + 48 \, b^{3} c e^{2} f - 72 \, a b^{2} d e^{2} f - 72 \, a b^{2} c e f^{2} + 90 \, a^{2} b d e f^{2} + 30 \, a^{2} b c f^{3} - 35 \, a^{3} d f^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {9}{2}}} \] Input:

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

Output:

1/48*((2*(4*d*f^3*x^2/b + (18*a*b^6*d*e*f^2 + 6*a*b^6*c*f^3 - 7*a^2*b^5*d* 
f^3)/(a*b^7))*x^2 + (72*a*b^6*d*e^2*f + 72*a*b^6*c*e*f^2 - 90*a^2*b^5*d*e* 
f^2 - 30*a^2*b^5*c*f^3 + 35*a^3*b^4*d*f^3)/(a*b^7))*x^2 + 3*(16*b^7*c*e^3 
- 16*a*b^6*d*e^3 - 48*a*b^6*c*e^2*f + 72*a^2*b^5*d*e^2*f + 72*a^2*b^5*c*e* 
f^2 - 90*a^3*b^4*d*e*f^2 - 30*a^3*b^4*c*f^3 + 35*a^4*b^3*d*f^3)/(a*b^7))*x 
/sqrt(b*x^2 + a) - 1/16*(16*b^3*d*e^3 + 48*b^3*c*e^2*f - 72*a*b^2*d*e^2*f 
- 72*a*b^2*c*e*f^2 + 90*a^2*b*d*e*f^2 + 30*a^2*b*c*f^3 - 35*a^3*d*f^3)*log 
(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 31.96 (sec) , antiderivative size = 1098, normalized size of antiderivative = 4.48 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(840*sqrt(a + b*x**2)*a**4*b*d*f**3*x - 720*sqrt(a + b*x**2)*a**3*b**2*c*f 
**3*x - 2160*sqrt(a + b*x**2)*a**3*b**2*d*e*f**2*x + 280*sqrt(a + b*x**2)* 
a**3*b**2*d*f**3*x**3 + 1728*sqrt(a + b*x**2)*a**2*b**3*c*e*f**2*x - 240*s 
qrt(a + b*x**2)*a**2*b**3*c*f**3*x**3 + 1728*sqrt(a + b*x**2)*a**2*b**3*d* 
e**2*f*x - 720*sqrt(a + b*x**2)*a**2*b**3*d*e*f**2*x**3 - 112*sqrt(a + b*x 
**2)*a**2*b**3*d*f**3*x**5 - 1152*sqrt(a + b*x**2)*a*b**4*c*e**2*f*x + 576 
*sqrt(a + b*x**2)*a*b**4*c*e*f**2*x**3 + 96*sqrt(a + b*x**2)*a*b**4*c*f**3 
*x**5 - 384*sqrt(a + b*x**2)*a*b**4*d*e**3*x + 576*sqrt(a + b*x**2)*a*b**4 
*d*e**2*f*x**3 + 288*sqrt(a + b*x**2)*a*b**4*d*e*f**2*x**5 + 64*sqrt(a + b 
*x**2)*a*b**4*d*f**3*x**7 + 384*sqrt(a + b*x**2)*b**5*c*e**3*x - 840*sqrt( 
b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**5*d*f**3 + 720*sqrt(b)*l 
og((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b*c*f**3 + 2160*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b*d*e*f**2 - 840*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b*d*f**3*x**2 - 1728*sqrt(b 
)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**2*c*e*f**2 + 720*sqr 
t(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**2*c*f**3*x**2 - 1 
728*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**2*d*e**2*f 
 + 2160*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**2*d*e* 
f**2*x**2 + 1152*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2* 
b**3*c*e**2*f - 1728*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a)...