\(\int \sqrt {a+b x^2} (c+d x^2) (e+f x^2)^2 \, dx\) [261]

Optimal result

Integrand size = 28, antiderivative size = 267 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {\left (64 b^3 c e^2-5 a^3 d f^2+8 a^2 b f (2 d e+c f)-16 a b^2 e (d e+2 c f)\right ) x \sqrt {a+b x^2}}{128 b^3}+\frac {\left (5 a^2 d f^2-8 a b f (2 d e+c f)+16 b^2 e (d e+2 c f)\right ) x \left (a+b x^2\right )^{3/2}}{64 b^3}-\frac {f (5 a d f-8 b (2 d e+c f)) x^3 \left (a+b x^2\right )^{3/2}}{48 b^2}+\frac {d f^2 x^5 \left (a+b x^2\right )^{3/2}}{8 b}+\frac {a \left (64 b^3 c e^2-5 a^3 d f^2+8 a^2 b f (2 d e+c f)-16 a b^2 e (d e+2 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{7/2}} \] Output:

1/128*(64*b^3*c*e^2-5*a^3*d*f^2+8*a^2*b*f*(c*f+2*d*e)-16*a*b^2*e*(2*c*f+d* 
e))*x*(b*x^2+a)^(1/2)/b^3+1/64*(5*a^2*d*f^2-8*a*b*f*(c*f+2*d*e)+16*b^2*e*( 
2*c*f+d*e))*x*(b*x^2+a)^(3/2)/b^3-1/48*f*(5*a*d*f-8*b*(c*f+2*d*e))*x^3*(b* 
x^2+a)^(3/2)/b^2+1/8*d*f^2*x^5*(b*x^2+a)^(3/2)/b+1/128*a*(64*b^3*c*e^2-5*a 
^3*d*f^2+8*a^2*b*f*(c*f+2*d*e)-16*a*b^2*e*(2*c*f+d*e))*arctanh(b^(1/2)*x/( 
b*x^2+a)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.87 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (15 a^3 d f^2-2 a^2 b f \left (24 d e+12 c f+5 d f x^2\right )+8 a b^2 \left (2 c f \left (6 e+f x^2\right )+d \left (6 e^2+4 e f x^2+f^2 x^4\right )\right )+16 b^3 \left (4 c \left (3 e^2+3 e f x^2+f^2 x^4\right )+d x^2 \left (6 e^2+8 e f x^2+3 f^2 x^4\right )\right )\right )+3 a \left (-64 b^3 c e^2+5 a^3 d f^2-8 a^2 b f (2 d e+c f)+16 a b^2 e (d e+2 c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{384 b^{7/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(15*a^3*d*f^2 - 2*a^2*b*f*(24*d*e + 12*c*f + 5* 
d*f*x^2) + 8*a*b^2*(2*c*f*(6*e + f*x^2) + d*(6*e^2 + 4*e*f*x^2 + f^2*x^4)) 
 + 16*b^3*(4*c*(3*e^2 + 3*e*f*x^2 + f^2*x^4) + d*x^2*(6*e^2 + 8*e*f*x^2 + 
3*f^2*x^4))) + 3*a*(-64*b^3*c*e^2 + 5*a^3*d*f^2 - 8*a^2*b*f*(2*d*e + c*f) 
+ 16*a*b^2*e*(d*e + 2*c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(384*b^(7 
/2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {403, 403, 299, 211, 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.

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 d f^2+8 a^2 b f (c f+2 d e)-16 a b^2 e (2 c f+d e)+64 b^3 c e^2\right ) \int \sqrt {b x^2+a}dx}{4 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (15 a^2 d f^2-4 a b f (6 c f+7 d e)+8 b^2 e (8 c f+d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (-5 a d f+8 b c f+4 b d e)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{8 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 d f^2+8 a^2 b f (c f+2 d e)-16 a b^2 e (2 c f+d e)+64 b^3 c e^2\right ) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (15 a^2 d f^2-4 a b f (6 c f+7 d e)+8 b^2 e (8 c f+d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (-5 a d f+8 b c f+4 b d e)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{8 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 d f^2+8 a^2 b f (c f+2 d e)-16 a b^2 e (2 c f+d e)+64 b^3 c e^2\right ) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (15 a^2 d f^2-4 a b f (6 c f+7 d e)+8 b^2 e (8 c f+d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (-5 a d f+8 b c f+4 b d e)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{8 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {x \left (a+b x^2\right )^{3/2} \left (15 a^2 d f^2-4 a b f (6 c f+7 d e)+8 b^2 e (8 c f+d e)\right )}{4 b}+\frac {3 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (-5 a^3 d f^2+8 a^2 b f (c f+2 d e)-16 a b^2 e (2 c f+d e)+64 b^3 c e^2\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) (-5 a d f+8 b c f+4 b d e)}{6 b}}{8 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{8 b}\)

Input:

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

Output:

(d*x*(a + b*x^2)^(3/2)*(e + f*x^2)^2)/(8*b) + (((4*b*d*e + 8*b*c*f - 5*a*d 
*f)*x*(a + b*x^2)^(3/2)*(e + f*x^2))/(6*b) + (((15*a^2*d*f^2 - 4*a*b*f*(7* 
d*e + 6*c*f) + 8*b^2*e*(d*e + 8*c*f))*x*(a + b*x^2)^(3/2))/(4*b) + (3*(64* 
b^3*c*e^2 - 5*a^3*d*f^2 + 8*a^2*b*f*(2*d*e + c*f) - 16*a*b^2*e*(d*e + 2*c* 
f))*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*S 
qrt[b])))/(4*b))/(6*b))/(8*b)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 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 = 0.68 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(-\frac {5 \left (a \left (a^{2} \left (a d -\frac {8 b c}{5}\right ) f^{2}-\frac {16 a b e \left (a d -2 b c \right ) f}{5}+\frac {16 b^{2} e^{2} \left (a d -4 b c \right )}{5}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\sqrt {b \,x^{2}+a}\, x \left (\frac {64 \left (\frac {\left (\frac {3 x^{2} d}{4}+c \right ) x^{4} f^{2}}{3}+\left (\frac {2 x^{2} d}{3}+c \right ) x^{2} e f +\left (\frac {x^{2} d}{2}+c \right ) e^{2}\right ) b^{\frac {7}{2}}}{5}+a \left (\frac {16 \left (\frac {\left (\frac {x^{2} d}{2}+c \right ) x^{2} f^{2}}{3}+2 \left (\frac {x^{2} d}{3}+c \right ) e f +d \,e^{2}\right ) b^{\frac {5}{2}}}{5}+a \left (2 \left (\left (-\frac {x^{2} d}{3}-\frac {4 c}{5}\right ) f -\frac {8 d e}{5}\right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right ) f \right )\right )\right )}{128 b^{\frac {7}{2}}}\)	\(208\)
risch	\(\frac {x \left (48 b^{3} d \,f^{2} x^{6}+8 a \,b^{2} x^{4} d \,f^{2}+64 b^{3} c \,f^{2} x^{4}+128 b^{3} d e f \,x^{4}-10 a^{2} b \,x^{2} d \,f^{2}+16 a c \,f^{2} b^{2} x^{2}+32 a \,b^{2} d e f \,x^{2}+192 b^{3} c e f \,x^{2}+96 b^{3} d \,e^{2} x^{2}+15 a^{3} d \,f^{2}-24 a^{2} c \,f^{2} b -48 a^{2} b d e f +96 a c e f \,b^{2}+48 a \,b^{2} d \,e^{2}+192 b^{3} c \,e^{2}\right ) \sqrt {b \,x^{2}+a}}{384 b^{3}}-\frac {a \left (5 a^{3} d \,f^{2}-8 a^{2} c \,f^{2} b -16 a^{2} b d e f +32 a c e f \,b^{2}+16 a \,b^{2} d \,e^{2}-64 b^{3} c \,e^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {7}{2}}}\)	\(262\)
default	\(c \,e^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+f \left (c f +2 d e \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )+e \left (2 c f +d e \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )+f^{2} d \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{8 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )}{8 b}\right )\)	\(310\)

Input:

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

Output:

-5/128/b^(7/2)*(a*(a^2*(a*d-8/5*b*c)*f^2-16/5*a*b*e*(a*d-2*b*c)*f+16/5*b^2 
*e^2*(a*d-4*b*c))*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))-(b*x^2+a)^(1/2)*x*(64 
/5*(1/3*(3/4*x^2*d+c)*x^4*f^2+(2/3*x^2*d+c)*x^2*e*f+(1/2*x^2*d+c)*e^2)*b^( 
7/2)+a*(16/5*(1/3*(1/2*x^2*d+c)*x^2*f^2+2*(1/3*x^2*d+c)*e*f+d*e^2)*b^(5/2) 
+a*(2*((-1/3*x^2*d-4/5*c)*f-8/5*d*e)*b^(3/2)+a*d*f*b^(1/2))*f)))
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.01 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\left [-\frac {3 \, {\left (16 \, {\left (4 \, a b^{3} c - a^{2} b^{2} d\right )} e^{2} - 16 \, {\left (2 \, a^{2} b^{2} c - a^{3} b d\right )} e f + {\left (8 \, a^{3} b c - 5 \, a^{4} d\right )} f^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (48 \, b^{4} d f^{2} x^{7} + 8 \, {\left (16 \, b^{4} d e f + {\left (8 \, b^{4} c + a b^{3} d\right )} f^{2}\right )} x^{5} + 2 \, {\left (48 \, b^{4} d e^{2} + 16 \, {\left (6 \, b^{4} c + a b^{3} d\right )} e f + {\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d\right )} f^{2}\right )} x^{3} + 3 \, {\left (16 \, {\left (4 \, b^{4} c + a b^{3} d\right )} e^{2} + 16 \, {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} e f - {\left (8 \, a^{2} b^{2} c - 5 \, a^{3} b d\right )} f^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{4}}, -\frac {3 \, {\left (16 \, {\left (4 \, a b^{3} c - a^{2} b^{2} d\right )} e^{2} - 16 \, {\left (2 \, a^{2} b^{2} c - a^{3} b d\right )} e f + {\left (8 \, a^{3} b c - 5 \, a^{4} d\right )} f^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (48 \, b^{4} d f^{2} x^{7} + 8 \, {\left (16 \, b^{4} d e f + {\left (8 \, b^{4} c + a b^{3} d\right )} f^{2}\right )} x^{5} + 2 \, {\left (48 \, b^{4} d e^{2} + 16 \, {\left (6 \, b^{4} c + a b^{3} d\right )} e f + {\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d\right )} f^{2}\right )} x^{3} + 3 \, {\left (16 \, {\left (4 \, b^{4} c + a b^{3} d\right )} e^{2} + 16 \, {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} e f - {\left (8 \, a^{2} b^{2} c - 5 \, a^{3} b d\right )} f^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{4}}\right ] \] Input:

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

Output:

[-1/768*(3*(16*(4*a*b^3*c - a^2*b^2*d)*e^2 - 16*(2*a^2*b^2*c - a^3*b*d)*e* 
f + (8*a^3*b*c - 5*a^4*d)*f^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sq 
rt(b)*x - a) - 2*(48*b^4*d*f^2*x^7 + 8*(16*b^4*d*e*f + (8*b^4*c + a*b^3*d) 
*f^2)*x^5 + 2*(48*b^4*d*e^2 + 16*(6*b^4*c + a*b^3*d)*e*f + (8*a*b^3*c - 5* 
a^2*b^2*d)*f^2)*x^3 + 3*(16*(4*b^4*c + a*b^3*d)*e^2 + 16*(2*a*b^3*c - a^2* 
b^2*d)*e*f - (8*a^2*b^2*c - 5*a^3*b*d)*f^2)*x)*sqrt(b*x^2 + a))/b^4, -1/38 
4*(3*(16*(4*a*b^3*c - a^2*b^2*d)*e^2 - 16*(2*a^2*b^2*c - a^3*b*d)*e*f + (8 
*a^3*b*c - 5*a^4*d)*f^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (48 
*b^4*d*f^2*x^7 + 8*(16*b^4*d*e*f + (8*b^4*c + a*b^3*d)*f^2)*x^5 + 2*(48*b^ 
4*d*e^2 + 16*(6*b^4*c + a*b^3*d)*e*f + (8*a*b^3*c - 5*a^2*b^2*d)*f^2)*x^3 
+ 3*(16*(4*b^4*c + a*b^3*d)*e^2 + 16*(2*a*b^3*c - a^2*b^2*d)*e*f - (8*a^2* 
b^2*c - 5*a^3*b*d)*f^2)*x)*sqrt(b*x^2 + a))/b^4]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.55 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d f^{2} x^{7}}{8} + \frac {x^{5} \left (\frac {a d f^{2}}{8} + b c f^{2} + 2 b d e f\right )}{6 b} + \frac {x^{3} \left (a c f^{2} + 2 a d e f - \frac {5 a \left (\frac {a d f^{2}}{8} + b c f^{2} + 2 b d e f\right )}{6 b} + 2 b c e f + b d e^{2}\right )}{4 b} + \frac {x \left (2 a c e f + a d e^{2} - \frac {3 a \left (a c f^{2} + 2 a d e f - \frac {5 a \left (\frac {a d f^{2}}{8} + b c f^{2} + 2 b d e f\right )}{6 b} + 2 b c e f + b d e^{2}\right )}{4 b} + b c e^{2}\right )}{2 b}\right ) + \left (a c e^{2} - \frac {a \left (2 a c e f + a d e^{2} - \frac {3 a \left (a c f^{2} + 2 a d e f - \frac {5 a \left (\frac {a d f^{2}}{8} + b c f^{2} + 2 b d e f\right )}{6 b} + 2 b c e f + b d e^{2}\right )}{4 b} + b c e^{2}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\sqrt {a} \left (c e^{2} x + \frac {d f^{2} x^{7}}{7} + \frac {x^{5} \left (c f^{2} + 2 d e f\right )}{5} + \frac {x^{3} \cdot \left (2 c e f + d e^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(a + b*x**2)*(d*f**2*x**7/8 + x**5*(a*d*f**2/8 + b*c*f**2 + 
 2*b*d*e*f)/(6*b) + x**3*(a*c*f**2 + 2*a*d*e*f - 5*a*(a*d*f**2/8 + b*c*f** 
2 + 2*b*d*e*f)/(6*b) + 2*b*c*e*f + b*d*e**2)/(4*b) + x*(2*a*c*e*f + a*d*e* 
*2 - 3*a*(a*c*f**2 + 2*a*d*e*f - 5*a*(a*d*f**2/8 + b*c*f**2 + 2*b*d*e*f)/( 
6*b) + 2*b*c*e*f + b*d*e**2)/(4*b) + b*c*e**2)/(2*b)) + (a*c*e**2 - a*(2*a 
*c*e*f + a*d*e**2 - 3*a*(a*c*f**2 + 2*a*d*e*f - 5*a*(a*d*f**2/8 + b*c*f**2 
 + 2*b*d*e*f)/(6*b) + 2*b*c*e*f + b*d*e**2)/(4*b) + b*c*e**2)/(2*b))*Piece 
wise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x 
)/sqrt(b*x**2), True)), Ne(b, 0)), (sqrt(a)*(c*e**2*x + d*f**2*x**7/7 + x* 
*5*(c*f**2 + 2*d*e*f)/5 + x**3*(2*c*e*f + d*e**2)/3), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.26 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} d f^{2} x^{5}}{8 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a d f^{2} x^{3}}{48 \, b^{2}} + \frac {1}{2} \, \sqrt {b x^{2} + a} c e^{2} x + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d f^{2} x}{64 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} d f^{2} x}{128 \, b^{3}} + \frac {{\left (2 \, d e f + c f^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{3}}{6 \, b} + \frac {a c e^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} - \frac {5 \, a^{4} d f^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {7}{2}}} - \frac {{\left (2 \, d e f + c f^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x}{8 \, b^{2}} + \frac {{\left (2 \, d e f + c f^{2}\right )} \sqrt {b x^{2} + a} a^{2} x}{16 \, b^{2}} + \frac {{\left (d e^{2} + 2 \, c e f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x}{4 \, b} - \frac {{\left (d e^{2} + 2 \, c e f\right )} \sqrt {b x^{2} + a} a x}{8 \, b} + \frac {{\left (2 \, d e f + c f^{2}\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {5}{2}}} - \frac {{\left (d e^{2} + 2 \, c e f\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} \] Input:

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

Output:

1/8*(b*x^2 + a)^(3/2)*d*f^2*x^5/b - 5/48*(b*x^2 + a)^(3/2)*a*d*f^2*x^3/b^2 
 + 1/2*sqrt(b*x^2 + a)*c*e^2*x + 5/64*(b*x^2 + a)^(3/2)*a^2*d*f^2*x/b^3 - 
5/128*sqrt(b*x^2 + a)*a^3*d*f^2*x/b^3 + 1/6*(2*d*e*f + c*f^2)*(b*x^2 + a)^ 
(3/2)*x^3/b + 1/2*a*c*e^2*arcsinh(b*x/sqrt(a*b))/sqrt(b) - 5/128*a^4*d*f^2 
*arcsinh(b*x/sqrt(a*b))/b^(7/2) - 1/8*(2*d*e*f + c*f^2)*(b*x^2 + a)^(3/2)* 
a*x/b^2 + 1/16*(2*d*e*f + c*f^2)*sqrt(b*x^2 + a)*a^2*x/b^2 + 1/4*(d*e^2 + 
2*c*e*f)*(b*x^2 + a)^(3/2)*x/b - 1/8*(d*e^2 + 2*c*e*f)*sqrt(b*x^2 + a)*a*x 
/b + 1/16*(2*d*e*f + c*f^2)*a^3*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 1/8*(d*e^ 
2 + 2*c*e*f)*a^2*arcsinh(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.02 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, d f^{2} x^{2} + \frac {16 \, b^{6} d e f + 8 \, b^{6} c f^{2} + a b^{5} d f^{2}}{b^{6}}\right )} x^{2} + \frac {48 \, b^{6} d e^{2} + 96 \, b^{6} c e f + 16 \, a b^{5} d e f + 8 \, a b^{5} c f^{2} - 5 \, a^{2} b^{4} d f^{2}}{b^{6}}\right )} x^{2} + \frac {3 \, {\left (64 \, b^{6} c e^{2} + 16 \, a b^{5} d e^{2} + 32 \, a b^{5} c e f - 16 \, a^{2} b^{4} d e f - 8 \, a^{2} b^{4} c f^{2} + 5 \, a^{3} b^{3} d f^{2}\right )}}{b^{6}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (64 \, a b^{3} c e^{2} - 16 \, a^{2} b^{2} d e^{2} - 32 \, a^{2} b^{2} c e f + 16 \, a^{3} b d e f + 8 \, a^{3} b c f^{2} - 5 \, a^{4} d f^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {7}{2}}} \] Input:

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

Output:

1/384*(2*(4*(6*d*f^2*x^2 + (16*b^6*d*e*f + 8*b^6*c*f^2 + a*b^5*d*f^2)/b^6) 
*x^2 + (48*b^6*d*e^2 + 96*b^6*c*e*f + 16*a*b^5*d*e*f + 8*a*b^5*c*f^2 - 5*a 
^2*b^4*d*f^2)/b^6)*x^2 + 3*(64*b^6*c*e^2 + 16*a*b^5*d*e^2 + 32*a*b^5*c*e*f 
 - 16*a^2*b^4*d*e*f - 8*a^2*b^4*c*f^2 + 5*a^3*b^3*d*f^2)/b^6)*sqrt(b*x^2 + 
 a)*x - 1/128*(64*a*b^3*c*e^2 - 16*a^2*b^2*d*e^2 - 32*a^2*b^2*c*e*f + 16*a 
^3*b*d*e*f + 8*a^3*b*c*f^2 - 5*a^4*d*f^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 
+ a)))/b^(7/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.84 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {15 \sqrt {b \,x^{2}+a}\, a^{3} b d \,f^{2} x -24 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,f^{2} x -48 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d e f x -10 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,f^{2} x^{3}+96 \sqrt {b \,x^{2}+a}\, a \,b^{3} c e f x +16 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,f^{2} x^{3}+48 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,e^{2} x +32 \sqrt {b \,x^{2}+a}\, a \,b^{3} d e f \,x^{3}+8 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,f^{2} x^{5}+192 \sqrt {b \,x^{2}+a}\, b^{4} c \,e^{2} x +192 \sqrt {b \,x^{2}+a}\, b^{4} c e f \,x^{3}+64 \sqrt {b \,x^{2}+a}\, b^{4} c \,f^{2} x^{5}+96 \sqrt {b \,x^{2}+a}\, b^{4} d \,e^{2} x^{3}+128 \sqrt {b \,x^{2}+a}\, b^{4} d e f \,x^{5}+48 \sqrt {b \,x^{2}+a}\, b^{4} d \,f^{2} x^{7}-15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d \,f^{2}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b c \,f^{2}+48 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b d e f -96 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c e f -48 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d \,e^{2}+192 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c \,e^{2}}{384 b^{4}} \] Input:

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

Output:

(15*sqrt(a + b*x**2)*a**3*b*d*f**2*x - 24*sqrt(a + b*x**2)*a**2*b**2*c*f** 
2*x - 48*sqrt(a + b*x**2)*a**2*b**2*d*e*f*x - 10*sqrt(a + b*x**2)*a**2*b** 
2*d*f**2*x**3 + 96*sqrt(a + b*x**2)*a*b**3*c*e*f*x + 16*sqrt(a + b*x**2)*a 
*b**3*c*f**2*x**3 + 48*sqrt(a + b*x**2)*a*b**3*d*e**2*x + 32*sqrt(a + b*x* 
*2)*a*b**3*d*e*f*x**3 + 8*sqrt(a + b*x**2)*a*b**3*d*f**2*x**5 + 192*sqrt(a 
 + b*x**2)*b**4*c*e**2*x + 192*sqrt(a + b*x**2)*b**4*c*e*f*x**3 + 64*sqrt( 
a + b*x**2)*b**4*c*f**2*x**5 + 96*sqrt(a + b*x**2)*b**4*d*e**2*x**3 + 128* 
sqrt(a + b*x**2)*b**4*d*e*f*x**5 + 48*sqrt(a + b*x**2)*b**4*d*f**2*x**7 - 
15*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*d*f**2 + 24*sq 
rt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*c*f**2 + 48*sqrt( 
b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*d*e*f - 96*sqrt(b)*l 
og((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*c*e*f - 48*sqrt(b)*lo 
g((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*d*e**2 + 192*sqrt(b)*l 
og((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**3*c*e**2)/(384*b**4)