\(\int \sqrt {a+b x^2} (c+d x^2)^2 (e+f x^2) \, dx\) [269]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.86 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right ) \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (15 a^3 d^2 f-2 a^2 b d \left (12 d e+24 c f+5 d f x^2\right )+8 a b^2 \left (6 c^2 f+d^2 x^2 \left (2 e+f x^2\right )+4 c d \left (3 e+f x^2\right )\right )+16 b^3 \left (6 c^2 \left (2 e+f x^2\right )+4 c d x^2 \left (3 e+2 f x^2\right )+d^2 x^4 \left (4 e+3 f x^2\right )\right )\right )+3 a \left (-64 b^3 c^2 e+5 a^3 d^2 f+16 a b^2 c (2 d e+c f)-8 a^2 b d (d e+2 c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{384 b^{7/2}} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(f*x*(a + b*x^2)^(3/2)*(c + d*x^2)^2)/(8*b) + (((8*b*d*e + 4*b*c*f - 5*a*d 
*f)*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(6*b) + (((15*a^2*d^2*f + 8*b^2*c*(8* 
d*e + c*f) - 4*a*b*d*(6*d*e + 7*c*f))*x*(a + b*x^2)^(3/2))/(4*b) + (3*(64* 
b^3*c^2*e - 5*a^3*d^2*f - 16*a*b^2*c*(2*d*e + c*f) + 8*a^2*b*d*(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 = 0.71 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.78

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.06 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right ) \, dx=\left [-\frac {3 \, {\left (8 \, {\left (8 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} e - {\left (16 \, a^{2} b^{2} c^{2} - 16 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (48 \, b^{4} d^{2} f x^{7} + 8 \, {\left (8 \, b^{4} d^{2} e + {\left (16 \, b^{4} c d + a b^{3} d^{2}\right )} f\right )} x^{5} + 2 \, {\left (8 \, {\left (12 \, b^{4} c d + a b^{3} d^{2}\right )} e + {\left (48 \, b^{4} c^{2} + 16 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} f\right )} x^{3} + 3 \, {\left (8 \, {\left (8 \, b^{4} c^{2} + 4 \, a b^{3} c d - a^{2} b^{2} d^{2}\right )} e + {\left (16 \, a b^{3} c^{2} - 16 \, a^{2} b^{2} c d + 5 \, a^{3} b d^{2}\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{4}}, -\frac {3 \, {\left (8 \, {\left (8 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} e - {\left (16 \, a^{2} b^{2} c^{2} - 16 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (48 \, b^{4} d^{2} f x^{7} + 8 \, {\left (8 \, b^{4} d^{2} e + {\left (16 \, b^{4} c d + a b^{3} d^{2}\right )} f\right )} x^{5} + 2 \, {\left (8 \, {\left (12 \, b^{4} c d + a b^{3} d^{2}\right )} e + {\left (48 \, b^{4} c^{2} + 16 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} f\right )} x^{3} + 3 \, {\left (8 \, {\left (8 \, b^{4} c^{2} + 4 \, a b^{3} c d - a^{2} b^{2} d^{2}\right )} e + {\left (16 \, a b^{3} c^{2} - 16 \, a^{2} b^{2} c d + 5 \, a^{3} b d^{2}\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{4}}\right ] \] Input:

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

Output:

[-1/768*(3*(8*(8*a*b^3*c^2 - 4*a^2*b^2*c*d + a^3*b*d^2)*e - (16*a^2*b^2*c^ 
2 - 16*a^3*b*c*d + 5*a^4*d^2)*f)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)* 
sqrt(b)*x - a) - 2*(48*b^4*d^2*f*x^7 + 8*(8*b^4*d^2*e + (16*b^4*c*d + a*b^ 
3*d^2)*f)*x^5 + 2*(8*(12*b^4*c*d + a*b^3*d^2)*e + (48*b^4*c^2 + 16*a*b^3*c 
*d - 5*a^2*b^2*d^2)*f)*x^3 + 3*(8*(8*b^4*c^2 + 4*a*b^3*c*d - a^2*b^2*d^2)* 
e + (16*a*b^3*c^2 - 16*a^2*b^2*c*d + 5*a^3*b*d^2)*f)*x)*sqrt(b*x^2 + a))/b 
^4, -1/384*(3*(8*(8*a*b^3*c^2 - 4*a^2*b^2*c*d + a^3*b*d^2)*e - (16*a^2*b^2 
*c^2 - 16*a^3*b*c*d + 5*a^4*d^2)*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 
+ a)) - (48*b^4*d^2*f*x^7 + 8*(8*b^4*d^2*e + (16*b^4*c*d + a*b^3*d^2)*f)*x 
^5 + 2*(8*(12*b^4*c*d + a*b^3*d^2)*e + (48*b^4*c^2 + 16*a*b^3*c*d - 5*a^2* 
b^2*d^2)*f)*x^3 + 3*(8*(8*b^4*c^2 + 4*a*b^3*c*d - a^2*b^2*d^2)*e + (16*a*b 
^3*c^2 - 16*a^2*b^2*c*d + 5*a^3*b*d^2)*f)*x)*sqrt(b*x^2 + a))/b^4]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.55 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right ) \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d^{2} f x^{7}}{8} + \frac {x^{5} \left (\frac {a d^{2} f}{8} + 2 b c d f + b d^{2} e\right )}{6 b} + \frac {x^{3} \cdot \left (2 a c d f + a d^{2} e - \frac {5 a \left (\frac {a d^{2} f}{8} + 2 b c d f + b d^{2} e\right )}{6 b} + b c^{2} f + 2 b c d e\right )}{4 b} + \frac {x \left (a c^{2} f + 2 a c d e - \frac {3 a \left (2 a c d f + a d^{2} e - \frac {5 a \left (\frac {a d^{2} f}{8} + 2 b c d f + b d^{2} e\right )}{6 b} + b c^{2} f + 2 b c d e\right )}{4 b} + b c^{2} e\right )}{2 b}\right ) + \left (a c^{2} e - \frac {a \left (a c^{2} f + 2 a c d e - \frac {3 a \left (2 a c d f + a d^{2} e - \frac {5 a \left (\frac {a d^{2} f}{8} + 2 b c d f + b d^{2} e\right )}{6 b} + b c^{2} f + 2 b c d e\right )}{4 b} + b c^{2} e\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\sqrt {a} \left (c^{2} e x + \frac {d^{2} f x^{7}}{7} + \frac {x^{5} \cdot \left (2 c d f + d^{2} e\right )}{5} + \frac {x^{3} \left (c^{2} f + 2 c d e\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.26 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} d^{2} f x^{5}}{8 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a d^{2} f x^{3}}{48 \, b^{2}} + \frac {1}{2} \, \sqrt {b x^{2} + a} c^{2} e x + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{2} f x}{64 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} d^{2} f x}{128 \, b^{3}} + \frac {{\left (d^{2} e + 2 \, c d f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{3}}{6 \, b} + \frac {a c^{2} e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} - \frac {5 \, a^{4} d^{2} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {7}{2}}} - \frac {{\left (d^{2} e + 2 \, c d f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x}{8 \, b^{2}} + \frac {{\left (d^{2} e + 2 \, c d f\right )} \sqrt {b x^{2} + a} a^{2} x}{16 \, b^{2}} + \frac {{\left (2 \, c d e + c^{2} f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x}{4 \, b} - \frac {{\left (2 \, c d e + c^{2} f\right )} \sqrt {b x^{2} + a} a x}{8 \, b} + \frac {{\left (d^{2} e + 2 \, c d f\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {5}{2}}} - \frac {{\left (2 \, c d e + c^{2} f\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.02 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right ) \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, d^{2} f x^{2} + \frac {8 \, b^{6} d^{2} e + 16 \, b^{6} c d f + a b^{5} d^{2} f}{b^{6}}\right )} x^{2} + \frac {96 \, b^{6} c d e + 8 \, a b^{5} d^{2} e + 48 \, b^{6} c^{2} f + 16 \, a b^{5} c d f - 5 \, a^{2} b^{4} d^{2} f}{b^{6}}\right )} x^{2} + \frac {3 \, {\left (64 \, b^{6} c^{2} e + 32 \, a b^{5} c d e - 8 \, a^{2} b^{4} d^{2} e + 16 \, a b^{5} c^{2} f - 16 \, a^{2} b^{4} c d f + 5 \, a^{3} b^{3} d^{2} f\right )}}{b^{6}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (64 \, a b^{3} c^{2} e - 32 \, a^{2} b^{2} c d e + 8 \, a^{3} b d^{2} e - 16 \, a^{2} b^{2} c^{2} f + 16 \, a^{3} b c d f - 5 \, a^{4} d^{2} f\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {7}{2}}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.84 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right ) \, dx=\frac {15 \sqrt {b \,x^{2}+a}\, a^{3} b \,d^{2} f x -48 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c d f x -24 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{2} e x -10 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{2} f \,x^{3}+48 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} f x +96 \sqrt {b \,x^{2}+a}\, a \,b^{3} c d e x +32 \sqrt {b \,x^{2}+a}\, a \,b^{3} c d f \,x^{3}+16 \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{2} e \,x^{3}+8 \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{2} f \,x^{5}+192 \sqrt {b \,x^{2}+a}\, b^{4} c^{2} e x +96 \sqrt {b \,x^{2}+a}\, b^{4} c^{2} f \,x^{3}+192 \sqrt {b \,x^{2}+a}\, b^{4} c d e \,x^{3}+128 \sqrt {b \,x^{2}+a}\, b^{4} c d f \,x^{5}+64 \sqrt {b \,x^{2}+a}\, b^{4} d^{2} e \,x^{5}+48 \sqrt {b \,x^{2}+a}\, b^{4} d^{2} f \,x^{7}-15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d^{2} f +48 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b c d f +24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b \,d^{2} e -48 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c^{2} f -96 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c d e +192 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c^{2} e}{384 b^{4}} \] Input:

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

Output:

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