\(\int \sqrt {a+b x^2} (c+d x^2) (e+f x^2)^3 \, dx\) [260]

Optimal result

Integrand size = 28, antiderivative size = 390 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {\left (128 b^4 c e^3+7 a^4 d f^3+48 a^2 b^2 e f (d e+c f)-10 a^3 b f^2 (3 d e+c f)-32 a b^3 e^2 (d e+3 c f)\right ) x \sqrt {a+b x^2}}{256 b^4}-\frac {\left (7 a^3 d f^3+48 a b^2 e f (d e+c f)-10 a^2 b f^2 (3 d e+c f)-32 b^3 e^2 (d e+3 c f)\right ) x \left (a+b x^2\right )^{3/2}}{128 b^4}+\frac {f \left (7 a^2 d f^2+48 b^2 e (d e+c f)-10 a b f (3 d e+c f)\right ) x^3 \left (a+b x^2\right )^{3/2}}{96 b^3}-\frac {f^2 (7 a d f-10 b (3 d e+c f)) x^5 \left (a+b x^2\right )^{3/2}}{80 b^2}+\frac {d f^3 x^7 \left (a+b x^2\right )^{3/2}}{10 b}+\frac {a \left (128 b^4 c e^3+7 a^4 d f^3+48 a^2 b^2 e f (d e+c f)-10 a^3 b f^2 (3 d e+c f)-32 a b^3 e^2 (d e+3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{9/2}} \] Output:

1/256*(128*b^4*c*e^3+7*a^4*d*f^3+48*a^2*b^2*e*f*(c*f+d*e)-10*a^3*b*f^2*(c* 
f+3*d*e)-32*a*b^3*e^2*(3*c*f+d*e))*x*(b*x^2+a)^(1/2)/b^4-1/128*(7*a^3*d*f^ 
3+48*a*b^2*e*f*(c*f+d*e)-10*a^2*b*f^2*(c*f+3*d*e)-32*b^3*e^2*(3*c*f+d*e))* 
x*(b*x^2+a)^(3/2)/b^4+1/96*f*(7*a^2*d*f^2+48*b^2*e*(c*f+d*e)-10*a*b*f*(c*f 
+3*d*e))*x^3*(b*x^2+a)^(3/2)/b^3-1/80*f^2*(7*a*d*f-10*b*(c*f+3*d*e))*x^5*( 
b*x^2+a)^(3/2)/b^2+1/10*d*f^3*x^7*(b*x^2+a)^(3/2)/b+1/256*a*(128*b^4*c*e^3 
+7*a^4*d*f^3+48*a^2*b^2*e*f*(c*f+d*e)-10*a^3*b*f^2*(c*f+3*d*e)-32*a*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 = 1.10 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.89 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^4 d f^3+10 a^3 b f^2 \left (45 d e+15 c f+7 d f x^2\right )-4 a^2 b^2 f \left (5 c f \left (36 e+5 f x^2\right )+d \left (180 e^2+75 e f x^2+14 f^2 x^4\right )\right )+16 a b^3 \left (5 c f \left (18 e^2+6 e f x^2+f^2 x^4\right )+3 d \left (10 e^3+10 e^2 f x^2+5 e f^2 x^4+f^3 x^6\right )\right )+96 b^4 \left (5 c \left (4 e^3+6 e^2 f x^2+4 e f^2 x^4+f^3 x^6\right )+d x^2 \left (10 e^3+20 e^2 f x^2+15 e f^2 x^4+4 f^3 x^6\right )\right )\right )-15 a \left (128 b^4 c e^3+7 a^4 d f^3+48 a^2 b^2 e f (d e+c f)-10 a^3 b f^2 (3 d e+c f)-32 a b^3 e^2 (d e+3 c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{3840 b^{9/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^4*d*f^3 + 10*a^3*b*f^2*(45*d*e + 15*c*f 
 + 7*d*f*x^2) - 4*a^2*b^2*f*(5*c*f*(36*e + 5*f*x^2) + d*(180*e^2 + 75*e*f* 
x^2 + 14*f^2*x^4)) + 16*a*b^3*(5*c*f*(18*e^2 + 6*e*f*x^2 + f^2*x^4) + 3*d* 
(10*e^3 + 10*e^2*f*x^2 + 5*e*f^2*x^4 + f^3*x^6)) + 96*b^4*(5*c*(4*e^3 + 6* 
e^2*f*x^2 + 4*e*f^2*x^4 + f^3*x^6) + d*x^2*(10*e^3 + 20*e^2*f*x^2 + 15*e*f 
^2*x^4 + 4*f^3*x^6))) - 15*a*(128*b^4*c*e^3 + 7*a^4*d*f^3 + 48*a^2*b^2*e*f 
*(d*e + c*f) - 10*a^3*b*f^2*(3*d*e + c*f) - 32*a*b^3*e^2*(d*e + 3*c*f))*Lo 
g[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(3840*b^(9/2))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {403, 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.

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\int \sqrt {b x^2+a} \left (f x^2+e\right ) \left (\left (24 e (d e+5 c f) b^2-2 a f (33 d e+25 c f) b+35 a^2 d f^2\right ) x^2+e \left (7 d f a^2-14 b d e a-10 b c f a+80 b^2 c e\right )\right )dx}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (-7 a d f+10 b c f+6 b d e)}{8 b}}{10 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\frac {\int \sqrt {b x^2+a} \left (e \left (-35 d f^2 a^3+2 b f (54 d e+25 c f) a^2-36 b^2 e (3 d e+5 c f) a+480 b^3 c e^2\right )-\left (-48 e^2 (d e+15 c f) b^3+8 a e f (36 d e+65 c f) b^2-10 a^2 f^2 (31 d e+15 c f) b+105 a^3 d f^3\right ) x^2\right )dx}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-2 a b f (25 c f+33 d e)+24 b^2 e (5 c f+d e)\right )}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (-7 a d f+10 b c f+6 b d e)}{8 b}}{10 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {\frac {15 \left (7 a^4 d f^3-10 a^3 b f^2 (c f+3 d e)+48 a^2 b^2 e f (c f+d e)-32 a b^3 e^2 (3 c f+d e)+128 b^4 c e^3\right ) \int \sqrt {b x^2+a}dx}{4 b}-\frac {x \left (a+b x^2\right )^{3/2} \left (105 a^3 d f^3-10 a^2 b f^2 (15 c f+31 d e)+8 a b^2 e f (65 c f+36 d e)-48 b^3 e^2 (15 c f+d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-2 a b f (25 c f+33 d e)+24 b^2 e (5 c f+d e)\right )}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (-7 a d f+10 b c f+6 b d e)}{8 b}}{10 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {15 \left (7 a^4 d f^3-10 a^3 b f^2 (c f+3 d e)+48 a^2 b^2 e f (c f+d e)-32 a b^3 e^2 (3 c f+d e)+128 b^4 c e^3\right ) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}-\frac {x \left (a+b x^2\right )^{3/2} \left (105 a^3 d f^3-10 a^2 b f^2 (15 c f+31 d e)+8 a b^2 e f (65 c f+36 d e)-48 b^3 e^2 (15 c f+d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-2 a b f (25 c f+33 d e)+24 b^2 e (5 c f+d e)\right )}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (-7 a d f+10 b c f+6 b d e)}{8 b}}{10 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {15 \left (7 a^4 d f^3-10 a^3 b f^2 (c f+3 d e)+48 a^2 b^2 e f (c f+d e)-32 a b^3 e^2 (3 c f+d e)+128 b^4 c e^3\right ) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}-\frac {x \left (a+b x^2\right )^{3/2} \left (105 a^3 d f^3-10 a^2 b f^2 (15 c f+31 d e)+8 a b^2 e f (65 c f+36 d e)-48 b^3 e^2 (15 c f+d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-2 a b f (25 c f+33 d e)+24 b^2 e (5 c f+d e)\right )}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (-7 a d f+10 b c f+6 b d e)}{8 b}}{10 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-2 a b f (25 c f+33 d e)+24 b^2 e (5 c f+d e)\right )}{6 b}+\frac {\frac {15 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (7 a^4 d f^3-10 a^3 b f^2 (c f+3 d e)+48 a^2 b^2 e f (c f+d e)-32 a b^3 e^2 (3 c f+d e)+128 b^4 c e^3\right )}{4 b}-\frac {x \left (a+b x^2\right )^{3/2} \left (105 a^3 d f^3-10 a^2 b f^2 (15 c f+31 d e)+8 a b^2 e f (65 c f+36 d e)-48 b^3 e^2 (15 c f+d e)\right )}{4 b}}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (-7 a d f+10 b c f+6 b d e)}{8 b}}{10 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^3}{10 b}\)

Input:

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

Output:

(d*x*(a + b*x^2)^(3/2)*(e + f*x^2)^3)/(10*b) + (((6*b*d*e + 10*b*c*f - 7*a 
*d*f)*x*(a + b*x^2)^(3/2)*(e + f*x^2)^2)/(8*b) + (((35*a^2*d*f^2 + 24*b^2* 
e*(d*e + 5*c*f) - 2*a*b*f*(33*d*e + 25*c*f))*x*(a + b*x^2)^(3/2)*(e + f*x^ 
2))/(6*b) + (-1/4*((105*a^3*d*f^3 - 48*b^3*e^2*(d*e + 15*c*f) - 10*a^2*b*f 
^2*(31*d*e + 15*c*f) + 8*a*b^2*e*f*(36*d*e + 65*c*f))*x*(a + b*x^2)^(3/2)) 
/b + (15*(128*b^4*c*e^3 + 7*a^4*d*f^3 + 48*a^2*b^2*e*f*(d*e + c*f) - 10*a^ 
3*b*f^2*(3*d*e + c*f) - 32*a*b^3*e^2*(d*e + 3*c*f))*((x*Sqrt[a + b*x^2])/2 
 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/(4*b))/(6*b))/(8 
*b))/(10*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 = 0.70 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(\frac {\frac {7 a \left (a^{3} \left (a d -\frac {10 b c}{7}\right ) f^{3}-\frac {30 a^{2} \left (a d -\frac {8 b c}{5}\right ) b e \,f^{2}}{7}+\frac {48 a \,b^{2} e^{2} \left (a d -2 b c \right ) f}{7}-\frac {32 b^{3} e^{3} \left (a d -4 b c \right )}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{256}-\frac {7 \left (\left (-\frac {32 \left (\frac {4 x^{2} d}{5}+c \right ) x^{6} f^{3}}{7}-\frac {128 \left (\frac {3 x^{2} d}{4}+c \right ) x^{4} e \,f^{2}}{7}-\frac {192 \left (\frac {2 x^{2} d}{3}+c \right ) x^{2} e^{2} f}{7}-\frac {128 \left (\frac {x^{2} d}{2}+c \right ) e^{3}}{7}\right ) b^{\frac {9}{2}}+\left (\left (-\frac {16 \left (\frac {3 x^{2} d}{5}+c \right ) x^{4} f^{3}}{21}-\frac {32 \left (\frac {x^{2} d}{2}+c \right ) x^{2} e \,f^{2}}{7}-\frac {96 \left (\frac {x^{2} d}{3}+c \right ) e^{2} f}{7}-\frac {32 d \,e^{3}}{7}\right ) b^{\frac {7}{2}}+a f \left (\left (\left (\frac {8}{15} d \,x^{4}+\frac {20}{21} c \,x^{2}\right ) f^{2}+\frac {48 \left (\frac {5 x^{2} d}{12}+c \right ) e f}{7}+\frac {48 d \,e^{2}}{7}\right ) b^{\frac {5}{2}}+a f \left (\left (\left (-\frac {2 x^{2} d}{3}-\frac {10 c}{7}\right ) f -\frac {30 d e}{7}\right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right )\right )\right ) a \right ) \sqrt {b \,x^{2}+a}\, x}{256}}{b^{\frac {9}{2}}}\)	\(312\)
risch	\(-\frac {x \left (-384 b^{4} d \,f^{3} x^{8}-48 a \,b^{3} x^{6} d \,f^{3}-480 b^{4} c \,f^{3} x^{6}-1440 b^{4} d e \,f^{2} x^{6}+56 a^{2} b^{2} x^{4} d \,f^{3}-80 a \,b^{3} c \,f^{3} x^{4}-240 a \,b^{3} d e \,f^{2} x^{4}-1920 b^{4} c e \,f^{2} x^{4}-1920 b^{4} d \,e^{2} f \,x^{4}-70 a^{3} b \,x^{2} d \,f^{3}+100 a^{2} b^{2} c \,f^{3} x^{2}+300 a^{2} b^{2} d e \,f^{2} x^{2}-480 a \,b^{3} c e \,f^{2} x^{2}-480 a \,b^{3} d \,e^{2} f \,x^{2}-2880 b^{4} c \,e^{2} f \,x^{2}-960 b^{4} d \,e^{3} x^{2}+105 a^{4} d \,f^{3}-150 a^{3} b c \,f^{3}-450 a^{3} b d e \,f^{2}+720 a^{2} b^{2} c e \,f^{2}+720 a^{2} b^{2} d \,e^{2} f -1440 a \,b^{3} c \,e^{2} f -480 a \,b^{3} d \,e^{3}-1920 b^{4} c \,e^{3}\right ) \sqrt {b \,x^{2}+a}}{3840 b^{4}}+\frac {a \left (7 a^{4} d \,f^{3}-10 a^{3} b c \,f^{3}-30 a^{3} b d e \,f^{2}+48 a^{2} b^{2} c e \,f^{2}+48 a^{2} b^{2} d \,e^{2} f -96 a \,b^{3} c \,e^{2} f -32 a \,b^{3} d \,e^{3}+128 b^{4} c \,e^{3}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {9}{2}}}\)	\(428\)
default	\(c \,e^{3} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+f^{2} \left (c f +3 d e \right ) \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{8 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )}{8 b}\right )+3 e f \left (c f +d e \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )+e^{2} \left (3 c f +d e \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )+d \,f^{3} \left (\frac {x^{7} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{10 b}-\frac {7 a \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{8 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )}{8 b}\right )}{10 b}\right )\)	\(454\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e)^3,x,method=_RETURNVERBOSE)
 

Output:

7/256*(a*(a^3*(a*d-10/7*b*c)*f^3-30/7*a^2*(a*d-8/5*b*c)*b*e*f^2+48/7*a*b^2 
*e^2*(a*d-2*b*c)*f-32/7*b^3*e^3*(a*d-4*b*c))*arctanh((b*x^2+a)^(1/2)/x/b^( 
1/2))-((-32/7*(4/5*x^2*d+c)*x^6*f^3-128/7*(3/4*x^2*d+c)*x^4*e*f^2-192/7*(2 
/3*x^2*d+c)*x^2*e^2*f-128/7*(1/2*x^2*d+c)*e^3)*b^(9/2)+((-16/21*(3/5*x^2*d 
+c)*x^4*f^3-32/7*(1/2*x^2*d+c)*x^2*e*f^2-96/7*(1/3*x^2*d+c)*e^2*f-32/7*d*e 
^3)*b^(7/2)+a*f*(((8/15*d*x^4+20/21*c*x^2)*f^2+48/7*(5/12*x^2*d+c)*e*f+48/ 
7*d*e^2)*b^(5/2)+a*f*(((-2/3*x^2*d-10/7*c)*f-30/7*d*e)*b^(3/2)+a*d*f*b^(1/ 
2))))*a)*(b*x^2+a)^(1/2)*x)/b^(9/2)
 

Fricas [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 818, normalized size of antiderivative = 2.10 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

[1/7680*(15*(32*(4*a*b^4*c - a^2*b^3*d)*e^3 - 48*(2*a^2*b^3*c - a^3*b^2*d) 
*e^2*f + 6*(8*a^3*b^2*c - 5*a^4*b*d)*e*f^2 - (10*a^4*b*c - 7*a^5*d)*f^3)*s 
qrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(384*b^5*d*f^3* 
x^9 + 48*(30*b^5*d*e*f^2 + (10*b^5*c + a*b^4*d)*f^3)*x^7 + 8*(240*b^5*d*e^ 
2*f + 30*(8*b^5*c + a*b^4*d)*e*f^2 + (10*a*b^4*c - 7*a^2*b^3*d)*f^3)*x^5 + 
 10*(96*b^5*d*e^3 + 48*(6*b^5*c + a*b^4*d)*e^2*f + 6*(8*a*b^4*c - 5*a^2*b^ 
3*d)*e*f^2 - (10*a^2*b^3*c - 7*a^3*b^2*d)*f^3)*x^3 + 15*(32*(4*b^5*c + a*b 
^4*d)*e^3 + 48*(2*a*b^4*c - a^2*b^3*d)*e^2*f - 6*(8*a^2*b^3*c - 5*a^3*b^2* 
d)*e*f^2 + (10*a^3*b^2*c - 7*a^4*b*d)*f^3)*x)*sqrt(b*x^2 + a))/b^5, -1/384 
0*(15*(32*(4*a*b^4*c - a^2*b^3*d)*e^3 - 48*(2*a^2*b^3*c - a^3*b^2*d)*e^2*f 
 + 6*(8*a^3*b^2*c - 5*a^4*b*d)*e*f^2 - (10*a^4*b*c - 7*a^5*d)*f^3)*sqrt(-b 
)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (384*b^5*d*f^3*x^9 + 48*(30*b^5*d*e 
*f^2 + (10*b^5*c + a*b^4*d)*f^3)*x^7 + 8*(240*b^5*d*e^2*f + 30*(8*b^5*c + 
a*b^4*d)*e*f^2 + (10*a*b^4*c - 7*a^2*b^3*d)*f^3)*x^5 + 10*(96*b^5*d*e^3 + 
48*(6*b^5*c + a*b^4*d)*e^2*f + 6*(8*a*b^4*c - 5*a^2*b^3*d)*e*f^2 - (10*a^2 
*b^3*c - 7*a^3*b^2*d)*f^3)*x^3 + 15*(32*(4*b^5*c + a*b^4*d)*e^3 + 48*(2*a* 
b^4*c - a^2*b^3*d)*e^2*f - 6*(8*a^2*b^3*c - 5*a^3*b^2*d)*e*f^2 + (10*a^3*b 
^2*c - 7*a^4*b*d)*f^3)*x)*sqrt(b*x^2 + a))/b^5]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.74 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d f^{3} x^{9}}{10} + \frac {x^{7} \left (\frac {a d f^{3}}{10} + b c f^{3} + 3 b d e f^{2}\right )}{8 b} + \frac {x^{5} \left (a c f^{3} + 3 a d e f^{2} - \frac {7 a \left (\frac {a d f^{3}}{10} + b c f^{3} + 3 b d e f^{2}\right )}{8 b} + 3 b c e f^{2} + 3 b d e^{2} f\right )}{6 b} + \frac {x^{3} \cdot \left (3 a c e f^{2} + 3 a d e^{2} f - \frac {5 a \left (a c f^{3} + 3 a d e f^{2} - \frac {7 a \left (\frac {a d f^{3}}{10} + b c f^{3} + 3 b d e f^{2}\right )}{8 b} + 3 b c e f^{2} + 3 b d e^{2} f\right )}{6 b} + 3 b c e^{2} f + b d e^{3}\right )}{4 b} + \frac {x \left (3 a c e^{2} f + a d e^{3} - \frac {3 a \left (3 a c e f^{2} + 3 a d e^{2} f - \frac {5 a \left (a c f^{3} + 3 a d e f^{2} - \frac {7 a \left (\frac {a d f^{3}}{10} + b c f^{3} + 3 b d e f^{2}\right )}{8 b} + 3 b c e f^{2} + 3 b d e^{2} f\right )}{6 b} + 3 b c e^{2} f + b d e^{3}\right )}{4 b} + b c e^{3}\right )}{2 b}\right ) + \left (a c e^{3} - \frac {a \left (3 a c e^{2} f + a d e^{3} - \frac {3 a \left (3 a c e f^{2} + 3 a d e^{2} f - \frac {5 a \left (a c f^{3} + 3 a d e f^{2} - \frac {7 a \left (\frac {a d f^{3}}{10} + b c f^{3} + 3 b d e f^{2}\right )}{8 b} + 3 b c e f^{2} + 3 b d e^{2} f\right )}{6 b} + 3 b c e^{2} f + b d e^{3}\right )}{4 b} + b c e^{3}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\sqrt {a} \left (c e^{3} x + \frac {d f^{3} x^{9}}{9} + \frac {x^{7} \left (c f^{3} + 3 d e f^{2}\right )}{7} + \frac {x^{5} \cdot \left (3 c e f^{2} + 3 d e^{2} f\right )}{5} + \frac {x^{3} \cdot \left (3 c e^{2} f + d e^{3}\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)*(f*x**2+e)**3,x)
 

Output:

Piecewise((sqrt(a + b*x**2)*(d*f**3*x**9/10 + x**7*(a*d*f**3/10 + b*c*f**3 
 + 3*b*d*e*f**2)/(8*b) + x**5*(a*c*f**3 + 3*a*d*e*f**2 - 7*a*(a*d*f**3/10 
+ b*c*f**3 + 3*b*d*e*f**2)/(8*b) + 3*b*c*e*f**2 + 3*b*d*e**2*f)/(6*b) + x* 
*3*(3*a*c*e*f**2 + 3*a*d*e**2*f - 5*a*(a*c*f**3 + 3*a*d*e*f**2 - 7*a*(a*d* 
f**3/10 + b*c*f**3 + 3*b*d*e*f**2)/(8*b) + 3*b*c*e*f**2 + 3*b*d*e**2*f)/(6 
*b) + 3*b*c*e**2*f + b*d*e**3)/(4*b) + x*(3*a*c*e**2*f + a*d*e**3 - 3*a*(3 
*a*c*e*f**2 + 3*a*d*e**2*f - 5*a*(a*c*f**3 + 3*a*d*e*f**2 - 7*a*(a*d*f**3/ 
10 + b*c*f**3 + 3*b*d*e*f**2)/(8*b) + 3*b*c*e*f**2 + 3*b*d*e**2*f)/(6*b) + 
 3*b*c*e**2*f + b*d*e**3)/(4*b) + b*c*e**3)/(2*b)) + (a*c*e**3 - a*(3*a*c* 
e**2*f + a*d*e**3 - 3*a*(3*a*c*e*f**2 + 3*a*d*e**2*f - 5*a*(a*c*f**3 + 3*a 
*d*e*f**2 - 7*a*(a*d*f**3/10 + b*c*f**3 + 3*b*d*e*f**2)/(8*b) + 3*b*c*e*f* 
*2 + 3*b*d*e**2*f)/(6*b) + 3*b*c*e**2*f + b*d*e**3)/(4*b) + b*c*e**3)/(2*b 
))*Piecewise((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**3*x + d*f**3*x** 
9/9 + x**7*(c*f**3 + 3*d*e*f**2)/7 + x**5*(3*c*e*f**2 + 3*d*e**2*f)/5 + x* 
*3*(3*c*e**2*f + d*e**3)/3), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.35 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} d f^{3} x^{7}}{10 \, b} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a d f^{3} x^{5}}{80 \, b^{2}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d f^{3} x^{3}}{96 \, b^{3}} + \frac {{\left (3 \, d e f^{2} + c f^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{5}}{8 \, b} + \frac {1}{2} \, \sqrt {b x^{2} + a} c e^{3} x - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} d f^{3} x}{128 \, b^{4}} + \frac {7 \, \sqrt {b x^{2} + a} a^{4} d f^{3} x}{256 \, b^{4}} + \frac {a c e^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} + \frac {7 \, a^{5} d f^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {9}{2}}} - \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{3}}{48 \, b^{2}} + \frac {{\left (d e^{2} f + c e f^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{3}}{2 \, b} + \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x}{64 \, b^{3}} - \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} \sqrt {b x^{2} + a} a^{3} x}{128 \, b^{3}} - \frac {3 \, {\left (d e^{2} f + c e f^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x}{8 \, b^{2}} + \frac {3 \, {\left (d e^{2} f + c e f^{2}\right )} \sqrt {b x^{2} + a} a^{2} x}{16 \, b^{2}} + \frac {{\left (d e^{3} + 3 \, c e^{2} f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x}{4 \, b} - \frac {{\left (d e^{3} + 3 \, c e^{2} f\right )} \sqrt {b x^{2} + a} a x}{8 \, b} - \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {7}{2}}} + \frac {3 \, {\left (d e^{2} f + c e f^{2}\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {5}{2}}} - \frac {{\left (d e^{3} + 3 \, c e^{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)*(f*x^2+e)^3,x, algorithm="maxima")
 

Output:

1/10*(b*x^2 + a)^(3/2)*d*f^3*x^7/b - 7/80*(b*x^2 + a)^(3/2)*a*d*f^3*x^5/b^ 
2 + 7/96*(b*x^2 + a)^(3/2)*a^2*d*f^3*x^3/b^3 + 1/8*(3*d*e*f^2 + c*f^3)*(b* 
x^2 + a)^(3/2)*x^5/b + 1/2*sqrt(b*x^2 + a)*c*e^3*x - 7/128*(b*x^2 + a)^(3/ 
2)*a^3*d*f^3*x/b^4 + 7/256*sqrt(b*x^2 + a)*a^4*d*f^3*x/b^4 + 1/2*a*c*e^3*a 
rcsinh(b*x/sqrt(a*b))/sqrt(b) + 7/256*a^5*d*f^3*arcsinh(b*x/sqrt(a*b))/b^( 
9/2) - 5/48*(3*d*e*f^2 + c*f^3)*(b*x^2 + a)^(3/2)*a*x^3/b^2 + 1/2*(d*e^2*f 
 + c*e*f^2)*(b*x^2 + a)^(3/2)*x^3/b + 5/64*(3*d*e*f^2 + c*f^3)*(b*x^2 + a) 
^(3/2)*a^2*x/b^3 - 5/128*(3*d*e*f^2 + c*f^3)*sqrt(b*x^2 + a)*a^3*x/b^3 - 3 
/8*(d*e^2*f + c*e*f^2)*(b*x^2 + a)^(3/2)*a*x/b^2 + 3/16*(d*e^2*f + c*e*f^2 
)*sqrt(b*x^2 + a)*a^2*x/b^2 + 1/4*(d*e^3 + 3*c*e^2*f)*(b*x^2 + a)^(3/2)*x/ 
b - 1/8*(d*e^3 + 3*c*e^2*f)*sqrt(b*x^2 + a)*a*x/b - 5/128*(3*d*e*f^2 + c*f 
^3)*a^4*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 3/16*(d*e^2*f + c*e*f^2)*a^3*arcs 
inh(b*x/sqrt(a*b))/b^(5/2) - 1/8*(d*e^3 + 3*c*e^2*f)*a^2*arcsinh(b*x/sqrt( 
a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.10 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, d f^{3} x^{2} + \frac {30 \, b^{8} d e f^{2} + 10 \, b^{8} c f^{3} + a b^{7} d f^{3}}{b^{8}}\right )} x^{2} + \frac {240 \, b^{8} d e^{2} f + 240 \, b^{8} c e f^{2} + 30 \, a b^{7} d e f^{2} + 10 \, a b^{7} c f^{3} - 7 \, a^{2} b^{6} d f^{3}}{b^{8}}\right )} x^{2} + \frac {5 \, {\left (96 \, b^{8} d e^{3} + 288 \, b^{8} c e^{2} f + 48 \, a b^{7} d e^{2} f + 48 \, a b^{7} c e f^{2} - 30 \, a^{2} b^{6} d e f^{2} - 10 \, a^{2} b^{6} c f^{3} + 7 \, a^{3} b^{5} d f^{3}\right )}}{b^{8}}\right )} x^{2} + \frac {15 \, {\left (128 \, b^{8} c e^{3} + 32 \, a b^{7} d e^{3} + 96 \, a b^{7} c e^{2} f - 48 \, a^{2} b^{6} d e^{2} f - 48 \, a^{2} b^{6} c e f^{2} + 30 \, a^{3} b^{5} d e f^{2} + 10 \, a^{3} b^{5} c f^{3} - 7 \, a^{4} b^{4} d f^{3}\right )}}{b^{8}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (128 \, a b^{4} c e^{3} - 32 \, a^{2} b^{3} d e^{3} - 96 \, a^{2} b^{3} c e^{2} f + 48 \, a^{3} b^{2} d e^{2} f + 48 \, a^{3} b^{2} c e f^{2} - 30 \, a^{4} b d e f^{2} - 10 \, a^{4} b c f^{3} + 7 \, a^{5} d f^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {9}{2}}} \] Input:

integrate((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e)^3,x, algorithm="giac")
 

Output:

1/3840*(2*(4*(6*(8*d*f^3*x^2 + (30*b^8*d*e*f^2 + 10*b^8*c*f^3 + a*b^7*d*f^ 
3)/b^8)*x^2 + (240*b^8*d*e^2*f + 240*b^8*c*e*f^2 + 30*a*b^7*d*e*f^2 + 10*a 
*b^7*c*f^3 - 7*a^2*b^6*d*f^3)/b^8)*x^2 + 5*(96*b^8*d*e^3 + 288*b^8*c*e^2*f 
 + 48*a*b^7*d*e^2*f + 48*a*b^7*c*e*f^2 - 30*a^2*b^6*d*e*f^2 - 10*a^2*b^6*c 
*f^3 + 7*a^3*b^5*d*f^3)/b^8)*x^2 + 15*(128*b^8*c*e^3 + 32*a*b^7*d*e^3 + 96 
*a*b^7*c*e^2*f - 48*a^2*b^6*d*e^2*f - 48*a^2*b^6*c*e*f^2 + 30*a^3*b^5*d*e* 
f^2 + 10*a^3*b^5*c*f^3 - 7*a^4*b^4*d*f^3)/b^8)*sqrt(b*x^2 + a)*x - 1/256*( 
128*a*b^4*c*e^3 - 32*a^2*b^3*d*e^3 - 96*a^2*b^3*c*e^2*f + 48*a^3*b^2*d*e^2 
*f + 48*a^3*b^2*c*e*f^2 - 30*a^4*b*d*e*f^2 - 10*a^4*b*c*f^3 + 7*a^5*d*f^3) 
*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

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

int((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2)^3,x)
 

Output:

int((a + b*x^2)^(1/2)*(c + d*x^2)*(e + f*x^2)^3, x)
 

Reduce [F]

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

int((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e)^3,x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)*(f*x^2+e)^3,x)