\(\int (a+b x^2)^{3/2} (c+d x^2) (e+f x^2)^3 \, dx\) [290]

Optimal result

Integrand size = 28, antiderivative size = 482 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {a \left (384 b^4 c e^3+7 a^4 d f^3+72 a^2 b^2 e f (d e+c f)-12 a^3 b f^2 (3 d e+c f)-64 a b^3 e^2 (d e+3 c f)\right ) x \sqrt {a+b x^2}}{1024 b^4}+\frac {\left (384 b^4 c e^3+7 a^4 d f^3+72 a^2 b^2 e f (d e+c f)-12 a^3 b f^2 (3 d e+c f)-64 a b^3 e^2 (d e+3 c f)\right ) x \left (a+b x^2\right )^{3/2}}{1536 b^4}-\frac {\left (7 a^3 d f^3+72 a b^2 e f (d e+c f)-12 a^2 b f^2 (3 d e+c f)-64 b^3 e^2 (d e+3 c f)\right ) x \left (a+b x^2\right )^{5/2}}{384 b^4}+\frac {f \left (7 a^2 d f^2+72 b^2 e (d e+c f)-12 a b f (3 d e+c f)\right ) x^3 \left (a+b x^2\right )^{5/2}}{192 b^3}-\frac {f^2 (7 a d f-12 b (3 d e+c f)) x^5 \left (a+b x^2\right )^{5/2}}{120 b^2}+\frac {d f^3 x^7 \left (a+b x^2\right )^{5/2}}{12 b}+\frac {a^2 \left (384 b^4 c e^3+7 a^4 d f^3+72 a^2 b^2 e f (d e+c f)-12 a^3 b f^2 (3 d e+c f)-64 a b^3 e^2 (d e+3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{9/2}} \] Output:

1/1024*a*(384*b^4*c*e^3+7*a^4*d*f^3+72*a^2*b^2*e*f*(c*f+d*e)-12*a^3*b*f^2* 
(c*f+3*d*e)-64*a*b^3*e^2*(3*c*f+d*e))*x*(b*x^2+a)^(1/2)/b^4+1/1536*(384*b^ 
4*c*e^3+7*a^4*d*f^3+72*a^2*b^2*e*f*(c*f+d*e)-12*a^3*b*f^2*(c*f+3*d*e)-64*a 
*b^3*e^2*(3*c*f+d*e))*x*(b*x^2+a)^(3/2)/b^4-1/384*(7*a^3*d*f^3+72*a*b^2*e* 
f*(c*f+d*e)-12*a^2*b*f^2*(c*f+3*d*e)-64*b^3*e^2*(3*c*f+d*e))*x*(b*x^2+a)^( 
5/2)/b^4+1/192*f*(7*a^2*d*f^2+72*b^2*e*(c*f+d*e)-12*a*b*f*(c*f+3*d*e))*x^3 
*(b*x^2+a)^(5/2)/b^3-1/120*f^2*(7*a*d*f-12*b*(c*f+3*d*e))*x^5*(b*x^2+a)^(5 
/2)/b^2+1/12*d*f^3*x^7*(b*x^2+a)^(5/2)/b+1/1024*a^2*(384*b^4*c*e^3+7*a^4*d 
*f^3+72*a^2*b^2*e*f*(c*f+d*e)-12*a^3*b*f^2*(c*f+3*d*e)-64*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 = 1.58 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.89 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^5 d f^3+10 a^4 b f^2 \left (54 d e+18 c f+7 d f x^2\right )-8 a^3 b^2 f \left (15 c f \left (9 e+f x^2\right )+d \left (135 e^2+45 e f x^2+7 f^2 x^4\right )\right )+48 a^2 b^3 \left (c f \left (60 e^2+15 e f x^2+2 f^2 x^4\right )+d \left (20 e^3+15 e^2 f x^2+6 e f^2 x^4+f^3 x^6\right )\right )+128 b^5 x^2 \left (3 c \left (10 e^3+20 e^2 f x^2+15 e f^2 x^4+4 f^3 x^6\right )+d x^2 \left (20 e^3+45 e^2 f x^2+36 e f^2 x^4+10 f^3 x^6\right )\right )+64 a b^4 \left (3 c \left (50 e^3+70 e^2 f x^2+45 e f^2 x^4+11 f^3 x^6\right )+d x^2 \left (70 e^3+135 e^2 f x^2+99 e f^2 x^4+26 f^3 x^6\right )\right )\right )-15 a^2 \left (384 b^4 c e^3+7 a^4 d f^3+72 a^2 b^2 e f (d e+c f)-12 a^3 b f^2 (3 d e+c f)-64 a b^3 e^2 (d e+3 c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{15360 b^{9/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^5*d*f^3 + 10*a^4*b*f^2*(54*d*e + 18*c*f 
 + 7*d*f*x^2) - 8*a^3*b^2*f*(15*c*f*(9*e + f*x^2) + d*(135*e^2 + 45*e*f*x^ 
2 + 7*f^2*x^4)) + 48*a^2*b^3*(c*f*(60*e^2 + 15*e*f*x^2 + 2*f^2*x^4) + d*(2 
0*e^3 + 15*e^2*f*x^2 + 6*e*f^2*x^4 + f^3*x^6)) + 128*b^5*x^2*(3*c*(10*e^3 
+ 20*e^2*f*x^2 + 15*e*f^2*x^4 + 4*f^3*x^6) + d*x^2*(20*e^3 + 45*e^2*f*x^2 
+ 36*e*f^2*x^4 + 10*f^3*x^6)) + 64*a*b^4*(3*c*(50*e^3 + 70*e^2*f*x^2 + 45* 
e*f^2*x^4 + 11*f^3*x^6) + d*x^2*(70*e^3 + 135*e^2*f*x^2 + 99*e*f^2*x^4 + 2 
6*f^3*x^6))) - 15*a^2*(384*b^4*c*e^3 + 7*a^4*d*f^3 + 72*a^2*b^2*e*f*(d*e + 
 c*f) - 12*a^3*b*f^2*(3*d*e + c*f) - 64*a*b^3*e^2*(d*e + 3*c*f))*Log[-(Sqr 
t[b]*x) + Sqrt[a + b*x^2]])/(15360*b^(9/2))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {403, 403, 403, 299, 211, 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.

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\int \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2 \left ((12 b c-a d) e-(7 a d f-6 b (d e+2 c f)) x^2\right )dx}{12 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3}{12 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\int \left (b x^2+a\right )^{3/2} \left (f x^2+e\right ) \left (\left (24 e (d e+7 c f) b^2-4 a f (17 d e+15 c f) b+35 a^2 d f^2\right ) x^2+e \left (7 d f a^2-16 b d e a-12 b c f a+120 b^2 c e\right )\right )dx}{10 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2 (-7 a d f+12 b c f+6 b d e)}{10 b}}{12 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3}{12 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\frac {\int \left (b x^2+a\right )^{3/2} \left (e \left (-35 d f^2 a^3+4 b f (31 d e+15 c f) a^2-8 b^2 e (19 d e+33 c f) a+960 b^3 c e^2\right )-3 \left (-16 e^2 (d e+27 c f) b^3+16 a e f (7 d e+15 c f) b^2-10 a^2 f^2 (11 d e+6 c f) b+35 a^3 d f^3\right ) x^2\right )dx}{8 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-4 a b f (15 c f+17 d e)+24 b^2 e (7 c f+d e)\right )}{8 b}}{10 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2 (-7 a d f+12 b c f+6 b d e)}{10 b}}{12 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3}{12 b}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {\frac {5 \left (7 a^4 d f^3-12 a^3 b f^2 (c f+3 d e)+72 a^2 b^2 e f (c f+d e)-64 a b^3 e^2 (3 c f+d e)+384 b^4 c e^3\right ) \int \left (b x^2+a\right )^{3/2}dx}{2 b}-\frac {x \left (a+b x^2\right )^{5/2} \left (35 a^3 d f^3-10 a^2 b f^2 (6 c f+11 d e)+16 a b^2 e f (15 c f+7 d e)-16 b^3 e^2 (27 c f+d e)\right )}{2 b}}{8 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-4 a b f (15 c f+17 d e)+24 b^2 e (7 c f+d e)\right )}{8 b}}{10 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2 (-7 a d f+12 b c f+6 b d e)}{10 b}}{12 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3}{12 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {5 \left (7 a^4 d f^3-12 a^3 b f^2 (c f+3 d e)+72 a^2 b^2 e f (c f+d e)-64 a b^3 e^2 (3 c f+d e)+384 b^4 c e^3\right ) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {x \left (a+b x^2\right )^{5/2} \left (35 a^3 d f^3-10 a^2 b f^2 (6 c f+11 d e)+16 a b^2 e f (15 c f+7 d e)-16 b^3 e^2 (27 c f+d e)\right )}{2 b}}{8 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-4 a b f (15 c f+17 d e)+24 b^2 e (7 c f+d e)\right )}{8 b}}{10 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2 (-7 a d f+12 b c f+6 b d e)}{10 b}}{12 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3}{12 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {5 \left (7 a^4 d f^3-12 a^3 b f^2 (c f+3 d e)+72 a^2 b^2 e f (c f+d e)-64 a b^3 e^2 (3 c f+d e)+384 b^4 c e^3\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {x \left (a+b x^2\right )^{5/2} \left (35 a^3 d f^3-10 a^2 b f^2 (6 c f+11 d e)+16 a b^2 e f (15 c f+7 d e)-16 b^3 e^2 (27 c f+d e)\right )}{2 b}}{8 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-4 a b f (15 c f+17 d e)+24 b^2 e (7 c f+d e)\right )}{8 b}}{10 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2 (-7 a d f+12 b c f+6 b d e)}{10 b}}{12 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3}{12 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {5 \left (7 a^4 d f^3-12 a^3 b f^2 (c f+3 d e)+72 a^2 b^2 e f (c f+d e)-64 a b^3 e^2 (3 c f+d e)+384 b^4 c e^3\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {x \left (a+b x^2\right )^{5/2} \left (35 a^3 d f^3-10 a^2 b f^2 (6 c f+11 d e)+16 a b^2 e f (15 c f+7 d e)-16 b^3 e^2 (27 c f+d e)\right )}{2 b}}{8 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-4 a b f (15 c f+17 d e)+24 b^2 e (7 c f+d e)\right )}{8 b}}{10 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2 (-7 a d f+12 b c f+6 b d e)}{10 b}}{12 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3}{12 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right ) \left (35 a^2 d f^2-4 a b f (15 c f+17 d e)+24 b^2 e (7 c f+d e)\right )}{8 b}+\frac {\frac {5 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) \left (7 a^4 d f^3-12 a^3 b f^2 (c f+3 d e)+72 a^2 b^2 e f (c f+d e)-64 a b^3 e^2 (3 c f+d e)+384 b^4 c e^3\right )}{2 b}-\frac {x \left (a+b x^2\right )^{5/2} \left (35 a^3 d f^3-10 a^2 b f^2 (6 c f+11 d e)+16 a b^2 e f (15 c f+7 d e)-16 b^3 e^2 (27 c f+d e)\right )}{2 b}}{8 b}}{10 b}+\frac {x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2 (-7 a d f+12 b c f+6 b d e)}{10 b}}{12 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^3}{12 b}\)

Input:

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

Output:

(d*x*(a + b*x^2)^(5/2)*(e + f*x^2)^3)/(12*b) + (((6*b*d*e + 12*b*c*f - 7*a 
*d*f)*x*(a + b*x^2)^(5/2)*(e + f*x^2)^2)/(10*b) + (((35*a^2*d*f^2 + 24*b^2 
*e*(d*e + 7*c*f) - 4*a*b*f*(17*d*e + 15*c*f))*x*(a + b*x^2)^(5/2)*(e + f*x 
^2))/(8*b) + (-1/2*((35*a^3*d*f^3 - 10*a^2*b*f^2*(11*d*e + 6*c*f) + 16*a*b 
^2*e*f*(7*d*e + 15*c*f) - 16*b^3*e^2*(d*e + 27*c*f))*x*(a + b*x^2)^(5/2))/ 
b + (5*(384*b^4*c*e^3 + 7*a^4*d*f^3 + 72*a^2*b^2*e*f*(d*e + c*f) - 12*a^3* 
b*f^2*(3*d*e + c*f) - 64*a*b^3*e^2*(d*e + 3*c*f))*((x*(a + b*x^2)^(3/2))/4 
 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/ 
(2*Sqrt[b])))/4))/(2*b))/(8*b))/(10*b))/(12*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 = 0.90 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {\frac {7 a^{2} \left (a^{3} \left (-\frac {12 b c}{7}+a d \right ) f^{3}-\frac {36 a^{2} b e \left (a d -2 b c \right ) f^{2}}{7}+\frac {72 a \left (a d -\frac {8 b c}{3}\right ) b^{2} e^{2} f}{7}-\frac {64 e^{3} b^{3} \left (a d -6 b c \right )}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{1024}-\frac {7 \left (-\frac {256 \left (\frac {2 \left (\frac {5 x^{2} d}{6}+c \right ) x^{6} f^{3}}{5}+\frac {3 \left (\frac {4 x^{2} d}{5}+c \right ) x^{4} e \,f^{2}}{2}+2 \left (\frac {3 x^{2} d}{4}+c \right ) x^{2} e^{2} f +e^{3} \left (\frac {2 x^{2} d}{3}+c \right )\right ) x^{2} b^{\frac {11}{2}}}{7}+a \left (64 \left (-\frac {11 x^{6} \left (\frac {26 x^{2} d}{33}+c \right ) f^{3}}{35}-\frac {9 \left (\frac {11 x^{2} d}{15}+c \right ) x^{4} e \,f^{2}}{7}-2 x^{2} \left (\frac {9 x^{2} d}{14}+c \right ) e^{2} f -\frac {10 \left (\frac {7 x^{2} d}{15}+c \right ) e^{3}}{7}\right ) b^{\frac {9}{2}}+a \left (\frac {16 \left (-\frac {2 \left (\frac {x^{2} d}{2}+c \right ) x^{4} f^{3}}{5}-3 \left (\frac {2 x^{2} d}{5}+c \right ) x^{2} e \,f^{2}-12 \left (\frac {x^{2} d}{4}+c \right ) e^{2} f -4 d \,e^{3}\right ) b^{\frac {7}{2}}}{7}+a \left (\frac {8 \left (\left (\frac {7 x^{2} d}{15}+c \right ) x^{2} f^{2}+9 \left (\frac {x^{2} d}{3}+c \right ) e f +9 d \,e^{2}\right ) b^{\frac {5}{2}}}{7}+a \left (2 \left (\left (-\frac {x^{2} d}{3}-\frac {6 c}{7}\right ) f -\frac {18 d e}{7}\right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right ) f \right ) f \right )\right )\right ) x \sqrt {b \,x^{2}+a}}{1024}}{b^{\frac {9}{2}}}\)	\(390\)
default	\(c \,e^{3} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 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}\right )+f^{2} \left (c f +3 d e \right ) \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 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}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )+3 e f \left (c f +d e \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 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}\right )}{6 b}\right )}{8 b}\right )+e^{2} \left (3 c f +d e \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 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}\right )}{6 b}\right )+d \,f^{3} \left (\frac {x^{7} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{12 b}-\frac {7 a \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 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}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )}{12 b}\right )\)	\(534\)
risch	\(-\frac {x \left (-1280 b^{5} d \,f^{3} x^{10}-1664 a \,b^{4} d \,f^{3} x^{8}-1536 b^{5} c \,f^{3} x^{8}-4608 b^{5} d e \,f^{2} x^{8}-48 a^{2} b^{3} d \,f^{3} x^{6}-2112 a \,b^{4} c \,f^{3} x^{6}-6336 a \,b^{4} d e \,f^{2} x^{6}-5760 b^{5} c e \,f^{2} x^{6}-5760 b^{5} d \,e^{2} f \,x^{6}+56 a^{3} b^{2} d \,f^{3} x^{4}-96 a^{2} b^{3} c \,f^{3} x^{4}-288 a^{2} b^{3} d e \,f^{2} x^{4}-8640 a \,b^{4} c e \,f^{2} x^{4}-8640 a \,b^{4} d \,e^{2} f \,x^{4}-7680 b^{5} c \,e^{2} f \,x^{4}-2560 b^{5} d \,e^{3} x^{4}-70 a^{4} b d \,f^{3} x^{2}+120 a^{3} b^{2} c \,f^{3} x^{2}+360 a^{3} b^{2} d e \,f^{2} x^{2}-720 a^{2} b^{3} c e \,f^{2} x^{2}-720 a^{2} b^{3} d \,e^{2} f \,x^{2}-13440 a \,b^{4} c \,e^{2} f \,x^{2}-4480 a \,b^{4} d \,e^{3} x^{2}-3840 b^{5} c \,e^{3} x^{2}+105 a^{5} d \,f^{3}-180 a^{4} b c \,f^{3}-540 a^{4} b d e \,f^{2}+1080 a^{3} b^{2} c e \,f^{2}+1080 a^{3} b^{2} d \,e^{2} f -2880 a^{2} b^{3} c \,e^{2} f -960 a^{2} b^{3} d \,e^{3}-9600 a \,b^{4} c \,e^{3}\right ) \sqrt {b \,x^{2}+a}}{15360 b^{4}}+\frac {a^{2} \left (7 a^{4} d \,f^{3}-12 a^{3} b c \,f^{3}-36 a^{3} b d e \,f^{2}+72 a^{2} b^{2} c e \,f^{2}+72 a^{2} b^{2} d \,e^{2} f -192 a \,b^{3} c \,e^{2} f -64 a \,b^{3} d \,e^{3}+384 b^{4} c \,e^{3}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {9}{2}}}\)	\(552\)

Input:

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

Output:

7/1024*(a^2*(a^3*(-12/7*b*c+a*d)*f^3-36/7*a^2*b*e*(a*d-2*b*c)*f^2+72/7*a*( 
a*d-8/3*b*c)*b^2*e^2*f-64/7*e^3*b^3*(a*d-6*b*c))*arctanh((b*x^2+a)^(1/2)/x 
/b^(1/2))-(-256/7*(2/5*(5/6*x^2*d+c)*x^6*f^3+3/2*(4/5*x^2*d+c)*x^4*e*f^2+2 
*(3/4*x^2*d+c)*x^2*e^2*f+e^3*(2/3*x^2*d+c))*x^2*b^(11/2)+a*(64*(-11/35*x^6 
*(26/33*x^2*d+c)*f^3-9/7*(11/15*x^2*d+c)*x^4*e*f^2-2*x^2*(9/14*x^2*d+c)*e^ 
2*f-10/7*(7/15*x^2*d+c)*e^3)*b^(9/2)+a*(16/7*(-2/5*(1/2*x^2*d+c)*x^4*f^3-3 
*(2/5*x^2*d+c)*x^2*e*f^2-12*(1/4*x^2*d+c)*e^2*f-4*d*e^3)*b^(7/2)+a*(8/7*(( 
7/15*x^2*d+c)*x^2*f^2+9*(1/3*x^2*d+c)*e*f+9*d*e^2)*b^(5/2)+a*(2*((-1/3*x^2 
*d-6/7*c)*f-18/7*d*e)*b^(3/2)+a*d*f*b^(1/2))*f)*f)))*x*(b*x^2+a)^(1/2))/b^ 
(9/2)
 

Fricas [A] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 1026, normalized size of antiderivative = 2.13 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

[1/30720*(15*(64*(6*a^2*b^4*c - a^3*b^3*d)*e^3 - 24*(8*a^3*b^3*c - 3*a^4*b 
^2*d)*e^2*f + 36*(2*a^4*b^2*c - a^5*b*d)*e*f^2 - (12*a^5*b*c - 7*a^6*d)*f^ 
3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(1280*b^6*d 
*f^3*x^11 + 128*(36*b^6*d*e*f^2 + (12*b^6*c + 13*a*b^5*d)*f^3)*x^9 + 48*(1 
20*b^6*d*e^2*f + 12*(10*b^6*c + 11*a*b^5*d)*e*f^2 + (44*a*b^5*c + a^2*b^4* 
d)*f^3)*x^7 + 8*(320*b^6*d*e^3 + 120*(8*b^6*c + 9*a*b^5*d)*e^2*f + 36*(30* 
a*b^5*c + a^2*b^4*d)*e*f^2 + (12*a^2*b^4*c - 7*a^3*b^3*d)*f^3)*x^5 + 10*(6 
4*(6*b^6*c + 7*a*b^5*d)*e^3 + 24*(56*a*b^5*c + 3*a^2*b^4*d)*e^2*f + 36*(2* 
a^2*b^4*c - a^3*b^3*d)*e*f^2 - (12*a^3*b^3*c - 7*a^4*b^2*d)*f^3)*x^3 + 15* 
(64*(10*a*b^5*c + a^2*b^4*d)*e^3 + 24*(8*a^2*b^4*c - 3*a^3*b^3*d)*e^2*f - 
36*(2*a^3*b^3*c - a^4*b^2*d)*e*f^2 + (12*a^4*b^2*c - 7*a^5*b*d)*f^3)*x)*sq 
rt(b*x^2 + a))/b^5, -1/15360*(15*(64*(6*a^2*b^4*c - a^3*b^3*d)*e^3 - 24*(8 
*a^3*b^3*c - 3*a^4*b^2*d)*e^2*f + 36*(2*a^4*b^2*c - a^5*b*d)*e*f^2 - (12*a 
^5*b*c - 7*a^6*d)*f^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (1280 
*b^6*d*f^3*x^11 + 128*(36*b^6*d*e*f^2 + (12*b^6*c + 13*a*b^5*d)*f^3)*x^9 + 
 48*(120*b^6*d*e^2*f + 12*(10*b^6*c + 11*a*b^5*d)*e*f^2 + (44*a*b^5*c + a^ 
2*b^4*d)*f^3)*x^7 + 8*(320*b^6*d*e^3 + 120*(8*b^6*c + 9*a*b^5*d)*e^2*f + 3 
6*(30*a*b^5*c + a^2*b^4*d)*e*f^2 + (12*a^2*b^4*c - 7*a^3*b^3*d)*f^3)*x^5 + 
 10*(64*(6*b^6*c + 7*a*b^5*d)*e^3 + 24*(56*a*b^5*c + 3*a^2*b^4*d)*e^2*f + 
36*(2*a^2*b^4*c - a^3*b^3*d)*e*f^2 - (12*a^3*b^3*c - 7*a^4*b^2*d)*f^3)*...
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1234 vs. \(2 (498) = 996\).

Time = 0.61 (sec) , antiderivative size = 1234, normalized size of antiderivative = 2.56 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x**2)*(b*d*f**3*x**11/12 + x**9*(13*a*b*d*f**3/12 + 
b**2*c*f**3 + 3*b**2*d*e*f**2)/(10*b) + x**7*(a**2*d*f**3 + 2*a*b*c*f**3 + 
 6*a*b*d*e*f**2 - 9*a*(13*a*b*d*f**3/12 + b**2*c*f**3 + 3*b**2*d*e*f**2)/( 
10*b) + 3*b**2*c*e*f**2 + 3*b**2*d*e**2*f)/(8*b) + x**5*(a**2*c*f**3 + 3*a 
**2*d*e*f**2 + 6*a*b*c*e*f**2 + 6*a*b*d*e**2*f - 7*a*(a**2*d*f**3 + 2*a*b* 
c*f**3 + 6*a*b*d*e*f**2 - 9*a*(13*a*b*d*f**3/12 + b**2*c*f**3 + 3*b**2*d*e 
*f**2)/(10*b) + 3*b**2*c*e*f**2 + 3*b**2*d*e**2*f)/(8*b) + 3*b**2*c*e**2*f 
 + b**2*d*e**3)/(6*b) + x**3*(3*a**2*c*e*f**2 + 3*a**2*d*e**2*f + 6*a*b*c* 
e**2*f + 2*a*b*d*e**3 - 5*a*(a**2*c*f**3 + 3*a**2*d*e*f**2 + 6*a*b*c*e*f** 
2 + 6*a*b*d*e**2*f - 7*a*(a**2*d*f**3 + 2*a*b*c*f**3 + 6*a*b*d*e*f**2 - 9* 
a*(13*a*b*d*f**3/12 + b**2*c*f**3 + 3*b**2*d*e*f**2)/(10*b) + 3*b**2*c*e*f 
**2 + 3*b**2*d*e**2*f)/(8*b) + 3*b**2*c*e**2*f + b**2*d*e**3)/(6*b) + b**2 
*c*e**3)/(4*b) + x*(3*a**2*c*e**2*f + a**2*d*e**3 + 2*a*b*c*e**3 - 3*a*(3* 
a**2*c*e*f**2 + 3*a**2*d*e**2*f + 6*a*b*c*e**2*f + 2*a*b*d*e**3 - 5*a*(a** 
2*c*f**3 + 3*a**2*d*e*f**2 + 6*a*b*c*e*f**2 + 6*a*b*d*e**2*f - 7*a*(a**2*d 
*f**3 + 2*a*b*c*f**3 + 6*a*b*d*e*f**2 - 9*a*(13*a*b*d*f**3/12 + b**2*c*f** 
3 + 3*b**2*d*e*f**2)/(10*b) + 3*b**2*c*e*f**2 + 3*b**2*d*e**2*f)/(8*b) + 3 
*b**2*c*e**2*f + b**2*d*e**3)/(6*b) + b**2*c*e**3)/(4*b))/(2*b)) + (a**2*c 
*e**3 - a*(3*a**2*c*e**2*f + a**2*d*e**3 + 2*a*b*c*e**3 - 3*a*(3*a**2*c*e* 
f**2 + 3*a**2*d*e**2*f + 6*a*b*c*e**2*f + 2*a*b*d*e**3 - 5*a*(a**2*c*f*...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.37 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/12*(b*x^2 + a)^(5/2)*d*f^3*x^7/b - 7/120*(b*x^2 + a)^(5/2)*a*d*f^3*x^5/b 
^2 + 7/192*(b*x^2 + a)^(5/2)*a^2*d*f^3*x^3/b^3 + 1/10*(3*d*e*f^2 + c*f^3)* 
(b*x^2 + a)^(5/2)*x^5/b + 1/4*(b*x^2 + a)^(3/2)*c*e^3*x + 3/8*sqrt(b*x^2 + 
 a)*a*c*e^3*x - 7/384*(b*x^2 + a)^(5/2)*a^3*d*f^3*x/b^4 + 7/1536*(b*x^2 + 
a)^(3/2)*a^4*d*f^3*x/b^4 + 7/1024*sqrt(b*x^2 + a)*a^5*d*f^3*x/b^4 + 3/8*a^ 
2*c*e^3*arcsinh(b*x/sqrt(a*b))/sqrt(b) + 7/1024*a^6*d*f^3*arcsinh(b*x/sqrt 
(a*b))/b^(9/2) - 1/16*(3*d*e*f^2 + c*f^3)*(b*x^2 + a)^(5/2)*a*x^3/b^2 + 3/ 
8*(d*e^2*f + c*e*f^2)*(b*x^2 + a)^(5/2)*x^3/b + 1/32*(3*d*e*f^2 + c*f^3)*( 
b*x^2 + a)^(5/2)*a^2*x/b^3 - 1/128*(3*d*e*f^2 + c*f^3)*(b*x^2 + a)^(3/2)*a 
^3*x/b^3 - 3/256*(3*d*e*f^2 + c*f^3)*sqrt(b*x^2 + a)*a^4*x/b^3 - 3/16*(d*e 
^2*f + c*e*f^2)*(b*x^2 + a)^(5/2)*a*x/b^2 + 3/64*(d*e^2*f + c*e*f^2)*(b*x^ 
2 + a)^(3/2)*a^2*x/b^2 + 9/128*(d*e^2*f + c*e*f^2)*sqrt(b*x^2 + a)*a^3*x/b 
^2 + 1/6*(d*e^3 + 3*c*e^2*f)*(b*x^2 + a)^(5/2)*x/b - 1/24*(d*e^3 + 3*c*e^2 
*f)*(b*x^2 + a)^(3/2)*a*x/b - 1/16*(d*e^3 + 3*c*e^2*f)*sqrt(b*x^2 + a)*a^2 
*x/b - 3/256*(3*d*e*f^2 + c*f^3)*a^5*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 9/12 
8*(d*e^2*f + c*e*f^2)*a^4*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 1/16*(d*e^3 + 3 
*c*e^2*f)*a^3*arcsinh(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.13 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b d f^{3} x^{2} + \frac {36 \, b^{11} d e f^{2} + 12 \, b^{11} c f^{3} + 13 \, a b^{10} d f^{3}}{b^{10}}\right )} x^{2} + \frac {3 \, {\left (120 \, b^{11} d e^{2} f + 120 \, b^{11} c e f^{2} + 132 \, a b^{10} d e f^{2} + 44 \, a b^{10} c f^{3} + a^{2} b^{9} d f^{3}\right )}}{b^{10}}\right )} x^{2} + \frac {320 \, b^{11} d e^{3} + 960 \, b^{11} c e^{2} f + 1080 \, a b^{10} d e^{2} f + 1080 \, a b^{10} c e f^{2} + 36 \, a^{2} b^{9} d e f^{2} + 12 \, a^{2} b^{9} c f^{3} - 7 \, a^{3} b^{8} d f^{3}}{b^{10}}\right )} x^{2} + \frac {5 \, {\left (384 \, b^{11} c e^{3} + 448 \, a b^{10} d e^{3} + 1344 \, a b^{10} c e^{2} f + 72 \, a^{2} b^{9} d e^{2} f + 72 \, a^{2} b^{9} c e f^{2} - 36 \, a^{3} b^{8} d e f^{2} - 12 \, a^{3} b^{8} c f^{3} + 7 \, a^{4} b^{7} d f^{3}\right )}}{b^{10}}\right )} x^{2} + \frac {15 \, {\left (640 \, a b^{10} c e^{3} + 64 \, a^{2} b^{9} d e^{3} + 192 \, a^{2} b^{9} c e^{2} f - 72 \, a^{3} b^{8} d e^{2} f - 72 \, a^{3} b^{8} c e f^{2} + 36 \, a^{4} b^{7} d e f^{2} + 12 \, a^{4} b^{7} c f^{3} - 7 \, a^{5} b^{6} d f^{3}\right )}}{b^{10}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (384 \, a^{2} b^{4} c e^{3} - 64 \, a^{3} b^{3} d e^{3} - 192 \, a^{3} b^{3} c e^{2} f + 72 \, a^{4} b^{2} d e^{2} f + 72 \, a^{4} b^{2} c e f^{2} - 36 \, a^{5} b d e f^{2} - 12 \, a^{5} b c f^{3} + 7 \, a^{6} d f^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{1024 \, b^{\frac {9}{2}}} \] Input:

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

Output:

1/15360*(2*(4*(2*(8*(10*b*d*f^3*x^2 + (36*b^11*d*e*f^2 + 12*b^11*c*f^3 + 1 
3*a*b^10*d*f^3)/b^10)*x^2 + 3*(120*b^11*d*e^2*f + 120*b^11*c*e*f^2 + 132*a 
*b^10*d*e*f^2 + 44*a*b^10*c*f^3 + a^2*b^9*d*f^3)/b^10)*x^2 + (320*b^11*d*e 
^3 + 960*b^11*c*e^2*f + 1080*a*b^10*d*e^2*f + 1080*a*b^10*c*e*f^2 + 36*a^2 
*b^9*d*e*f^2 + 12*a^2*b^9*c*f^3 - 7*a^3*b^8*d*f^3)/b^10)*x^2 + 5*(384*b^11 
*c*e^3 + 448*a*b^10*d*e^3 + 1344*a*b^10*c*e^2*f + 72*a^2*b^9*d*e^2*f + 72* 
a^2*b^9*c*e*f^2 - 36*a^3*b^8*d*e*f^2 - 12*a^3*b^8*c*f^3 + 7*a^4*b^7*d*f^3) 
/b^10)*x^2 + 15*(640*a*b^10*c*e^3 + 64*a^2*b^9*d*e^3 + 192*a^2*b^9*c*e^2*f 
 - 72*a^3*b^8*d*e^2*f - 72*a^3*b^8*c*e*f^2 + 36*a^4*b^7*d*e*f^2 + 12*a^4*b 
^7*c*f^3 - 7*a^5*b^6*d*f^3)/b^10)*sqrt(b*x^2 + a)*x - 1/1024*(384*a^2*b^4* 
c*e^3 - 64*a^3*b^3*d*e^3 - 192*a^3*b^3*c*e^2*f + 72*a^4*b^2*d*e^2*f + 72*a 
^4*b^2*c*e*f^2 - 36*a^5*b*d*e*f^2 - 12*a^5*b*c*f^3 + 7*a^6*d*f^3)*log(abs( 
-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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