\(\int (a+b x^2)^{3/2} (c+d x^2)^2 (e+f x^2)^3 \, dx\) [298]

Optimal result

Integrand size = 30, antiderivative size = 821 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {a \left (768 b^5 c^2 e^3-9 a^5 d^2 f^3+14 a^4 b d f^2 (3 d e+2 c f)-128 a b^4 c e^2 (2 d e+3 c f)-24 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+48 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{2048 b^5}+\frac {\left (768 b^5 c^2 e^3-9 a^5 d^2 f^3+14 a^4 b d f^2 (3 d e+2 c f)-128 a b^4 c e^2 (2 d e+3 c f)-24 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+48 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x \left (a+b x^2\right )^{3/2}}{3072 b^5}+\frac {\left (9 a^4 d^2 f^3-14 a^3 b d f^2 (3 d e+2 c f)+128 b^4 c e^2 (2 d e+3 c f)+24 a^2 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-48 a b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x \left (a+b x^2\right )^{5/2}}{768 b^5}-\frac {\left (9 a^3 d^2 f^3-14 a^2 b d f^2 (3 d e+2 c f)+24 a b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-48 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^3 \left (a+b x^2\right )^{5/2}}{384 b^4}+\frac {f \left (9 a^2 d^2 f^2-14 a b d f (3 d e+2 c f)+24 b^2 \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^5 \left (a+b x^2\right )^{5/2}}{240 b^3}+\frac {d f^2 (42 b d e+28 b c f-9 a d f) x^7 \left (a+b x^2\right )^{5/2}}{168 b^2}+\frac {d^2 f^3 x^9 \left (a+b x^2\right )^{5/2}}{14 b}+\frac {a^2 \left (768 b^5 c^2 e^3-9 a^5 d^2 f^3+14 a^4 b d f^2 (3 d e+2 c f)-128 a b^4 c e^2 (2 d e+3 c f)-24 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+48 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2048 b^{11/2}} \] Output:

1/2048*a*(768*b^5*c^2*e^3-9*a^5*d^2*f^3+14*a^4*b*d*f^2*(2*c*f+3*d*e)-128*a 
*b^4*c*e^2*(3*c*f+2*d*e)-24*a^3*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)+48*a^2 
*b^3*e*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)/b^5+1/3072*(768*b^ 
5*c^2*e^3-9*a^5*d^2*f^3+14*a^4*b*d*f^2*(2*c*f+3*d*e)-128*a*b^4*c*e^2*(3*c* 
f+2*d*e)-24*a^3*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)+48*a^2*b^3*e*(3*c^2*f^ 
2+6*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(3/2)/b^5+1/768*(9*a^4*d^2*f^3-14*a^3*b* 
d*f^2*(2*c*f+3*d*e)+128*b^4*c*e^2*(3*c*f+2*d*e)+24*a^2*b^2*f*(c^2*f^2+6*c* 
d*e*f+3*d^2*e^2)-48*a*b^3*e*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(5/ 
2)/b^5-1/384*(9*a^3*d^2*f^3-14*a^2*b*d*f^2*(2*c*f+3*d*e)+24*a*b^2*f*(c^2*f 
^2+6*c*d*e*f+3*d^2*e^2)-48*b^3*e*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*x^3*(b*x^2 
+a)^(5/2)/b^4+1/240*f*(9*a^2*d^2*f^2-14*a*b*d*f*(2*c*f+3*d*e)+24*b^2*(c^2* 
f^2+6*c*d*e*f+3*d^2*e^2))*x^5*(b*x^2+a)^(5/2)/b^3+1/168*d*f^2*(-9*a*d*f+28 
*b*c*f+42*b*d*e)*x^7*(b*x^2+a)^(5/2)/b^2+1/14*d^2*f^3*x^9*(b*x^2+a)^(5/2)/ 
b+1/2048*a^2*(768*b^5*c^2*e^3-9*a^5*d^2*f^3+14*a^4*b*d*f^2*(2*c*f+3*d*e)-1 
28*a*b^4*c*e^2*(3*c*f+2*d*e)-24*a^3*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)+48 
*a^2*b^3*e*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2 
))/b^(11/2)
 

Mathematica [A] (verified)

Time = 3.07 (sec) , antiderivative size = 722, normalized size of antiderivative = 0.88 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (945 a^6 d^2 f^3-210 a^5 b d f^2 \left (14 c f+3 d \left (7 e+f x^2\right )\right )+28 a^4 b^2 f \left (90 c^2 f^2+10 c d f \left (54 e+7 f x^2\right )+3 d^2 \left (90 e^2+35 e f x^2+6 f^2 x^4\right )\right )+96 a^2 b^4 \left (7 c^2 f \left (60 e^2+15 e f x^2+2 f^2 x^4\right )+14 c d \left (20 e^3+15 e^2 f x^2+6 e f^2 x^4+f^3 x^6\right )+d^2 x^2 \left (35 e^3+42 e^2 f x^2+21 e f^2 x^4+4 f^3 x^6\right )\right )-16 a^3 b^3 \left (105 c^2 f^2 \left (9 e+f x^2\right )+14 c d f \left (135 e^2+45 e f x^2+7 f^2 x^4\right )+3 d^2 \left (105 e^3+105 e^2 f x^2+49 e f^2 x^4+9 f^3 x^6\right )\right )+256 b^6 x^2 \left (21 c^2 \left (10 e^3+20 e^2 f x^2+15 e f^2 x^4+4 f^3 x^6\right )+14 c d x^2 \left (20 e^3+45 e^2 f x^2+36 e f^2 x^4+10 f^3 x^6\right )+3 d^2 x^4 \left (35 e^3+84 e^2 f x^2+70 e f^2 x^4+20 f^3 x^6\right )\right )+128 a b^5 \left (21 c^2 \left (50 e^3+70 e^2 f x^2+45 e f^2 x^4+11 f^3 x^6\right )+14 c d x^2 \left (70 e^3+135 e^2 f x^2+99 e f^2 x^4+26 f^3 x^6\right )+3 d^2 x^4 \left (105 e^3+231 e^2 f x^2+182 e f^2 x^4+50 f^3 x^6\right )\right )\right )+105 a^2 \left (-768 b^5 c^2 e^3+9 a^5 d^2 f^3-14 a^4 b d f^2 (3 d e+2 c f)+128 a b^4 c e^2 (2 d e+3 c f)+24 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-48 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{215040 b^{11/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(945*a^6*d^2*f^3 - 210*a^5*b*d*f^2*(14*c*f + 3* 
d*(7*e + f*x^2)) + 28*a^4*b^2*f*(90*c^2*f^2 + 10*c*d*f*(54*e + 7*f*x^2) + 
3*d^2*(90*e^2 + 35*e*f*x^2 + 6*f^2*x^4)) + 96*a^2*b^4*(7*c^2*f*(60*e^2 + 1 
5*e*f*x^2 + 2*f^2*x^4) + 14*c*d*(20*e^3 + 15*e^2*f*x^2 + 6*e*f^2*x^4 + f^3 
*x^6) + d^2*x^2*(35*e^3 + 42*e^2*f*x^2 + 21*e*f^2*x^4 + 4*f^3*x^6)) - 16*a 
^3*b^3*(105*c^2*f^2*(9*e + f*x^2) + 14*c*d*f*(135*e^2 + 45*e*f*x^2 + 7*f^2 
*x^4) + 3*d^2*(105*e^3 + 105*e^2*f*x^2 + 49*e*f^2*x^4 + 9*f^3*x^6)) + 256* 
b^6*x^2*(21*c^2*(10*e^3 + 20*e^2*f*x^2 + 15*e*f^2*x^4 + 4*f^3*x^6) + 14*c* 
d*x^2*(20*e^3 + 45*e^2*f*x^2 + 36*e*f^2*x^4 + 10*f^3*x^6) + 3*d^2*x^4*(35* 
e^3 + 84*e^2*f*x^2 + 70*e*f^2*x^4 + 20*f^3*x^6)) + 128*a*b^5*(21*c^2*(50*e 
^3 + 70*e^2*f*x^2 + 45*e*f^2*x^4 + 11*f^3*x^6) + 14*c*d*x^2*(70*e^3 + 135* 
e^2*f*x^2 + 99*e*f^2*x^4 + 26*f^3*x^6) + 3*d^2*x^4*(105*e^3 + 231*e^2*f*x^ 
2 + 182*e*f^2*x^4 + 50*f^3*x^6))) + 105*a^2*(-768*b^5*c^2*e^3 + 9*a^5*d^2* 
f^3 - 14*a^4*b*d*f^2*(3*d*e + 2*c*f) + 128*a*b^4*c*e^2*(2*d*e + 3*c*f) + 2 
4*a^3*b^2*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) - 48*a^2*b^3*e*(d^2*e^2 + 6* 
c*d*e*f + 3*c^2*f^2))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(215040*b^(11/2 
))
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 1217, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {433, 2009}

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.

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

\(\Big \downarrow \) 433

\(\displaystyle \int \left (f x^6 \left (a+b x^2\right )^{3/2} \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+e x^4 \left (a+b x^2\right )^{3/2} \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+c^2 e^3 \left (a+b x^2\right )^{3/2}+c e^2 x^2 \left (a+b x^2\right )^{3/2} (3 c f+2 d e)+d f^2 x^8 \left (a+b x^2\right )^{3/2} (2 c f+3 d e)+d^2 f^3 x^{10} \left (a+b x^2\right )^{3/2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{14} d^2 f^3 \left (b x^2+a\right )^{3/2} x^{11}+\frac {1}{56} a d^2 f^3 \sqrt {b x^2+a} x^{11}+\frac {1}{12} d f^2 (3 d e+2 c f) \left (b x^2+a\right )^{3/2} x^9+\frac {a^2 d^2 f^3 \sqrt {b x^2+a} x^9}{560 b}+\frac {1}{40} a d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^9+\frac {1}{10} f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \left (b x^2+a\right )^{3/2} x^7-\frac {9 a^3 d^2 f^3 \sqrt {b x^2+a} x^7}{4480 b^2}+\frac {a^2 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^7}{320 b}+\frac {3}{80} a f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \sqrt {b x^2+a} x^7+\frac {1}{8} e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \left (b x^2+a\right )^{3/2} x^5+\frac {3 a^4 d^2 f^3 \sqrt {b x^2+a} x^5}{1280 b^3}-\frac {7 a^3 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^5}{1920 b^2}+\frac {a^2 f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \sqrt {b x^2+a} x^5}{160 b}+\frac {1}{16} a e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \sqrt {b x^2+a} x^5+\frac {1}{6} c e^2 (2 d e+3 c f) \left (b x^2+a\right )^{3/2} x^3-\frac {3 a^5 d^2 f^3 \sqrt {b x^2+a} x^3}{1024 b^4}+\frac {7 a^4 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^3}{1536 b^3}+\frac {1}{8} a c e^2 (2 d e+3 c f) \sqrt {b x^2+a} x^3-\frac {a^3 f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \sqrt {b x^2+a} x^3}{128 b^2}+\frac {a^2 e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \sqrt {b x^2+a} x^3}{64 b}+\frac {1}{4} c^2 e^3 \left (b x^2+a\right )^{3/2} x+\frac {3}{8} a c^2 e^3 \sqrt {b x^2+a} x+\frac {9 a^6 d^2 f^3 \sqrt {b x^2+a} x}{2048 b^5}-\frac {7 a^5 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x}{1024 b^4}+\frac {a^2 c e^2 (2 d e+3 c f) \sqrt {b x^2+a} x}{16 b}+\frac {3 a^4 f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \sqrt {b x^2+a} x}{256 b^3}-\frac {3 a^3 e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \sqrt {b x^2+a} x}{128 b^2}+\frac {3 a^2 c^2 e^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{8 \sqrt {b}}-\frac {9 a^7 d^2 f^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2048 b^{11/2}}+\frac {7 a^6 d f^2 (3 d e+2 c f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{1024 b^{9/2}}-\frac {a^3 c e^2 (2 d e+3 c f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{16 b^{3/2}}-\frac {3 a^5 f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{256 b^{7/2}}+\frac {3 a^4 e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{128 b^{5/2}}\)

Input:

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

Output:

(3*a*c^2*e^3*x*Sqrt[a + b*x^2])/8 + (9*a^6*d^2*f^3*x*Sqrt[a + b*x^2])/(204 
8*b^5) - (7*a^5*d*f^2*(3*d*e + 2*c*f)*x*Sqrt[a + b*x^2])/(1024*b^4) + (a^2 
*c*e^2*(2*d*e + 3*c*f)*x*Sqrt[a + b*x^2])/(16*b) + (3*a^4*f*(3*d^2*e^2 + 6 
*c*d*e*f + c^2*f^2)*x*Sqrt[a + b*x^2])/(256*b^3) - (3*a^3*e*(d^2*e^2 + 6*c 
*d*e*f + 3*c^2*f^2)*x*Sqrt[a + b*x^2])/(128*b^2) - (3*a^5*d^2*f^3*x^3*Sqrt 
[a + b*x^2])/(1024*b^4) + (7*a^4*d*f^2*(3*d*e + 2*c*f)*x^3*Sqrt[a + b*x^2] 
)/(1536*b^3) + (a*c*e^2*(2*d*e + 3*c*f)*x^3*Sqrt[a + b*x^2])/8 - (a^3*f*(3 
*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^3*Sqrt[a + b*x^2])/(128*b^2) + (a^2*e*(d 
^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)*x^3*Sqrt[a + b*x^2])/(64*b) + (3*a^4*d^2*f 
^3*x^5*Sqrt[a + b*x^2])/(1280*b^3) - (7*a^3*d*f^2*(3*d*e + 2*c*f)*x^5*Sqrt 
[a + b*x^2])/(1920*b^2) + (a^2*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^5*Sqr 
t[a + b*x^2])/(160*b) + (a*e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)*x^5*Sqrt[a 
+ b*x^2])/16 - (9*a^3*d^2*f^3*x^7*Sqrt[a + b*x^2])/(4480*b^2) + (a^2*d*f^2 
*(3*d*e + 2*c*f)*x^7*Sqrt[a + b*x^2])/(320*b) + (3*a*f*(3*d^2*e^2 + 6*c*d* 
e*f + c^2*f^2)*x^7*Sqrt[a + b*x^2])/80 + (a^2*d^2*f^3*x^9*Sqrt[a + b*x^2]) 
/(560*b) + (a*d*f^2*(3*d*e + 2*c*f)*x^9*Sqrt[a + b*x^2])/40 + (a*d^2*f^3*x 
^11*Sqrt[a + b*x^2])/56 + (c^2*e^3*x*(a + b*x^2)^(3/2))/4 + (c*e^2*(2*d*e 
+ 3*c*f)*x^3*(a + b*x^2)^(3/2))/6 + (e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)*x 
^5*(a + b*x^2)^(3/2))/8 + (f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^7*(a + b* 
x^2)^(3/2))/10 + (d*f^2*(3*d*e + 2*c*f)*x^9*(a + b*x^2)^(3/2))/12 + (d^...
 

Defintions of rubi rules used

rule 433

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) 
^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, 
 q, r}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 667, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(-\frac {9 \left (a^{2} \left (a^{5} d^{2} f^{3}-\frac {28 \left (c f +\frac {3 d e}{2}\right ) d b \,f^{2} a^{4}}{9}+\frac {8 a^{3} b^{2} f \left (c^{2} f^{2}+6 c d e f +3 d^{2} e^{2}\right )}{3}-16 b^{3} e \left (c^{2} f^{2}+2 c d e f +\frac {1}{3} d^{2} e^{2}\right ) a^{2}+\frac {128 c \,b^{4} e^{2} \left (c f +\frac {2 d e}{3}\right ) a}{3}-\frac {256 b^{5} c^{2} e^{3}}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\sqrt {b \,x^{2}+a}\, x \left (\frac {1280 a \left (\left (\frac {1}{7} d^{2} x^{10}+\frac {26}{75} c d \,x^{8}+\frac {11}{50} c^{2} x^{6}\right ) f^{3}+\frac {9 \left (\frac {26}{45} d^{2} x^{4}+\frac {22}{15} c d \,x^{2}+c^{2}\right ) x^{4} e \,f^{2}}{10}+\frac {7 \left (\frac {33}{70} d^{2} x^{4}+\frac {9}{7} c d \,x^{2}+c^{2}\right ) x^{2} e^{2} f}{5}+e^{3} \left (\frac {3}{10} d^{2} x^{4}+\frac {14}{15} c d \,x^{2}+c^{2}\right )\right ) b^{\frac {11}{2}}}{9}+\frac {512 \left (2 \left (\frac {1}{7} d^{2} x^{10}+\frac {1}{3} c d \,x^{8}+\frac {1}{5} c^{2} x^{6}\right ) f^{3}+\frac {3 \left (\frac {2}{3} d^{2} x^{4}+\frac {8}{5} c d \,x^{2}+c^{2}\right ) x^{4} e \,f^{2}}{2}+2 \left (\frac {3}{5} d^{2} x^{4}+\frac {3}{2} c d \,x^{2}+c^{2}\right ) x^{2} e^{2} f +e^{3} \left (\frac {1}{2} d^{2} x^{4}+\frac {4}{3} c d \,x^{2}+c^{2}\right )\right ) x^{2} b^{\frac {13}{2}}}{9}+a^{2} \left (\frac {32 \left (\frac {2 \left (\frac {2}{7} d^{2} x^{4}+c d \,x^{2}+c^{2}\right ) x^{4} f^{3}}{15}+x^{2} e \left (\frac {1}{5} d^{2} x^{4}+\frac {4}{5} c d \,x^{2}+c^{2}\right ) f^{2}+4 e^{2} \left (\frac {1}{10} d^{2} x^{4}+\frac {1}{2} c d \,x^{2}+c^{2}\right ) f +\frac {8 d \left (\frac {x^{2} d}{8}+c \right ) e^{3}}{3}\right ) b^{\frac {9}{2}}}{3}+a \left (16 \left (-\frac {\left (\frac {9}{35} d^{2} x^{4}+\frac {14}{15} c d \,x^{2}+c^{2}\right ) x^{2} f^{3}}{9}-\left (\frac {7}{45} d^{2} x^{4}+\frac {2}{3} c d \,x^{2}+c^{2}\right ) e \,f^{2}-2 \left (\frac {x^{2} d}{6}+c \right ) d \,e^{2} f -\frac {d^{2} e^{3}}{3}\right ) b^{\frac {7}{2}}+a \left (8 \left (\frac {\left (\frac {7}{9} c d \,x^{2}+c^{2}+\frac {1}{5} d^{2} x^{4}\right ) f^{2}}{3}+2 \left (\frac {7 x^{2} d}{36}+c \right ) d e f +d^{2} e^{2}\right ) b^{\frac {5}{2}}+a d \left (\frac {2 \left (\left (-\frac {14 c}{3}-x^{2} d \right ) f -7 d e \right ) b^{\frac {3}{2}}}{3}+a d f \sqrt {b}\right ) f \right ) f \right )\right )\right )\right )}{2048 b^{\frac {11}{2}}}\)	\(667\)
default	\(c^{2} 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} d \left (2 c f +3 d e \right ) \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 )+c \,e^{2} \left (3 c f +2 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 )+f \left (c^{2} f^{2}+6 c d e f +3 d^{2} e^{2}\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 )+e \left (3 c^{2} f^{2}+6 c d e f +d^{2} e^{2}\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 )+d^{2} f^{3} \left (\frac {x^{9} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{14 b}-\frac {9 a \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 )}{14 b}\right )\)	\(748\)
risch	\(\text {Expression too large to display}\)	\(1021\)

Input:

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

Output:

-9/2048/b^(11/2)*(a^2*(a^5*d^2*f^3-28/9*(c*f+3/2*d*e)*d*b*f^2*a^4+8/3*a^3* 
b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)-16*b^3*e*(c^2*f^2+2*c*d*e*f+1/3*d^2*e^ 
2)*a^2+128/3*c*b^4*e^2*(c*f+2/3*d*e)*a-256/3*b^5*c^2*e^3)*arctanh((b*x^2+a 
)^(1/2)/x/b^(1/2))-(b*x^2+a)^(1/2)*x*(1280/9*a*((1/7*d^2*x^10+26/75*c*d*x^ 
8+11/50*c^2*x^6)*f^3+9/10*(26/45*d^2*x^4+22/15*c*d*x^2+c^2)*x^4*e*f^2+7/5* 
(33/70*d^2*x^4+9/7*c*d*x^2+c^2)*x^2*e^2*f+e^3*(3/10*d^2*x^4+14/15*c*d*x^2+ 
c^2))*b^(11/2)+512/9*(2*(1/7*d^2*x^10+1/3*c*d*x^8+1/5*c^2*x^6)*f^3+3/2*(2/ 
3*d^2*x^4+8/5*c*d*x^2+c^2)*x^4*e*f^2+2*(3/5*d^2*x^4+3/2*c*d*x^2+c^2)*x^2*e 
^2*f+e^3*(1/2*d^2*x^4+4/3*c*d*x^2+c^2))*x^2*b^(13/2)+a^2*(32/3*(2/15*(2/7* 
d^2*x^4+c*d*x^2+c^2)*x^4*f^3+x^2*e*(1/5*d^2*x^4+4/5*c*d*x^2+c^2)*f^2+4*e^2 
*(1/10*d^2*x^4+1/2*c*d*x^2+c^2)*f+8/3*d*(1/8*x^2*d+c)*e^3)*b^(9/2)+a*(16*( 
-1/9*(9/35*d^2*x^4+14/15*c*d*x^2+c^2)*x^2*f^3-(7/45*d^2*x^4+2/3*c*d*x^2+c^ 
2)*e*f^2-2*(1/6*x^2*d+c)*d*e^2*f-1/3*d^2*e^3)*b^(7/2)+a*(8*(1/3*(7/9*c*d*x 
^2+c^2+1/5*d^2*x^4)*f^2+2*(7/36*x^2*d+c)*d*e*f+d^2*e^2)*b^(5/2)+a*d*(2/3*( 
(-14/3*c-x^2*d)*f-7*d*e)*b^(3/2)+a*d*f*b^(1/2))*f)*f))))
 

Fricas [A] (verification not implemented)

Time = 2.40 (sec) , antiderivative size = 1740, normalized size of antiderivative = 2.12 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/430080*(105*(16*(48*a^2*b^5*c^2 - 16*a^3*b^4*c*d + 3*a^4*b^3*d^2)*e^3 
- 24*(16*a^3*b^4*c^2 - 12*a^4*b^3*c*d + 3*a^5*b^2*d^2)*e^2*f + 6*(24*a^4*b 
^3*c^2 - 24*a^5*b^2*c*d + 7*a^6*b*d^2)*e*f^2 - (24*a^5*b^2*c^2 - 28*a^6*b* 
c*d + 9*a^7*d^2)*f^3)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - 
 a) - 2*(15360*b^7*d^2*f^3*x^13 + 1280*(42*b^7*d^2*e*f^2 + (28*b^7*c*d + 1 
5*a*b^6*d^2)*f^3)*x^11 + 128*(504*b^7*d^2*e^2*f + 42*(24*b^7*c*d + 13*a*b^ 
6*d^2)*e*f^2 + (168*b^7*c^2 + 364*a*b^6*c*d + 3*a^2*b^5*d^2)*f^3)*x^9 + 48 
*(560*b^7*d^2*e^3 + 168*(20*b^7*c*d + 11*a*b^6*d^2)*e^2*f + 42*(40*b^7*c^2 
 + 88*a*b^6*c*d + a^2*b^5*d^2)*e*f^2 + (616*a*b^6*c^2 + 28*a^2*b^5*c*d - 9 
*a^3*b^4*d^2)*f^3)*x^7 + 56*(80*(16*b^7*c*d + 9*a*b^6*d^2)*e^3 + 24*(80*b^ 
7*c^2 + 180*a*b^6*c*d + 3*a^2*b^5*d^2)*e^2*f + 6*(360*a*b^6*c^2 + 24*a^2*b 
^5*c*d - 7*a^3*b^4*d^2)*e*f^2 + (24*a^2*b^5*c^2 - 28*a^3*b^4*c*d + 9*a^4*b 
^3*d^2)*f^3)*x^5 + 70*(16*(48*b^7*c^2 + 112*a*b^6*c*d + 3*a^2*b^5*d^2)*e^3 
 + 24*(112*a*b^6*c^2 + 12*a^2*b^5*c*d - 3*a^3*b^4*d^2)*e^2*f + 6*(24*a^2*b 
^5*c^2 - 24*a^3*b^4*c*d + 7*a^4*b^3*d^2)*e*f^2 - (24*a^3*b^4*c^2 - 28*a^4* 
b^3*c*d + 9*a^5*b^2*d^2)*f^3)*x^3 + 105*(16*(80*a*b^6*c^2 + 16*a^2*b^5*c*d 
 - 3*a^3*b^4*d^2)*e^3 + 24*(16*a^2*b^5*c^2 - 12*a^3*b^4*c*d + 3*a^4*b^3*d^ 
2)*e^2*f - 6*(24*a^3*b^4*c^2 - 24*a^4*b^3*c*d + 7*a^5*b^2*d^2)*e*f^2 + (24 
*a^4*b^3*c^2 - 28*a^5*b^2*c*d + 9*a^6*b*d^2)*f^3)*x)*sqrt(b*x^2 + a))/b^6, 
 -1/215040*(105*(16*(48*a^2*b^5*c^2 - 16*a^3*b^4*c*d + 3*a^4*b^3*d^2)*e...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2317 vs. \(2 (877) = 1754\).

Time = 0.71 (sec) , antiderivative size = 2317, normalized size of antiderivative = 2.82 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x**2)*(b*d**2*f**3*x**13/14 + x**11*(15*a*b*d**2*f** 
3/14 + 2*b**2*c*d*f**3 + 3*b**2*d**2*e*f**2)/(12*b) + x**9*(a**2*d**2*f**3 
 + 4*a*b*c*d*f**3 + 6*a*b*d**2*e*f**2 - 11*a*(15*a*b*d**2*f**3/14 + 2*b**2 
*c*d*f**3 + 3*b**2*d**2*e*f**2)/(12*b) + b**2*c**2*f**3 + 6*b**2*c*d*e*f** 
2 + 3*b**2*d**2*e**2*f)/(10*b) + x**7*(2*a**2*c*d*f**3 + 3*a**2*d**2*e*f** 
2 + 2*a*b*c**2*f**3 + 12*a*b*c*d*e*f**2 + 6*a*b*d**2*e**2*f - 9*a*(a**2*d* 
*2*f**3 + 4*a*b*c*d*f**3 + 6*a*b*d**2*e*f**2 - 11*a*(15*a*b*d**2*f**3/14 + 
 2*b**2*c*d*f**3 + 3*b**2*d**2*e*f**2)/(12*b) + b**2*c**2*f**3 + 6*b**2*c* 
d*e*f**2 + 3*b**2*d**2*e**2*f)/(10*b) + 3*b**2*c**2*e*f**2 + 6*b**2*c*d*e* 
*2*f + b**2*d**2*e**3)/(8*b) + x**5*(a**2*c**2*f**3 + 6*a**2*c*d*e*f**2 + 
3*a**2*d**2*e**2*f + 6*a*b*c**2*e*f**2 + 12*a*b*c*d*e**2*f + 2*a*b*d**2*e* 
*3 - 7*a*(2*a**2*c*d*f**3 + 3*a**2*d**2*e*f**2 + 2*a*b*c**2*f**3 + 12*a*b* 
c*d*e*f**2 + 6*a*b*d**2*e**2*f - 9*a*(a**2*d**2*f**3 + 4*a*b*c*d*f**3 + 6* 
a*b*d**2*e*f**2 - 11*a*(15*a*b*d**2*f**3/14 + 2*b**2*c*d*f**3 + 3*b**2*d** 
2*e*f**2)/(12*b) + b**2*c**2*f**3 + 6*b**2*c*d*e*f**2 + 3*b**2*d**2*e**2*f 
)/(10*b) + 3*b**2*c**2*e*f**2 + 6*b**2*c*d*e**2*f + b**2*d**2*e**3)/(8*b) 
+ 3*b**2*c**2*e**2*f + 2*b**2*c*d*e**3)/(6*b) + x**3*(3*a**2*c**2*e*f**2 + 
 6*a**2*c*d*e**2*f + a**2*d**2*e**3 + 6*a*b*c**2*e**2*f + 4*a*b*c*d*e**3 - 
 5*a*(a**2*c**2*f**3 + 6*a**2*c*d*e*f**2 + 3*a**2*d**2*e**2*f + 6*a*b*c**2 
*e*f**2 + 12*a*b*c*d*e**2*f + 2*a*b*d**2*e**3 - 7*a*(2*a**2*c*d*f**3 + ...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 1101, normalized size of antiderivative = 1.34 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/14*(b*x^2 + a)^(5/2)*d^2*f^3*x^9/b - 3/56*(b*x^2 + a)^(5/2)*a*d^2*f^3*x^ 
7/b^2 + 3/80*(b*x^2 + a)^(5/2)*a^2*d^2*f^3*x^5/b^3 - 3/128*(b*x^2 + a)^(5/ 
2)*a^3*d^2*f^3*x^3/b^4 + 1/12*(3*d^2*e*f^2 + 2*c*d*f^3)*(b*x^2 + a)^(5/2)* 
x^7/b + 1/4*(b*x^2 + a)^(3/2)*c^2*e^3*x + 3/8*sqrt(b*x^2 + a)*a*c^2*e^3*x 
+ 3/256*(b*x^2 + a)^(5/2)*a^4*d^2*f^3*x/b^5 - 3/1024*(b*x^2 + a)^(3/2)*a^5 
*d^2*f^3*x/b^5 - 9/2048*sqrt(b*x^2 + a)*a^6*d^2*f^3*x/b^5 - 7/120*(3*d^2*e 
*f^2 + 2*c*d*f^3)*(b*x^2 + a)^(5/2)*a*x^5/b^2 + 1/10*(3*d^2*e^2*f + 6*c*d* 
e*f^2 + c^2*f^3)*(b*x^2 + a)^(5/2)*x^5/b + 3/8*a^2*c^2*e^3*arcsinh(b*x/sqr 
t(a*b))/sqrt(b) - 9/2048*a^7*d^2*f^3*arcsinh(b*x/sqrt(a*b))/b^(11/2) + 7/1 
92*(3*d^2*e*f^2 + 2*c*d*f^3)*(b*x^2 + a)^(5/2)*a^2*x^3/b^3 - 1/16*(3*d^2*e 
^2*f + 6*c*d*e*f^2 + c^2*f^3)*(b*x^2 + a)^(5/2)*a*x^3/b^2 + 1/8*(d^2*e^3 + 
 6*c*d*e^2*f + 3*c^2*e*f^2)*(b*x^2 + a)^(5/2)*x^3/b - 7/384*(3*d^2*e*f^2 + 
 2*c*d*f^3)*(b*x^2 + a)^(5/2)*a^3*x/b^4 + 7/1536*(3*d^2*e*f^2 + 2*c*d*f^3) 
*(b*x^2 + a)^(3/2)*a^4*x/b^4 + 7/1024*(3*d^2*e*f^2 + 2*c*d*f^3)*sqrt(b*x^2 
 + a)*a^5*x/b^4 + 1/32*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*(b*x^2 + a)^( 
5/2)*a^2*x/b^3 - 1/128*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*(b*x^2 + a)^( 
3/2)*a^3*x/b^3 - 3/256*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*sqrt(b*x^2 + 
a)*a^4*x/b^3 - 1/16*(d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*(b*x^2 + a)^(5/2 
)*a*x/b^2 + 1/64*(d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*(b*x^2 + a)^(3/2)*a 
^2*x/b^2 + 3/128*(d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*sqrt(b*x^2 + a)*...
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 971, normalized size of antiderivative = 1.18 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/215040*(2*(4*(2*(8*(10*(12*b*d^2*f^3*x^2 + (42*b^13*d^2*e*f^2 + 28*b^13* 
c*d*f^3 + 15*a*b^12*d^2*f^3)/b^12)*x^2 + (504*b^13*d^2*e^2*f + 1008*b^13*c 
*d*e*f^2 + 546*a*b^12*d^2*e*f^2 + 168*b^13*c^2*f^3 + 364*a*b^12*c*d*f^3 + 
3*a^2*b^11*d^2*f^3)/b^12)*x^2 + 3*(560*b^13*d^2*e^3 + 3360*b^13*c*d*e^2*f 
+ 1848*a*b^12*d^2*e^2*f + 1680*b^13*c^2*e*f^2 + 3696*a*b^12*c*d*e*f^2 + 42 
*a^2*b^11*d^2*e*f^2 + 616*a*b^12*c^2*f^3 + 28*a^2*b^11*c*d*f^3 - 9*a^3*b^1 
0*d^2*f^3)/b^12)*x^2 + 7*(1280*b^13*c*d*e^3 + 720*a*b^12*d^2*e^3 + 1920*b^ 
13*c^2*e^2*f + 4320*a*b^12*c*d*e^2*f + 72*a^2*b^11*d^2*e^2*f + 2160*a*b^12 
*c^2*e*f^2 + 144*a^2*b^11*c*d*e*f^2 - 42*a^3*b^10*d^2*e*f^2 + 24*a^2*b^11* 
c^2*f^3 - 28*a^3*b^10*c*d*f^3 + 9*a^4*b^9*d^2*f^3)/b^12)*x^2 + 35*(768*b^1 
3*c^2*e^3 + 1792*a*b^12*c*d*e^3 + 48*a^2*b^11*d^2*e^3 + 2688*a*b^12*c^2*e^ 
2*f + 288*a^2*b^11*c*d*e^2*f - 72*a^3*b^10*d^2*e^2*f + 144*a^2*b^11*c^2*e* 
f^2 - 144*a^3*b^10*c*d*e*f^2 + 42*a^4*b^9*d^2*e*f^2 - 24*a^3*b^10*c^2*f^3 
+ 28*a^4*b^9*c*d*f^3 - 9*a^5*b^8*d^2*f^3)/b^12)*x^2 + 105*(1280*a*b^12*c^2 
*e^3 + 256*a^2*b^11*c*d*e^3 - 48*a^3*b^10*d^2*e^3 + 384*a^2*b^11*c^2*e^2*f 
 - 288*a^3*b^10*c*d*e^2*f + 72*a^4*b^9*d^2*e^2*f - 144*a^3*b^10*c^2*e*f^2 
+ 144*a^4*b^9*c*d*e*f^2 - 42*a^5*b^8*d^2*e*f^2 + 24*a^4*b^9*c^2*f^3 - 28*a 
^5*b^8*c*d*f^3 + 9*a^6*b^7*d^2*f^3)/b^12)*sqrt(b*x^2 + a)*x - 1/2048*(768* 
a^2*b^5*c^2*e^3 - 256*a^3*b^4*c*d*e^3 + 48*a^4*b^3*d^2*e^3 - 384*a^3*b^4*c 
^2*e^2*f + 288*a^4*b^3*c*d*e^2*f - 72*a^5*b^2*d^2*e^2*f + 144*a^4*b^3*c...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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