\(\int \sqrt {a+b x^2} (c+d x^2)^2 (e+f x^2)^3 \, dx\) [267]

Optimal result

Integrand size = 30, antiderivative size = 676 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {\left (512 b^5 c^2 e^3-21 a^5 d^2 f^3+28 a^4 b d f^2 (3 d e+2 c f)-128 a b^4 c e^2 (2 d e+3 c f)-40 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+64 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{1024 b^5}+\frac {\left (21 a^4 d^2 f^3-28 a^3 b d f^2 (3 d e+2 c f)+128 b^4 c e^2 (2 d e+3 c f)+40 a^2 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-64 a b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x \left (a+b x^2\right )^{3/2}}{512 b^5}-\frac {\left (21 a^3 d^2 f^3-28 a^2 b d f^2 (3 d e+2 c f)+40 a b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-64 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^3 \left (a+b x^2\right )^{3/2}}{384 b^4}+\frac {f \left (21 a^2 d^2 f^2-28 a b d f (3 d e+2 c f)+40 b^2 \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^5 \left (a+b x^2\right )^{3/2}}{320 b^3}+\frac {d f^2 (12 b d e+8 b c f-3 a d f) x^7 \left (a+b x^2\right )^{3/2}}{40 b^2}+\frac {d^2 f^3 x^9 \left (a+b x^2\right )^{3/2}}{12 b}+\frac {a \left (512 b^5 c^2 e^3-21 a^5 d^2 f^3+28 a^4 b d f^2 (3 d e+2 c f)-128 a b^4 c e^2 (2 d e+3 c f)-40 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+64 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{11/2}} \] Output:

1/1024*(512*b^5*c^2*e^3-21*a^5*d^2*f^3+28*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)-40*a^3*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)+64*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+1/512*(21*a^4*d 
^2*f^3-28*a^3*b*d*f^2*(2*c*f+3*d*e)+128*b^4*c*e^2*(3*c*f+2*d*e)+40*a^2*b^2 
*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)-64*a*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^5-1/384*(21*a^3*d^2*f^3-28*a^2*b*d*f^2*(2*c*f+3*d*e)+ 
40*a*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)-64*b^3*e*(3*c^2*f^2+6*c*d*e*f+d^2 
*e^2))*x^3*(b*x^2+a)^(3/2)/b^4+1/320*f*(21*a^2*d^2*f^2-28*a*b*d*f*(2*c*f+3 
*d*e)+40*b^2*(c^2*f^2+6*c*d*e*f+3*d^2*e^2))*x^5*(b*x^2+a)^(3/2)/b^3+1/40*d 
*f^2*(-3*a*d*f+8*b*c*f+12*b*d*e)*x^7*(b*x^2+a)^(3/2)/b^2+1/12*d^2*f^3*x^9* 
(b*x^2+a)^(3/2)/b+1/1024*a*(512*b^5*c^2*e^3-21*a^5*d^2*f^3+28*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)-40*a^3*b^2*f*(c^2*f^2+6*c*d*e* 
f+3*d^2*e^2)+64*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 = 2.15 (sec) , antiderivative size = 587, normalized size of antiderivative = 0.87 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (315 a^5 d^2 f^3-210 a^4 b d f^2 \left (6 d e+4 c f+d f x^2\right )+8 a^3 b^2 f \left (75 c^2 f^2+10 c d f \left (45 e+7 f x^2\right )+3 d^2 \left (75 e^2+35 e f x^2+7 f^2 x^4\right )\right )+64 a b^4 \left (5 c^2 f \left (18 e^2+6 e f x^2+f^2 x^4\right )+6 c d \left (10 e^3+10 e^2 f x^2+5 e f^2 x^4+f^3 x^6\right )+d^2 x^2 \left (10 e^3+15 e^2 f x^2+9 e f^2 x^4+2 f^3 x^6\right )\right )-16 a^2 b^3 \left (5 c^2 f^2 \left (36 e+5 f x^2\right )+2 c d f \left (180 e^2+75 e f x^2+14 f^2 x^4\right )+d^2 \left (60 e^3+75 e^2 f x^2+42 e f^2 x^4+9 f^3 x^6\right )\right )+128 b^5 \left (15 c^2 \left (4 e^3+6 e^2 f x^2+4 e f^2 x^4+f^3 x^6\right )+6 c d x^2 \left (10 e^3+20 e^2 f x^2+15 e f^2 x^4+4 f^3 x^6\right )+d^2 x^4 \left (20 e^3+45 e^2 f x^2+36 e f^2 x^4+10 f^3 x^6\right )\right )\right )+15 a \left (-512 b^5 c^2 e^3+21 a^5 d^2 f^3-28 a^4 b d f^2 (3 d e+2 c f)+128 a b^4 c e^2 (2 d e+3 c f)+40 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-64 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{15360 b^{11/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(315*a^5*d^2*f^3 - 210*a^4*b*d*f^2*(6*d*e + 4*c 
*f + d*f*x^2) + 8*a^3*b^2*f*(75*c^2*f^2 + 10*c*d*f*(45*e + 7*f*x^2) + 3*d^ 
2*(75*e^2 + 35*e*f*x^2 + 7*f^2*x^4)) + 64*a*b^4*(5*c^2*f*(18*e^2 + 6*e*f*x 
^2 + f^2*x^4) + 6*c*d*(10*e^3 + 10*e^2*f*x^2 + 5*e*f^2*x^4 + f^3*x^6) + d^ 
2*x^2*(10*e^3 + 15*e^2*f*x^2 + 9*e*f^2*x^4 + 2*f^3*x^6)) - 16*a^2*b^3*(5*c 
^2*f^2*(36*e + 5*f*x^2) + 2*c*d*f*(180*e^2 + 75*e*f*x^2 + 14*f^2*x^4) + d^ 
2*(60*e^3 + 75*e^2*f*x^2 + 42*e*f^2*x^4 + 9*f^3*x^6)) + 128*b^5*(15*c^2*(4 
*e^3 + 6*e^2*f*x^2 + 4*e*f^2*x^4 + f^3*x^6) + 6*c*d*x^2*(10*e^3 + 20*e^2*f 
*x^2 + 15*e*f^2*x^4 + 4*f^3*x^6) + d^2*x^4*(20*e^3 + 45*e^2*f*x^2 + 36*e*f 
^2*x^4 + 10*f^3*x^6))) + 15*a*(-512*b^5*c^2*e^3 + 21*a^5*d^2*f^3 - 28*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) + 40*a^3*b^2*f*( 
3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) - 64*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]])/(15360*b^(11/2))
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 1009, normalized size of antiderivative = 1.49, 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.

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

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{12} d^2 f^3 \sqrt {b x^2+a} x^{11}+\frac {a d^2 f^3 \sqrt {b x^2+a} x^9}{120 b}+\frac {1}{10} d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^9-\frac {3 a^2 d^2 f^3 \sqrt {b x^2+a} x^7}{320 b^2}+\frac {a d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^7}{80 b}+\frac {1}{8} 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 {7 a^3 d^2 f^3 \sqrt {b x^2+a} x^5}{640 b^3}-\frac {7 a^2 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^5}{480 b^2}+\frac {a f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \sqrt {b x^2+a} x^5}{48 b}+\frac {1}{6} 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 {7 a^4 d^2 f^3 \sqrt {b x^2+a} x^3}{512 b^4}+\frac {7 a^3 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^3}{384 b^3}+\frac {1}{4} c e^2 (2 d e+3 c f) \sqrt {b x^2+a} x^3-\frac {5 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^3}{192 b^2}+\frac {a e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \sqrt {b x^2+a} x^3}{24 b}+\frac {1}{2} c^2 e^3 \sqrt {b x^2+a} x+\frac {21 a^5 d^2 f^3 \sqrt {b x^2+a} x}{1024 b^5}-\frac {7 a^4 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x}{256 b^4}+\frac {a c e^2 (2 d e+3 c f) \sqrt {b x^2+a} x}{8 b}+\frac {5 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}{128 b^3}-\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}{16 b^2}+\frac {a c^2 e^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 \sqrt {b}}-\frac {21 a^6 d^2 f^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{1024 b^{11/2}}+\frac {7 a^5 d f^2 (3 d e+2 c f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{256 b^{9/2}}-\frac {a^2 c e^2 (2 d e+3 c f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{8 b^{3/2}}-\frac {5 a^4 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 )}{128 b^{7/2}}+\frac {a^3 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 )}{16 b^{5/2}}\)

Input:

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

Output:

(c^2*e^3*x*Sqrt[a + b*x^2])/2 + (21*a^5*d^2*f^3*x*Sqrt[a + b*x^2])/(1024*b 
^5) - (7*a^4*d*f^2*(3*d*e + 2*c*f)*x*Sqrt[a + b*x^2])/(256*b^4) + (a*c*e^2 
*(2*d*e + 3*c*f)*x*Sqrt[a + b*x^2])/(8*b) + (5*a^3*f*(3*d^2*e^2 + 6*c*d*e* 
f + c^2*f^2)*x*Sqrt[a + b*x^2])/(128*b^3) - (a^2*e*(d^2*e^2 + 6*c*d*e*f + 
3*c^2*f^2)*x*Sqrt[a + b*x^2])/(16*b^2) - (7*a^4*d^2*f^3*x^3*Sqrt[a + b*x^2 
])/(512*b^4) + (7*a^3*d*f^2*(3*d*e + 2*c*f)*x^3*Sqrt[a + b*x^2])/(384*b^3) 
 + (c*e^2*(2*d*e + 3*c*f)*x^3*Sqrt[a + b*x^2])/4 - (5*a^2*f*(3*d^2*e^2 + 6 
*c*d*e*f + c^2*f^2)*x^3*Sqrt[a + b*x^2])/(192*b^2) + (a*e*(d^2*e^2 + 6*c*d 
*e*f + 3*c^2*f^2)*x^3*Sqrt[a + b*x^2])/(24*b) + (7*a^3*d^2*f^3*x^5*Sqrt[a 
+ b*x^2])/(640*b^3) - (7*a^2*d*f^2*(3*d*e + 2*c*f)*x^5*Sqrt[a + b*x^2])/(4 
80*b^2) + (a*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^5*Sqrt[a + b*x^2])/(48* 
b) + (e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)*x^5*Sqrt[a + b*x^2])/6 - (3*a^2* 
d^2*f^3*x^7*Sqrt[a + b*x^2])/(320*b^2) + (a*d*f^2*(3*d*e + 2*c*f)*x^7*Sqrt 
[a + b*x^2])/(80*b) + (f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^7*Sqrt[a + b* 
x^2])/8 + (a*d^2*f^3*x^9*Sqrt[a + b*x^2])/(120*b) + (d*f^2*(3*d*e + 2*c*f) 
*x^9*Sqrt[a + b*x^2])/10 + (d^2*f^3*x^11*Sqrt[a + b*x^2])/12 + (a*c^2*e^3* 
ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b]) - (21*a^6*d^2*f^3*ArcTan 
h[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(11/2)) + (7*a^5*d*f^2*(3*d*e + 2* 
c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*b^(9/2)) - (a^2*c*e^2*(2*d 
*e + 3*c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(3/2)) - (5*a^4*...
 

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 = 0.94 (sec) , antiderivative size = 546, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(-\frac {21 \left (a \left (a^{3} \left (-\frac {8}{3} a b c d +a^{2} d^{2}+\frac {40}{21} b^{2} c^{2}\right ) f^{3}-4 a^{2} b e \left (a^{2} d^{2}-\frac {20}{7} a b c d +\frac {16}{7} b^{2} c^{2}\right ) f^{2}+\frac {40 a \left (a^{2} d^{2}-\frac {16}{5} a b c d +\frac {16}{5} b^{2} c^{2}\right ) b^{2} e^{2} f}{7}-\frac {64 b^{3} e^{3} \left (a^{2} d^{2}-4 a b c d +8 b^{2} c^{2}\right )}{21}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\left (\left (\frac {128 \left (\frac {2}{3} d^{2} x^{4}+\frac {8}{5} c d \,x^{2}+c^{2}\right ) x^{6} f^{3}}{21}+\frac {512 x^{4} \left (\frac {3}{5} d^{2} x^{4}+\frac {3}{2} c d \,x^{2}+c^{2}\right ) e \,f^{2}}{21}+\frac {256 \left (\frac {1}{2} d^{2} x^{4}+\frac {4}{3} c d \,x^{2}+c^{2}\right ) x^{2} e^{2} f}{7}+\frac {512 \left (\frac {1}{3} d^{2} x^{4}+c d \,x^{2}+c^{2}\right ) e^{3}}{21}\right ) b^{\frac {11}{2}}+a \left (\left (\frac {64 \left (\frac {2}{5} d^{2} x^{4}+\frac {6}{5} c d \,x^{2}+c^{2}\right ) x^{4} f^{3}}{63}+\frac {128 \left (\frac {3}{10} d^{2} x^{4}+c d \,x^{2}+c^{2}\right ) x^{2} e \,f^{2}}{21}+\frac {128 \left (\frac {1}{6} d^{2} x^{4}+\frac {2}{3} c d \,x^{2}+c^{2}\right ) e^{2} f}{7}+\frac {256 d \left (\frac {x^{2} d}{6}+c \right ) e^{3}}{21}\right ) b^{\frac {9}{2}}+a \left (\left (-\frac {80 \left (\frac {9}{25} d^{2} x^{4}+\frac {28}{25} c d \,x^{2}+c^{2}\right ) x^{2} f^{3}}{63}-\frac {64 \left (\frac {7}{30} d^{2} x^{4}+\frac {5}{6} c d \,x^{2}+c^{2}\right ) e \,f^{2}}{7}-\frac {128 \left (\frac {5 x^{2} d}{24}+c \right ) d \,e^{2} f}{7}-\frac {64 d^{2} e^{3}}{21}\right ) b^{\frac {7}{2}}+a \left (\left (\left (\frac {16}{9} c d \,x^{2}+\frac {40}{21} c^{2}+\frac {8}{15} d^{2} x^{4}\right ) f^{2}+\frac {80 d \left (\frac {7 x^{2} d}{30}+c \right ) e f}{7}+\frac {40 d^{2} e^{2}}{7}\right ) b^{\frac {5}{2}}+a d f \left (\left (\left (-\frac {8 c}{3}-\frac {2 x^{2} d}{3}\right ) f -4 d e \right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right )\right ) f \right )\right )\right ) \sqrt {b \,x^{2}+a}\, x \right )}{1024 b^{\frac {11}{2}}}\)	\(546\)
default	\(c^{2} 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} d \left (2 c f +3 d e \right ) \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 )+c \,e^{2} \left (3 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 )+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 {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 )+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 {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 )+d^{2} f^{3} \left (\frac {x^{9} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{12 b}-\frac {3 a \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 )}{4 b}\right )\)	\(652\)
risch	\(\frac {x \left (1280 d^{2} f^{3} b^{5} x^{10}+128 a \,b^{4} d^{2} f^{3} x^{8}+3072 b^{5} c d \,f^{3} x^{8}+4608 b^{5} d^{2} e \,f^{2} x^{8}-144 a^{2} b^{3} d^{2} f^{3} x^{6}+384 a \,b^{4} c d \,f^{3} x^{6}+576 a \,b^{4} d^{2} e \,f^{2} x^{6}+1920 b^{5} c^{2} f^{3} x^{6}+11520 b^{5} c d e \,f^{2} x^{6}+5760 b^{5} d^{2} e^{2} f \,x^{6}+168 a^{3} b^{2} d^{2} f^{3} x^{4}-448 a^{2} b^{3} c d \,f^{3} x^{4}-672 a^{2} b^{3} d^{2} e \,f^{2} x^{4}+320 a \,b^{4} c^{2} f^{3} x^{4}+1920 a \,b^{4} c d e \,f^{2} x^{4}+960 a \,b^{4} d^{2} e^{2} f \,x^{4}+7680 b^{5} c^{2} e \,f^{2} x^{4}+15360 b^{5} c d \,e^{2} f \,x^{4}+2560 b^{5} d^{2} e^{3} x^{4}-210 a^{4} b \,d^{2} f^{3} x^{2}+560 a^{3} b^{2} c d \,f^{3} x^{2}+840 a^{3} b^{2} d^{2} e \,f^{2} x^{2}-400 a^{2} b^{3} c^{2} f^{3} x^{2}-2400 a^{2} b^{3} c d e \,f^{2} x^{2}-1200 a^{2} b^{3} d^{2} e^{2} f \,x^{2}+1920 a \,b^{4} c^{2} e \,f^{2} x^{2}+3840 a \,b^{4} c d \,e^{2} f \,x^{2}+640 a \,b^{4} d^{2} e^{3} x^{2}+11520 b^{5} c^{2} e^{2} f \,x^{2}+7680 b^{5} c d \,e^{3} x^{2}+315 a^{5} d^{2} f^{3}-840 a^{4} b c d \,f^{3}-1260 a^{4} b \,d^{2} e \,f^{2}+600 a^{3} b^{2} c^{2} f^{3}+3600 a^{3} b^{2} c d e \,f^{2}+1800 a^{3} b^{2} d^{2} e^{2} f -2880 a^{2} b^{3} c^{2} e \,f^{2}-5760 a^{2} b^{3} c d \,e^{2} f -960 a^{2} b^{3} d^{2} e^{3}+5760 a \,b^{4} c^{2} e^{2} f +3840 a \,b^{4} c d \,e^{3}+7680 b^{5} c^{2} e^{3}\right ) \sqrt {b \,x^{2}+a}}{15360 b^{5}}-\frac {a \left (21 a^{5} d^{2} f^{3}-56 a^{4} b c d \,f^{3}-84 a^{4} b \,d^{2} e \,f^{2}+40 a^{3} b^{2} c^{2} f^{3}+240 a^{3} b^{2} c d e \,f^{2}+120 a^{3} b^{2} d^{2} e^{2} f -192 a^{2} b^{3} c^{2} e \,f^{2}-384 a^{2} b^{3} c d \,e^{2} f -64 a^{2} b^{3} d^{2} e^{3}+384 a \,b^{4} c^{2} e^{2} f +256 a \,b^{4} c d \,e^{3}-512 b^{5} c^{2} e^{3}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {11}{2}}}\)	\(815\)

Input:

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

Output:

-21/1024*(a*(a^3*(-8/3*a*b*c*d+a^2*d^2+40/21*b^2*c^2)*f^3-4*a^2*b*e*(a^2*d 
^2-20/7*a*b*c*d+16/7*b^2*c^2)*f^2+40/7*a*(a^2*d^2-16/5*a*b*c*d+16/5*b^2*c^ 
2)*b^2*e^2*f-64/21*b^3*e^3*(a^2*d^2-4*a*b*c*d+8*b^2*c^2))*arctanh((b*x^2+a 
)^(1/2)/x/b^(1/2))-((128/21*(2/3*d^2*x^4+8/5*c*d*x^2+c^2)*x^6*f^3+512/21*x 
^4*(3/5*d^2*x^4+3/2*c*d*x^2+c^2)*e*f^2+256/7*(1/2*d^2*x^4+4/3*c*d*x^2+c^2) 
*x^2*e^2*f+512/21*(1/3*d^2*x^4+c*d*x^2+c^2)*e^3)*b^(11/2)+a*((64/63*(2/5*d 
^2*x^4+6/5*c*d*x^2+c^2)*x^4*f^3+128/21*(3/10*d^2*x^4+c*d*x^2+c^2)*x^2*e*f^ 
2+128/7*(1/6*d^2*x^4+2/3*c*d*x^2+c^2)*e^2*f+256/21*d*(1/6*x^2*d+c)*e^3)*b^ 
(9/2)+a*((-80/63*(9/25*d^2*x^4+28/25*c*d*x^2+c^2)*x^2*f^3-64/7*(7/30*d^2*x 
^4+5/6*c*d*x^2+c^2)*e*f^2-128/7*(5/24*x^2*d+c)*d*e^2*f-64/21*d^2*e^3)*b^(7 
/2)+a*(((16/9*c*d*x^2+40/21*c^2+8/15*d^2*x^4)*f^2+80/7*d*(7/30*x^2*d+c)*e* 
f+40/7*d^2*e^2)*b^(5/2)+a*d*f*(((-8/3*c-2/3*x^2*d)*f-4*d*e)*b^(3/2)+a*d*f* 
b^(1/2)))*f)))*(b*x^2+a)^(1/2)*x)/b^(11/2)
 

Fricas [A] (verification not implemented)

Time = 1.41 (sec) , antiderivative size = 1414, normalized size of antiderivative = 2.09 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/30720*(15*(64*(8*a*b^5*c^2 - 4*a^2*b^4*c*d + a^3*b^3*d^2)*e^3 - 24*(16 
*a^2*b^4*c^2 - 16*a^3*b^3*c*d + 5*a^4*b^2*d^2)*e^2*f + 12*(16*a^3*b^3*c^2 
- 20*a^4*b^2*c*d + 7*a^5*b*d^2)*e*f^2 - (40*a^4*b^2*c^2 - 56*a^5*b*c*d + 2 
1*a^6*d^2)*f^3)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 
2*(1280*b^6*d^2*f^3*x^11 + 128*(36*b^6*d^2*e*f^2 + (24*b^6*c*d + a*b^5*d^2 
)*f^3)*x^9 + 48*(120*b^6*d^2*e^2*f + 12*(20*b^6*c*d + a*b^5*d^2)*e*f^2 + ( 
40*b^6*c^2 + 8*a*b^5*c*d - 3*a^2*b^4*d^2)*f^3)*x^7 + 8*(320*b^6*d^2*e^3 + 
120*(16*b^6*c*d + a*b^5*d^2)*e^2*f + 12*(80*b^6*c^2 + 20*a*b^5*c*d - 7*a^2 
*b^4*d^2)*e*f^2 + (40*a*b^5*c^2 - 56*a^2*b^4*c*d + 21*a^3*b^3*d^2)*f^3)*x^ 
5 + 10*(64*(12*b^6*c*d + a*b^5*d^2)*e^3 + 24*(48*b^6*c^2 + 16*a*b^5*c*d - 
5*a^2*b^4*d^2)*e^2*f + 12*(16*a*b^5*c^2 - 20*a^2*b^4*c*d + 7*a^3*b^3*d^2)* 
e*f^2 - (40*a^2*b^4*c^2 - 56*a^3*b^3*c*d + 21*a^4*b^2*d^2)*f^3)*x^3 + 15*( 
64*(8*b^6*c^2 + 4*a*b^5*c*d - a^2*b^4*d^2)*e^3 + 24*(16*a*b^5*c^2 - 16*a^2 
*b^4*c*d + 5*a^3*b^3*d^2)*e^2*f - 12*(16*a^2*b^4*c^2 - 20*a^3*b^3*c*d + 7* 
a^4*b^2*d^2)*e*f^2 + (40*a^3*b^3*c^2 - 56*a^4*b^2*c*d + 21*a^5*b*d^2)*f^3) 
*x)*sqrt(b*x^2 + a))/b^6, -1/15360*(15*(64*(8*a*b^5*c^2 - 4*a^2*b^4*c*d + 
a^3*b^3*d^2)*e^3 - 24*(16*a^2*b^4*c^2 - 16*a^3*b^3*c*d + 5*a^4*b^2*d^2)*e^ 
2*f + 12*(16*a^3*b^3*c^2 - 20*a^4*b^2*c*d + 7*a^5*b*d^2)*e*f^2 - (40*a^4*b 
^2*c^2 - 56*a^5*b*c*d + 21*a^6*d^2)*f^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b 
*x^2 + a)) - (1280*b^6*d^2*f^3*x^11 + 128*(36*b^6*d^2*e*f^2 + (24*b^6*c...
 

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 1290, normalized size of antiderivative = 1.91 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.33 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(b*x^2 + a)^(3/2)*d^2*f^3*x^9/b - 3/40*(b*x^2 + a)^(3/2)*a*d^2*f^3*x^ 
7/b^2 + 21/320*(b*x^2 + a)^(3/2)*a^2*d^2*f^3*x^5/b^3 - 7/128*(b*x^2 + a)^( 
3/2)*a^3*d^2*f^3*x^3/b^4 + 1/10*(3*d^2*e*f^2 + 2*c*d*f^3)*(b*x^2 + a)^(3/2 
)*x^7/b + 1/2*sqrt(b*x^2 + a)*c^2*e^3*x + 21/512*(b*x^2 + a)^(3/2)*a^4*d^2 
*f^3*x/b^5 - 21/1024*sqrt(b*x^2 + a)*a^5*d^2*f^3*x/b^5 - 7/80*(3*d^2*e*f^2 
 + 2*c*d*f^3)*(b*x^2 + a)^(3/2)*a*x^5/b^2 + 1/8*(3*d^2*e^2*f + 6*c*d*e*f^2 
 + c^2*f^3)*(b*x^2 + a)^(3/2)*x^5/b + 1/2*a*c^2*e^3*arcsinh(b*x/sqrt(a*b)) 
/sqrt(b) - 21/1024*a^6*d^2*f^3*arcsinh(b*x/sqrt(a*b))/b^(11/2) + 7/96*(3*d 
^2*e*f^2 + 2*c*d*f^3)*(b*x^2 + a)^(3/2)*a^2*x^3/b^3 - 5/48*(3*d^2*e^2*f + 
6*c*d*e*f^2 + c^2*f^3)*(b*x^2 + a)^(3/2)*a*x^3/b^2 + 1/6*(d^2*e^3 + 6*c*d* 
e^2*f + 3*c^2*e*f^2)*(b*x^2 + a)^(3/2)*x^3/b - 7/128*(3*d^2*e*f^2 + 2*c*d* 
f^3)*(b*x^2 + a)^(3/2)*a^3*x/b^4 + 7/256*(3*d^2*e*f^2 + 2*c*d*f^3)*sqrt(b* 
x^2 + a)*a^4*x/b^4 + 5/64*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*(b*x^2 + a 
)^(3/2)*a^2*x/b^3 - 5/128*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*sqrt(b*x^2 
 + a)*a^3*x/b^3 - 1/8*(d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*(b*x^2 + a)^(3 
/2)*a*x/b^2 + 1/16*(d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*sqrt(b*x^2 + a)*a 
^2*x/b^2 + 1/4*(2*c*d*e^3 + 3*c^2*e^2*f)*(b*x^2 + a)^(3/2)*x/b - 1/8*(2*c* 
d*e^3 + 3*c^2*e^2*f)*sqrt(b*x^2 + a)*a*x/b + 7/256*(3*d^2*e*f^2 + 2*c*d*f^ 
3)*a^5*arcsinh(b*x/sqrt(a*b))/b^(9/2) - 5/128*(3*d^2*e^2*f + 6*c*d*e*f^2 + 
 c^2*f^3)*a^4*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 1/16*(d^2*e^3 + 6*c*d*e^...
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.16 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/15360*(2*(4*(2*(8*(10*d^2*f^3*x^2 + (36*b^10*d^2*e*f^2 + 24*b^10*c*d*f^3 
 + a*b^9*d^2*f^3)/b^10)*x^2 + 3*(120*b^10*d^2*e^2*f + 240*b^10*c*d*e*f^2 + 
 12*a*b^9*d^2*e*f^2 + 40*b^10*c^2*f^3 + 8*a*b^9*c*d*f^3 - 3*a^2*b^8*d^2*f^ 
3)/b^10)*x^2 + (320*b^10*d^2*e^3 + 1920*b^10*c*d*e^2*f + 120*a*b^9*d^2*e^2 
*f + 960*b^10*c^2*e*f^2 + 240*a*b^9*c*d*e*f^2 - 84*a^2*b^8*d^2*e*f^2 + 40* 
a*b^9*c^2*f^3 - 56*a^2*b^8*c*d*f^3 + 21*a^3*b^7*d^2*f^3)/b^10)*x^2 + 5*(76 
8*b^10*c*d*e^3 + 64*a*b^9*d^2*e^3 + 1152*b^10*c^2*e^2*f + 384*a*b^9*c*d*e^ 
2*f - 120*a^2*b^8*d^2*e^2*f + 192*a*b^9*c^2*e*f^2 - 240*a^2*b^8*c*d*e*f^2 
+ 84*a^3*b^7*d^2*e*f^2 - 40*a^2*b^8*c^2*f^3 + 56*a^3*b^7*c*d*f^3 - 21*a^4* 
b^6*d^2*f^3)/b^10)*x^2 + 15*(512*b^10*c^2*e^3 + 256*a*b^9*c*d*e^3 - 64*a^2 
*b^8*d^2*e^3 + 384*a*b^9*c^2*e^2*f - 384*a^2*b^8*c*d*e^2*f + 120*a^3*b^7*d 
^2*e^2*f - 192*a^2*b^8*c^2*e*f^2 + 240*a^3*b^7*c*d*e*f^2 - 84*a^4*b^6*d^2* 
e*f^2 + 40*a^3*b^7*c^2*f^3 - 56*a^4*b^6*c*d*f^3 + 21*a^5*b^5*d^2*f^3)/b^10 
)*sqrt(b*x^2 + a)*x - 1/1024*(512*a*b^5*c^2*e^3 - 256*a^2*b^4*c*d*e^3 + 64 
*a^3*b^3*d^2*e^3 - 384*a^2*b^4*c^2*e^2*f + 384*a^3*b^3*c*d*e^2*f - 120*a^4 
*b^2*d^2*e^2*f + 192*a^3*b^3*c^2*e*f^2 - 240*a^4*b^2*c*d*e*f^2 + 84*a^5*b* 
d^2*e*f^2 - 40*a^4*b^2*c^2*f^3 + 56*a^5*b*c*d*f^3 - 21*a^6*d^2*f^3)*log(ab 
s(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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