\(\int \sqrt {a+b x^2} (c+d x^2)^3 (e+f x^2)^2 \, dx\) [274]

Optimal result

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

1/1024*(512*b^5*c^3*e^2-21*a^5*d^3*f^2-128*a*b^4*c^2*e*(2*c*f+3*d*e)+28*a^ 
4*b*d^2*f*(3*c*f+2*d*e)+64*a^2*b^3*c*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)-40*a^3* 
b^2*d*(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 
^3*f^2+128*b^4*c^2*e*(2*c*f+3*d*e)-28*a^3*b*d^2*f*(3*c*f+2*d*e)-64*a*b^3*c 
*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)+40*a^2*b^2*d*(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^3*f^2-28*a^2*b*d^2*f*(3*c*f+2*d*e)- 
64*b^3*c*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)+40*a*b^2*d*(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*d*(21*a^2*d^2*f^2-28*a*b*d*f*(3*c*f+2 
*d*e)+40*b^2*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*x^5*(b*x^2+a)^(3/2)/b^3+1/40*d 
^2*f*(-3*a*d*f+12*b*c*f+8*b*d*e)*x^7*(b*x^2+a)^(3/2)/b^2+1/12*d^3*f^2*x^9* 
(b*x^2+a)^(3/2)/b+1/1024*a*(512*b^5*c^3*e^2-21*a^5*d^3*f^2-128*a*b^4*c^2*e 
*(2*c*f+3*d*e)+28*a^4*b*d^2*f*(3*c*f+2*d*e)+64*a^2*b^3*c*(c^2*f^2+6*c*d*e* 
f+3*d^2*e^2)-40*a^3*b^2*d*(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 = 2.13 (sec) , antiderivative size = 584, normalized size of antiderivative = 0.86 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^2 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (315 a^5 d^3 f^2-210 a^4 b d^2 f \left (4 d e+6 c f+d f x^2\right )+64 a b^4 \left (10 c^3 f \left (6 e+f x^2\right )+15 c^2 d \left (6 e^2+4 e f x^2+f^2 x^4\right )+d^3 x^4 \left (5 e^2+6 e f x^2+2 f^2 x^4\right )+3 c d^2 x^2 \left (10 e^2+10 e f x^2+3 f^2 x^4\right )\right )-16 a^2 b^3 \left (60 c^3 f^2+15 c^2 d f \left (24 e+5 f x^2\right )+6 c d^2 \left (30 e^2+25 e f x^2+7 f^2 x^4\right )+d^3 x^2 \left (25 e^2+28 e f x^2+9 f^2 x^4\right )\right )+128 b^5 \left (20 c^3 \left (3 e^2+3 e f x^2+f^2 x^4\right )+15 c^2 d x^2 \left (6 e^2+8 e f x^2+3 f^2 x^4\right )+6 c d^2 x^4 \left (10 e^2+15 e f x^2+6 f^2 x^4\right )+d^3 x^6 \left (15 e^2+24 e f x^2+10 f^2 x^4\right )\right )+8 a^3 b^2 d \left (225 c^2 f^2+15 c d f \left (30 e+7 f x^2\right )+d^2 \left (75 e^2+70 e f x^2+21 f^2 x^4\right )\right )\right )+15 a \left (-512 b^5 c^3 e^2+21 a^5 d^3 f^2+128 a b^4 c^2 e (3 d e+2 c f)-28 a^4 b d^2 f (2 d e+3 c f)-64 a^2 b^3 c \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+40 a^3 b^2 d \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{15360 b^{11/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(315*a^5*d^3*f^2 - 210*a^4*b*d^2*f*(4*d*e + 6*c 
*f + d*f*x^2) + 64*a*b^4*(10*c^3*f*(6*e + f*x^2) + 15*c^2*d*(6*e^2 + 4*e*f 
*x^2 + f^2*x^4) + d^3*x^4*(5*e^2 + 6*e*f*x^2 + 2*f^2*x^4) + 3*c*d^2*x^2*(1 
0*e^2 + 10*e*f*x^2 + 3*f^2*x^4)) - 16*a^2*b^3*(60*c^3*f^2 + 15*c^2*d*f*(24 
*e + 5*f*x^2) + 6*c*d^2*(30*e^2 + 25*e*f*x^2 + 7*f^2*x^4) + d^3*x^2*(25*e^ 
2 + 28*e*f*x^2 + 9*f^2*x^4)) + 128*b^5*(20*c^3*(3*e^2 + 3*e*f*x^2 + f^2*x^ 
4) + 15*c^2*d*x^2*(6*e^2 + 8*e*f*x^2 + 3*f^2*x^4) + 6*c*d^2*x^4*(10*e^2 + 
15*e*f*x^2 + 6*f^2*x^4) + d^3*x^6*(15*e^2 + 24*e*f*x^2 + 10*f^2*x^4)) + 8* 
a^3*b^2*d*(225*c^2*f^2 + 15*c*d*f*(30*e + 7*f*x^2) + d^2*(75*e^2 + 70*e*f* 
x^2 + 21*f^2*x^4))) + 15*a*(-512*b^5*c^3*e^2 + 21*a^5*d^3*f^2 + 128*a*b^4* 
c^2*e*(3*d*e + 2*c*f) - 28*a^4*b*d^2*f*(2*d*e + 3*c*f) - 64*a^2*b^3*c*(3*d 
^2*e^2 + 6*c*d*e*f + c^2*f^2) + 40*a^3*b^2*d*(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 = 1.02 (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.

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

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 0.91 (sec) , antiderivative size = 546, normalized size of antiderivative = 0.81

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 1432, normalized size of antiderivative = 2.12 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/30720*(15*(8*(64*a*b^5*c^3 - 48*a^2*b^4*c^2*d + 24*a^3*b^3*c*d^2 - 5*a 
^4*b^2*d^3)*e^2 - 8*(32*a^2*b^4*c^3 - 48*a^3*b^3*c^2*d + 30*a^4*b^2*c*d^2 
- 7*a^5*b*d^3)*e*f + (64*a^3*b^3*c^3 - 120*a^4*b^2*c^2*d + 84*a^5*b*c*d^2 
- 21*a^6*d^3)*f^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) 
 - 2*(1280*b^6*d^3*f^2*x^11 + 128*(24*b^6*d^3*e*f + (36*b^6*c*d^2 + a*b^5* 
d^3)*f^2)*x^9 + 48*(40*b^6*d^3*e^2 + 8*(30*b^6*c*d^2 + a*b^5*d^3)*e*f + 3* 
(40*b^6*c^2*d + 4*a*b^5*c*d^2 - a^2*b^4*d^3)*f^2)*x^7 + 8*(40*(24*b^6*c*d^ 
2 + a*b^5*d^3)*e^2 + 8*(240*b^6*c^2*d + 30*a*b^5*c*d^2 - 7*a^2*b^4*d^3)*e* 
f + (320*b^6*c^3 + 120*a*b^5*c^2*d - 84*a^2*b^4*c*d^2 + 21*a^3*b^3*d^3)*f^ 
2)*x^5 + 10*(8*(144*b^6*c^2*d + 24*a*b^5*c*d^2 - 5*a^2*b^4*d^3)*e^2 + 8*(9 
6*b^6*c^3 + 48*a*b^5*c^2*d - 30*a^2*b^4*c*d^2 + 7*a^3*b^3*d^3)*e*f + (64*a 
*b^5*c^3 - 120*a^2*b^4*c^2*d + 84*a^3*b^3*c*d^2 - 21*a^4*b^2*d^3)*f^2)*x^3 
 + 15*(8*(64*b^6*c^3 + 48*a*b^5*c^2*d - 24*a^2*b^4*c*d^2 + 5*a^3*b^3*d^3)* 
e^2 + 8*(32*a*b^5*c^3 - 48*a^2*b^4*c^2*d + 30*a^3*b^3*c*d^2 - 7*a^4*b^2*d^ 
3)*e*f - (64*a^2*b^4*c^3 - 120*a^3*b^3*c^2*d + 84*a^4*b^2*c*d^2 - 21*a^5*b 
*d^3)*f^2)*x)*sqrt(b*x^2 + a))/b^6, -1/15360*(15*(8*(64*a*b^5*c^3 - 48*a^2 
*b^4*c^2*d + 24*a^3*b^3*c*d^2 - 5*a^4*b^2*d^3)*e^2 - 8*(32*a^2*b^4*c^3 - 4 
8*a^3*b^3*c^2*d + 30*a^4*b^2*c*d^2 - 7*a^5*b*d^3)*e*f + (64*a^3*b^3*c^3 - 
120*a^4*b^2*c^2*d + 84*a^5*b*c*d^2 - 21*a^6*d^3)*f^2)*sqrt(-b)*arctan(sqrt 
(-b)*x/sqrt(b*x^2 + a)) - (1280*b^6*d^3*f^2*x^11 + 128*(24*b^6*d^3*e*f ...
 

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 1290, normalized size of antiderivative = 1.91 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

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

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

Output:

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

Giac [A] (verification not implemented)

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

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

Output:

1/15360*(2*(4*(2*(8*(10*d^3*f^2*x^2 + (24*b^10*d^3*e*f + 36*b^10*c*d^2*f^2 
 + a*b^9*d^3*f^2)/b^10)*x^2 + 3*(40*b^10*d^3*e^2 + 240*b^10*c*d^2*e*f + 8* 
a*b^9*d^3*e*f + 120*b^10*c^2*d*f^2 + 12*a*b^9*c*d^2*f^2 - 3*a^2*b^8*d^3*f^ 
2)/b^10)*x^2 + (960*b^10*c*d^2*e^2 + 40*a*b^9*d^3*e^2 + 1920*b^10*c^2*d*e* 
f + 240*a*b^9*c*d^2*e*f - 56*a^2*b^8*d^3*e*f + 320*b^10*c^3*f^2 + 120*a*b^ 
9*c^2*d*f^2 - 84*a^2*b^8*c*d^2*f^2 + 21*a^3*b^7*d^3*f^2)/b^10)*x^2 + 5*(11 
52*b^10*c^2*d*e^2 + 192*a*b^9*c*d^2*e^2 - 40*a^2*b^8*d^3*e^2 + 768*b^10*c^ 
3*e*f + 384*a*b^9*c^2*d*e*f - 240*a^2*b^8*c*d^2*e*f + 56*a^3*b^7*d^3*e*f + 
 64*a*b^9*c^3*f^2 - 120*a^2*b^8*c^2*d*f^2 + 84*a^3*b^7*c*d^2*f^2 - 21*a^4* 
b^6*d^3*f^2)/b^10)*x^2 + 15*(512*b^10*c^3*e^2 + 384*a*b^9*c^2*d*e^2 - 192* 
a^2*b^8*c*d^2*e^2 + 40*a^3*b^7*d^3*e^2 + 256*a*b^9*c^3*e*f - 384*a^2*b^8*c 
^2*d*e*f + 240*a^3*b^7*c*d^2*e*f - 56*a^4*b^6*d^3*e*f - 64*a^2*b^8*c^3*f^2 
 + 120*a^3*b^7*c^2*d*f^2 - 84*a^4*b^6*c*d^2*f^2 + 21*a^5*b^5*d^3*f^2)/b^10 
)*sqrt(b*x^2 + a)*x - 1/1024*(512*a*b^5*c^3*e^2 - 384*a^2*b^4*c^2*d*e^2 + 
192*a^3*b^3*c*d^2*e^2 - 40*a^4*b^2*d^3*e^2 - 256*a^2*b^4*c^3*e*f + 384*a^3 
*b^3*c^2*d*e*f - 240*a^4*b^2*c*d^2*e*f + 56*a^5*b*d^3*e*f + 64*a^3*b^3*c^3 
*f^2 - 120*a^4*b^2*c^2*d*f^2 + 84*a^5*b*c*d^2*f^2 - 21*a^6*d^3*f^2)*log(ab 
s(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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