\(\int (a+b x^2)^{3/2} (c+d x^2)^2 (e+f x^2)^2 \, dx\) [299]

Optimal result

Integrand size = 30, antiderivative size = 542 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {a \left (384 b^4 c^2 e^2+7 a^4 d^2 f^2-128 a b^3 c e (d e+c f)-24 a^3 b d f (d e+c f)+24 a^2 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{1024 b^4}+\frac {\left (384 b^4 c^2 e^2+7 a^4 d^2 f^2-128 a b^3 c e (d e+c f)-24 a^3 b d f (d e+c f)+24 a^2 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x \left (a+b x^2\right )^{3/2}}{1536 b^4}-\frac {\left (7 a^3 d^2 f^2-128 b^3 c e (d e+c f)-24 a^2 b d f (d e+c f)+24 a b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x \left (a+b x^2\right )^{5/2}}{384 b^4}+\frac {\left (7 a^2 d^2 f^2-24 a b d f (d e+c f)+24 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^3 \left (a+b x^2\right )^{5/2}}{192 b^3}-\frac {d f (7 a d f-24 b (d e+c f)) x^5 \left (a+b x^2\right )^{5/2}}{120 b^2}+\frac {d^2 f^2 x^7 \left (a+b x^2\right )^{5/2}}{12 b}+\frac {a^2 \left (384 b^4 c^2 e^2+7 a^4 d^2 f^2-128 a b^3 c e (d e+c f)-24 a^3 b d f (d e+c f)+24 a^2 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{9/2}} \] Output:

1/1024*a*(384*b^4*c^2*e^2+7*a^4*d^2*f^2-128*a*b^3*c*e*(c*f+d*e)-24*a^3*b*d 
*f*(c*f+d*e)+24*a^2*b^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)/b^4 
+1/1536*(384*b^4*c^2*e^2+7*a^4*d^2*f^2-128*a*b^3*c*e*(c*f+d*e)-24*a^3*b*d* 
f*(c*f+d*e)+24*a^2*b^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(3/2)/b^4- 
1/384*(7*a^3*d^2*f^2-128*b^3*c*e*(c*f+d*e)-24*a^2*b*d*f*(c*f+d*e)+24*a*b^2 
*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(5/2)/b^4+1/192*(7*a^2*d^2*f^2-2 
4*a*b*d*f*(c*f+d*e)+24*b^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x^3*(b*x^2+a)^(5/2 
)/b^3-1/120*d*f*(7*a*d*f-24*b*(c*f+d*e))*x^5*(b*x^2+a)^(5/2)/b^2+1/12*d^2* 
f^2*x^7*(b*x^2+a)^(5/2)/b+1/1024*a^2*(384*b^4*c^2*e^2+7*a^4*d^2*f^2-128*a* 
b^3*c*e*(c*f+d*e)-24*a^3*b*d*f*(c*f+d*e)+24*a^2*b^2*(c^2*f^2+4*c*d*e*f+d^2 
*e^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
 

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.89 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^5 d^2 f^2+10 a^4 b d f \left (36 d e+36 c f+7 d f x^2\right )+48 a^2 b^3 \left (5 c^2 f \left (8 e+f x^2\right )+d^2 x^2 \left (5 e^2+4 e f x^2+f^2 x^4\right )+4 c d \left (10 e^2+5 e f x^2+f^2 x^4\right )\right )-8 a^3 b^2 \left (45 c^2 f^2+30 c d f \left (6 e+f x^2\right )+d^2 \left (45 e^2+30 e f x^2+7 f^2 x^4\right )\right )+128 b^5 x^2 \left (5 c^2 \left (6 e^2+8 e f x^2+3 f^2 x^4\right )+4 c d x^2 \left (10 e^2+15 e f x^2+6 f^2 x^4\right )+d^2 x^4 \left (15 e^2+24 e f x^2+10 f^2 x^4\right )\right )+64 a b^4 \left (5 c^2 \left (30 e^2+28 e f x^2+9 f^2 x^4\right )+d^2 x^4 \left (45 e^2+66 e f x^2+26 f^2 x^4\right )+2 c d x^2 \left (70 e^2+90 e f x^2+33 f^2 x^4\right )\right )\right )-15 a^2 \left (384 b^4 c^2 e^2+7 a^4 d^2 f^2-128 a b^3 c e (d e+c f)-24 a^3 b d f (d e+c f)+24 a^2 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{15360 b^{9/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^5*d^2*f^2 + 10*a^4*b*d*f*(36*d*e + 36*c 
*f + 7*d*f*x^2) + 48*a^2*b^3*(5*c^2*f*(8*e + f*x^2) + d^2*x^2*(5*e^2 + 4*e 
*f*x^2 + f^2*x^4) + 4*c*d*(10*e^2 + 5*e*f*x^2 + f^2*x^4)) - 8*a^3*b^2*(45* 
c^2*f^2 + 30*c*d*f*(6*e + f*x^2) + d^2*(45*e^2 + 30*e*f*x^2 + 7*f^2*x^4)) 
+ 128*b^5*x^2*(5*c^2*(6*e^2 + 8*e*f*x^2 + 3*f^2*x^4) + 4*c*d*x^2*(10*e^2 + 
 15*e*f*x^2 + 6*f^2*x^4) + d^2*x^4*(15*e^2 + 24*e*f*x^2 + 10*f^2*x^4)) + 6 
4*a*b^4*(5*c^2*(30*e^2 + 28*e*f*x^2 + 9*f^2*x^4) + d^2*x^4*(45*e^2 + 66*e* 
f*x^2 + 26*f^2*x^4) + 2*c*d*x^2*(70*e^2 + 90*e*f*x^2 + 33*f^2*x^4))) - 15* 
a^2*(384*b^4*c^2*e^2 + 7*a^4*d^2*f^2 - 128*a*b^3*c*e*(d*e + c*f) - 24*a^3* 
b*d*f*(d*e + c*f) + 24*a^2*b^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*Log[-(Sqrt 
[b]*x) + Sqrt[a + b*x^2]])/(15360*b^(9/2))
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.52, 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.

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

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(3*a*c^2*e^2*x*Sqrt[a + b*x^2])/8 - (7*a^5*d^2*f^2*x*Sqrt[a + b*x^2])/(102 
4*b^4) + (a^2*c*e*(d*e + c*f)*x*Sqrt[a + b*x^2])/(8*b) + (3*a^4*d*f*(d*e + 
 c*f)*x*Sqrt[a + b*x^2])/(128*b^3) - (3*a^3*(d^2*e^2 + 4*c*d*e*f + c^2*f^2 
)*x*Sqrt[a + b*x^2])/(128*b^2) + (7*a^4*d^2*f^2*x^3*Sqrt[a + b*x^2])/(1536 
*b^3) + (a*c*e*(d*e + c*f)*x^3*Sqrt[a + b*x^2])/4 - (a^3*d*f*(d*e + c*f)*x 
^3*Sqrt[a + b*x^2])/(64*b^2) + (a^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*x^3*Sq 
rt[a + b*x^2])/(64*b) - (7*a^3*d^2*f^2*x^5*Sqrt[a + b*x^2])/(1920*b^2) + ( 
a^2*d*f*(d*e + c*f)*x^5*Sqrt[a + b*x^2])/(80*b) + (a*(d^2*e^2 + 4*c*d*e*f 
+ c^2*f^2)*x^5*Sqrt[a + b*x^2])/16 + (a^2*d^2*f^2*x^7*Sqrt[a + b*x^2])/(32 
0*b) + (3*a*d*f*(d*e + c*f)*x^7*Sqrt[a + b*x^2])/40 + (a*d^2*f^2*x^9*Sqrt[ 
a + b*x^2])/40 + (c^2*e^2*x*(a + b*x^2)^(3/2))/4 + (c*e*(d*e + c*f)*x^3*(a 
 + b*x^2)^(3/2))/3 + ((d^2*e^2 + 4*c*d*e*f + c^2*f^2)*x^5*(a + b*x^2)^(3/2 
))/8 + (d*f*(d*e + c*f)*x^7*(a + b*x^2)^(3/2))/5 + (d^2*f^2*x^9*(a + b*x^2 
)^(3/2))/12 + (3*a^2*c^2*e^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*Sqrt 
[b]) + (7*a^6*d^2*f^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(9/2)) 
 - (a^3*c*e*(d*e + c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(3/2)) 
- (3*a^5*d*f*(d*e + c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(7/2 
)) + (3*a^4*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b 
*x^2]])/(128*b^(5/2))
 

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 = 0.84 (sec) , antiderivative size = 449, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {\frac {7 a^{2} \left (a^{4} d^{2} f^{2}-\frac {24 a^{3} b d f \left (c f +d e \right )}{7}+\frac {24 a^{2} b^{2} \left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}\right )}{7}-\frac {128 a \,b^{3} c e \left (c f +d e \right )}{7}+\frac {384 b^{4} c^{2} e^{2}}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{1024}-\frac {7 \left (-\frac {256 \left (\frac {d^{2} f^{2} x^{8}}{3}+\frac {4 d f \left (c f +d e \right ) x^{6}}{5}+\left (\frac {1}{2} c^{2} f^{2}+2 c d e f +\frac {1}{2} d^{2} e^{2}\right ) x^{4}+\frac {4 c e \left (c f +d e \right ) x^{2}}{3}+c^{2} e^{2}\right ) x^{2} b^{\frac {11}{2}}}{7}+a \left (64 \left (-\frac {26 d^{2} f^{2} x^{8}}{105}-\frac {22 d f \left (c f +d e \right ) x^{6}}{35}+\frac {3 \left (-c^{2} f^{2}-4 c d e f -d^{2} e^{2}\right ) x^{4}}{7}-\frac {4 c e \left (c f +d e \right ) x^{2}}{3}-\frac {10 c^{2} e^{2}}{7}\right ) b^{\frac {9}{2}}+a \left (\frac {16 \left (-\frac {d^{2} f^{2} x^{6}}{5}-\frac {4 d f \left (c f +d e \right ) x^{4}}{5}+\left (-c^{2} f^{2}-4 c d e f -d^{2} e^{2}\right ) x^{2}-8 c e \left (c f +d e \right )\right ) b^{\frac {7}{2}}}{7}+a \left (8 \left (\frac {d^{2} f^{2} x^{4}}{15}+\frac {2 d f \left (c f +d e \right ) x^{2}}{7}+\frac {12 c d e f}{7}+\frac {3 c^{2} f^{2}}{7}+\frac {3 d^{2} e^{2}}{7}\right ) b^{\frac {5}{2}}+\left (2 \left (-\frac {12}{7} c f -\frac {12}{7} d e -\frac {1}{3} d f \,x^{2}\right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right ) a d f \right )\right )\right )\right ) x \sqrt {b \,x^{2}+a}}{1024}}{b^{\frac {9}{2}}}\)	\(449\)
default	\(c^{2} e^{2} \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 )+2 d f \left (c f +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 )+2 c e \left (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 )+\left (c^{2} f^{2}+4 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^{2} \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 )\)	\(547\)
risch	\(-\frac {x \left (-1280 b^{5} d^{2} f^{2} x^{10}-1664 a \,b^{4} d^{2} f^{2} x^{8}-3072 b^{5} c d \,f^{2} x^{8}-3072 b^{5} d^{2} e f \,x^{8}-48 a^{2} b^{3} d^{2} f^{2} x^{6}-4224 a \,b^{4} c d \,f^{2} x^{6}-4224 a \,b^{4} d^{2} e f \,x^{6}-1920 b^{5} c^{2} f^{2} x^{6}-7680 b^{5} c d e f \,x^{6}-1920 b^{5} d^{2} e^{2} x^{6}+56 a^{3} b^{2} d^{2} f^{2} x^{4}-192 a^{2} b^{3} c d \,f^{2} x^{4}-192 a^{2} b^{3} d^{2} e f \,x^{4}-2880 a \,b^{4} c^{2} f^{2} x^{4}-11520 a \,b^{4} c d e f \,x^{4}-2880 a \,b^{4} d^{2} e^{2} x^{4}-5120 b^{5} c^{2} e f \,x^{4}-5120 b^{5} c d \,e^{2} x^{4}-70 a^{4} b \,d^{2} f^{2} x^{2}+240 a^{3} b^{2} c d \,f^{2} x^{2}+240 a^{3} b^{2} d^{2} e f \,x^{2}-240 a^{2} b^{3} c^{2} f^{2} x^{2}-960 a^{2} b^{3} c d e f \,x^{2}-240 a^{2} b^{3} d^{2} e^{2} x^{2}-8960 a \,b^{4} c^{2} e f \,x^{2}-8960 a \,b^{4} c d \,e^{2} x^{2}-3840 b^{5} c^{2} e^{2} x^{2}+105 a^{5} d^{2} f^{2}-360 a^{4} b c d \,f^{2}-360 a^{4} b \,d^{2} e f +360 a^{3} b^{2} c^{2} f^{2}+1440 a^{3} b^{2} c d e f +360 a^{3} b^{2} d^{2} e^{2}-1920 a^{2} b^{3} c^{2} e f -1920 a^{2} b^{3} c d \,e^{2}-9600 a \,b^{4} c^{2} e^{2}\right ) \sqrt {b \,x^{2}+a}}{15360 b^{4}}+\frac {a^{2} \left (7 a^{4} d^{2} f^{2}-24 a^{3} b c d \,f^{2}-24 a^{3} b \,d^{2} e f +24 a^{2} b^{2} c^{2} f^{2}+96 a^{2} b^{2} c d e f +24 a^{2} b^{2} d^{2} e^{2}-128 a \,b^{3} c^{2} e f -128 a \,b^{3} c d \,e^{2}+384 b^{4} c^{2} e^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {9}{2}}}\)	\(656\)

Input:

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

Output:

7/1024*(a^2*(a^4*d^2*f^2-24/7*a^3*b*d*f*(c*f+d*e)+24/7*a^2*b^2*(c^2*f^2+4* 
c*d*e*f+d^2*e^2)-128/7*a*b^3*c*e*(c*f+d*e)+384/7*b^4*c^2*e^2)*arctanh((b*x 
^2+a)^(1/2)/x/b^(1/2))-(-256/7*(1/3*d^2*f^2*x^8+4/5*d*f*(c*f+d*e)*x^6+(1/2 
*c^2*f^2+2*c*d*e*f+1/2*d^2*e^2)*x^4+4/3*c*e*(c*f+d*e)*x^2+c^2*e^2)*x^2*b^( 
11/2)+a*(64*(-26/105*d^2*f^2*x^8-22/35*d*f*(c*f+d*e)*x^6+3/7*(-c^2*f^2-4*c 
*d*e*f-d^2*e^2)*x^4-4/3*c*e*(c*f+d*e)*x^2-10/7*c^2*e^2)*b^(9/2)+a*(16/7*(- 
1/5*d^2*f^2*x^6-4/5*d*f*(c*f+d*e)*x^4+(-c^2*f^2-4*c*d*e*f-d^2*e^2)*x^2-8*c 
*e*(c*f+d*e))*b^(7/2)+a*(8*(1/15*d^2*f^2*x^4+2/7*d*f*(c*f+d*e)*x^2+12/7*c* 
d*e*f+3/7*c^2*f^2+3/7*d^2*e^2)*b^(5/2)+(2*(-12/7*c*f-12/7*d*e-1/3*d*f*x^2) 
*b^(3/2)+a*d*f*b^(1/2))*a*d*f))))*x*(b*x^2+a)^(1/2))/b^(9/2)
 

Fricas [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 1182, normalized size of antiderivative = 2.18 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

[1/30720*(15*(8*(48*a^2*b^4*c^2 - 16*a^3*b^3*c*d + 3*a^4*b^2*d^2)*e^2 - 8* 
(16*a^3*b^3*c^2 - 12*a^4*b^2*c*d + 3*a^5*b*d^2)*e*f + (24*a^4*b^2*c^2 - 24 
*a^5*b*c*d + 7*a^6*d^2)*f^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt 
(b)*x - a) + 2*(1280*b^6*d^2*f^2*x^11 + 128*(24*b^6*d^2*e*f + (24*b^6*c*d 
+ 13*a*b^5*d^2)*f^2)*x^9 + 48*(40*b^6*d^2*e^2 + 8*(20*b^6*c*d + 11*a*b^5*d 
^2)*e*f + (40*b^6*c^2 + 88*a*b^5*c*d + a^2*b^4*d^2)*f^2)*x^7 + 8*(40*(16*b 
^6*c*d + 9*a*b^5*d^2)*e^2 + 8*(80*b^6*c^2 + 180*a*b^5*c*d + 3*a^2*b^4*d^2) 
*e*f + (360*a*b^5*c^2 + 24*a^2*b^4*c*d - 7*a^3*b^3*d^2)*f^2)*x^5 + 10*(8*( 
48*b^6*c^2 + 112*a*b^5*c*d + 3*a^2*b^4*d^2)*e^2 + 8*(112*a*b^5*c^2 + 12*a^ 
2*b^4*c*d - 3*a^3*b^3*d^2)*e*f + (24*a^2*b^4*c^2 - 24*a^3*b^3*c*d + 7*a^4* 
b^2*d^2)*f^2)*x^3 + 15*(8*(80*a*b^5*c^2 + 16*a^2*b^4*c*d - 3*a^3*b^3*d^2)* 
e^2 + 8*(16*a^2*b^4*c^2 - 12*a^3*b^3*c*d + 3*a^4*b^2*d^2)*e*f - (24*a^3*b^ 
3*c^2 - 24*a^4*b^2*c*d + 7*a^5*b*d^2)*f^2)*x)*sqrt(b*x^2 + a))/b^5, -1/153 
60*(15*(8*(48*a^2*b^4*c^2 - 16*a^3*b^3*c*d + 3*a^4*b^2*d^2)*e^2 - 8*(16*a^ 
3*b^3*c^2 - 12*a^4*b^2*c*d + 3*a^5*b*d^2)*e*f + (24*a^4*b^2*c^2 - 24*a^5*b 
*c*d + 7*a^6*d^2)*f^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (1280 
*b^6*d^2*f^2*x^11 + 128*(24*b^6*d^2*e*f + (24*b^6*c*d + 13*a*b^5*d^2)*f^2) 
*x^9 + 48*(40*b^6*d^2*e^2 + 8*(20*b^6*c*d + 11*a*b^5*d^2)*e*f + (40*b^6*c^ 
2 + 88*a*b^5*c*d + a^2*b^4*d^2)*f^2)*x^7 + 8*(40*(16*b^6*c*d + 9*a*b^5*d^2 
)*e^2 + 8*(80*b^6*c^2 + 180*a*b^5*c*d + 3*a^2*b^4*d^2)*e*f + (360*a*b^5...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1464 vs. \(2 (554) = 1108\).

Time = 0.62 (sec) , antiderivative size = 1464, normalized size of antiderivative = 2.70 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.33 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/12*(b*x^2 + a)^(5/2)*d^2*f^2*x^7/b - 7/120*(b*x^2 + a)^(5/2)*a*d^2*f^2*x 
^5/b^2 + 7/192*(b*x^2 + a)^(5/2)*a^2*d^2*f^2*x^3/b^3 + 1/5*(d^2*e*f + c*d* 
f^2)*(b*x^2 + a)^(5/2)*x^5/b + 1/4*(b*x^2 + a)^(3/2)*c^2*e^2*x + 3/8*sqrt( 
b*x^2 + a)*a*c^2*e^2*x - 7/384*(b*x^2 + a)^(5/2)*a^3*d^2*f^2*x/b^4 + 7/153 
6*(b*x^2 + a)^(3/2)*a^4*d^2*f^2*x/b^4 + 7/1024*sqrt(b*x^2 + a)*a^5*d^2*f^2 
*x/b^4 + 3/8*a^2*c^2*e^2*arcsinh(b*x/sqrt(a*b))/sqrt(b) + 7/1024*a^6*d^2*f 
^2*arcsinh(b*x/sqrt(a*b))/b^(9/2) - 1/8*(d^2*e*f + c*d*f^2)*(b*x^2 + a)^(5 
/2)*a*x^3/b^2 + 1/8*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*(b*x^2 + a)^(5/2)*x^3/ 
b + 1/16*(d^2*e*f + c*d*f^2)*(b*x^2 + a)^(5/2)*a^2*x/b^3 - 1/64*(d^2*e*f + 
 c*d*f^2)*(b*x^2 + a)^(3/2)*a^3*x/b^3 - 3/128*(d^2*e*f + c*d*f^2)*sqrt(b*x 
^2 + a)*a^4*x/b^3 - 1/16*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*(b*x^2 + a)^(5/2) 
*a*x/b^2 + 1/64*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*(b*x^2 + a)^(3/2)*a^2*x/b^ 
2 + 3/128*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*sqrt(b*x^2 + a)*a^3*x/b^2 + 1/3* 
(c*d*e^2 + c^2*e*f)*(b*x^2 + a)^(5/2)*x/b - 1/12*(c*d*e^2 + c^2*e*f)*(b*x^ 
2 + a)^(3/2)*a*x/b - 1/8*(c*d*e^2 + c^2*e*f)*sqrt(b*x^2 + a)*a^2*x/b - 3/1 
28*(d^2*e*f + c*d*f^2)*a^5*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 3/128*(d^2*e^2 
 + 4*c*d*e*f + c^2*f^2)*a^4*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 1/8*(c*d*e^2 
+ c^2*e*f)*a^3*arcsinh(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.18 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, b d^{2} f^{2} x^{2} + \frac {24 \, b^{11} d^{2} e f + 24 \, b^{11} c d f^{2} + 13 \, a b^{10} d^{2} f^{2}}{b^{10}}\right )} x^{2} + \frac {3 \, {\left (40 \, b^{11} d^{2} e^{2} + 160 \, b^{11} c d e f + 88 \, a b^{10} d^{2} e f + 40 \, b^{11} c^{2} f^{2} + 88 \, a b^{10} c d f^{2} + a^{2} b^{9} d^{2} f^{2}\right )}}{b^{10}}\right )} x^{2} + \frac {640 \, b^{11} c d e^{2} + 360 \, a b^{10} d^{2} e^{2} + 640 \, b^{11} c^{2} e f + 1440 \, a b^{10} c d e f + 24 \, a^{2} b^{9} d^{2} e f + 360 \, a b^{10} c^{2} f^{2} + 24 \, a^{2} b^{9} c d f^{2} - 7 \, a^{3} b^{8} d^{2} f^{2}}{b^{10}}\right )} x^{2} + \frac {5 \, {\left (384 \, b^{11} c^{2} e^{2} + 896 \, a b^{10} c d e^{2} + 24 \, a^{2} b^{9} d^{2} e^{2} + 896 \, a b^{10} c^{2} e f + 96 \, a^{2} b^{9} c d e f - 24 \, a^{3} b^{8} d^{2} e f + 24 \, a^{2} b^{9} c^{2} f^{2} - 24 \, a^{3} b^{8} c d f^{2} + 7 \, a^{4} b^{7} d^{2} f^{2}\right )}}{b^{10}}\right )} x^{2} + \frac {15 \, {\left (640 \, a b^{10} c^{2} e^{2} + 128 \, a^{2} b^{9} c d e^{2} - 24 \, a^{3} b^{8} d^{2} e^{2} + 128 \, a^{2} b^{9} c^{2} e f - 96 \, a^{3} b^{8} c d e f + 24 \, a^{4} b^{7} d^{2} e f - 24 \, a^{3} b^{8} c^{2} f^{2} + 24 \, a^{4} b^{7} c d f^{2} - 7 \, a^{5} b^{6} d^{2} f^{2}\right )}}{b^{10}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (384 \, a^{2} b^{4} c^{2} e^{2} - 128 \, a^{3} b^{3} c d e^{2} + 24 \, a^{4} b^{2} d^{2} e^{2} - 128 \, a^{3} b^{3} c^{2} e f + 96 \, a^{4} b^{2} c d e f - 24 \, a^{5} b d^{2} e f + 24 \, a^{4} b^{2} c^{2} f^{2} - 24 \, a^{5} b c d f^{2} + 7 \, a^{6} d^{2} f^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{1024 \, b^{\frac {9}{2}}} \] Input:

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

Output:

1/15360*(2*(4*(2*(8*(10*b*d^2*f^2*x^2 + (24*b^11*d^2*e*f + 24*b^11*c*d*f^2 
 + 13*a*b^10*d^2*f^2)/b^10)*x^2 + 3*(40*b^11*d^2*e^2 + 160*b^11*c*d*e*f + 
88*a*b^10*d^2*e*f + 40*b^11*c^2*f^2 + 88*a*b^10*c*d*f^2 + a^2*b^9*d^2*f^2) 
/b^10)*x^2 + (640*b^11*c*d*e^2 + 360*a*b^10*d^2*e^2 + 640*b^11*c^2*e*f + 1 
440*a*b^10*c*d*e*f + 24*a^2*b^9*d^2*e*f + 360*a*b^10*c^2*f^2 + 24*a^2*b^9* 
c*d*f^2 - 7*a^3*b^8*d^2*f^2)/b^10)*x^2 + 5*(384*b^11*c^2*e^2 + 896*a*b^10* 
c*d*e^2 + 24*a^2*b^9*d^2*e^2 + 896*a*b^10*c^2*e*f + 96*a^2*b^9*c*d*e*f - 2 
4*a^3*b^8*d^2*e*f + 24*a^2*b^9*c^2*f^2 - 24*a^3*b^8*c*d*f^2 + 7*a^4*b^7*d^ 
2*f^2)/b^10)*x^2 + 15*(640*a*b^10*c^2*e^2 + 128*a^2*b^9*c*d*e^2 - 24*a^3*b 
^8*d^2*e^2 + 128*a^2*b^9*c^2*e*f - 96*a^3*b^8*c*d*e*f + 24*a^4*b^7*d^2*e*f 
 - 24*a^3*b^8*c^2*f^2 + 24*a^4*b^7*c*d*f^2 - 7*a^5*b^6*d^2*f^2)/b^10)*sqrt 
(b*x^2 + a)*x - 1/1024*(384*a^2*b^4*c^2*e^2 - 128*a^3*b^3*c*d*e^2 + 24*a^4 
*b^2*d^2*e^2 - 128*a^3*b^3*c^2*e*f + 96*a^4*b^2*c*d*e*f - 24*a^5*b*d^2*e*f 
 + 24*a^4*b^2*c^2*f^2 - 24*a^5*b*c*d*f^2 + 7*a^6*d^2*f^2)*log(abs(-sqrt(b) 
*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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