\(\int \frac {(c+d x^2)^2 (e+f x^2)^2}{\sqrt {a+b x^2}} \, dx\) [324]

Optimal result

Integrand size = 30, antiderivative size = 336 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=-\frac {\left (35 a^3 d^2 f^2-128 b^3 c e (d e+c f)-80 a^2 b d f (d e+c f)+48 a b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{128 b^4}+\frac {\left (35 a^2 d^2 f^2-80 a b d f (d e+c f)+48 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^3 \sqrt {a+b x^2}}{192 b^3}-\frac {d f (7 a d f-16 b (d e+c f)) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {d^2 f^2 x^7 \sqrt {a+b x^2}}{8 b}+\frac {\left (128 b^4 c^2 e^2+35 a^4 d^2 f^2-128 a b^3 c e (d e+c f)-80 a^3 b d f (d e+c f)+48 a^2 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{9/2}} \] Output:

-1/128*(35*a^3*d^2*f^2-128*b^3*c*e*(c*f+d*e)-80*a^2*b*d*f*(c*f+d*e)+48*a*b 
^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)/b^4+1/192*(35*a^2*d^2*f^ 
2-80*a*b*d*f*(c*f+d*e)+48*b^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x^3*(b*x^2+a)^( 
1/2)/b^3-1/48*d*f*(7*a*d*f-16*b*(c*f+d*e))*x^5*(b*x^2+a)^(1/2)/b^2+1/8*d^2 
*f^2*x^7*(b*x^2+a)^(1/2)/b+1/128*(128*b^4*c^2*e^2+35*a^4*d^2*f^2-128*a*b^3 
*c*e*(c*f+d*e)-80*a^3*b*d*f*(c*f+d*e)+48*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 = 0.82 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^3 d^2 f^2+10 a^2 b d f \left (24 d e+24 c f+7 d f x^2\right )+16 b^3 \left (6 c^2 f \left (4 e+f x^2\right )+8 c d \left (3 e^2+3 e f x^2+f^2 x^4\right )+d^2 x^2 \left (6 e^2+8 e f x^2+3 f^2 x^4\right )\right )-8 a b^2 \left (18 c^2 f^2+4 c d f \left (18 e+5 f x^2\right )+d^2 \left (18 e^2+20 e f x^2+7 f^2 x^4\right )\right )\right )-3 \left (128 b^4 c^2 e^2+35 a^4 d^2 f^2-128 a b^3 c e (d e+c f)-80 a^3 b d f (d e+c f)+48 a^2 b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{384 b^{9/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^3*d^2*f^2 + 10*a^2*b*d*f*(24*d*e + 24*c 
*f + 7*d*f*x^2) + 16*b^3*(6*c^2*f*(4*e + f*x^2) + 8*c*d*(3*e^2 + 3*e*f*x^2 
 + f^2*x^4) + d^2*x^2*(6*e^2 + 8*e*f*x^2 + 3*f^2*x^4)) - 8*a*b^2*(18*c^2*f 
^2 + 4*c*d*f*(18*e + 5*f*x^2) + d^2*(18*e^2 + 20*e*f*x^2 + 7*f^2*x^4))) - 
3*(128*b^4*c^2*e^2 + 35*a^4*d^2*f^2 - 128*a*b^3*c*e*(d*e + c*f) - 80*a^3*b 
*d*f*(d*e + c*f) + 48*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]])/(384*b^(9/2))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 510, 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.

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

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {35 a^4 d^2 f^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{9/2}}-\frac {5 a^3 d f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (c f+d e)}{8 b^{7/2}}-\frac {35 a^3 d^2 f^2 x \sqrt {a+b x^2}}{128 b^4}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{8 b^{5/2}}+\frac {5 a^2 d f x \sqrt {a+b x^2} (c f+d e)}{8 b^3}+\frac {35 a^2 d^2 f^2 x^3 \sqrt {a+b x^2}}{192 b^3}-\frac {a c e \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (c f+d e)}{b^{3/2}}+\frac {c^2 e^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-\frac {3 a x \sqrt {a+b x^2} \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{8 b^2}-\frac {5 a d f x^3 \sqrt {a+b x^2} (c f+d e)}{12 b^2}-\frac {7 a d^2 f^2 x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {x^3 \sqrt {a+b x^2} \left (c^2 f^2+4 c d e f+d^2 e^2\right )}{4 b}+\frac {c e x \sqrt {a+b x^2} (c f+d e)}{b}+\frac {d f x^5 \sqrt {a+b x^2} (c f+d e)}{3 b}+\frac {d^2 f^2 x^7 \sqrt {a+b x^2}}{8 b}\)

Input:

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

Output:

(-35*a^3*d^2*f^2*x*Sqrt[a + b*x^2])/(128*b^4) + (c*e*(d*e + c*f)*x*Sqrt[a 
+ b*x^2])/b + (5*a^2*d*f*(d*e + c*f)*x*Sqrt[a + b*x^2])/(8*b^3) - (3*a*(d^ 
2*e^2 + 4*c*d*e*f + c^2*f^2)*x*Sqrt[a + b*x^2])/(8*b^2) + (35*a^2*d^2*f^2* 
x^3*Sqrt[a + b*x^2])/(192*b^3) - (5*a*d*f*(d*e + c*f)*x^3*Sqrt[a + b*x^2]) 
/(12*b^2) + ((d^2*e^2 + 4*c*d*e*f + c^2*f^2)*x^3*Sqrt[a + b*x^2])/(4*b) - 
(7*a*d^2*f^2*x^5*Sqrt[a + b*x^2])/(48*b^2) + (d*f*(d*e + c*f)*x^5*Sqrt[a + 
 b*x^2])/(3*b) + (d^2*f^2*x^7*Sqrt[a + b*x^2])/(8*b) + (c^2*e^2*ArcTanh[(S 
qrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] + (35*a^4*d^2*f^2*ArcTanh[(Sqrt[b]*x)/ 
Sqrt[a + b*x^2]])/(128*b^(9/2)) - (a*c*e*(d*e + c*f)*ArcTanh[(Sqrt[b]*x)/S 
qrt[a + b*x^2]])/b^(3/2) - (5*a^3*d*f*(d*e + c*f)*ArcTanh[(Sqrt[b]*x)/Sqrt 
[a + b*x^2]])/(8*b^(7/2)) + (3*a^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*ArcTanh 
[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*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 = 0.72 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {\frac {35 \left (a^{2} \left (a^{2} f^{2}+\frac {48}{35} b^{2} e^{2}-\frac {16}{7} a b f e \right ) d^{2}-\frac {16 \left (a^{2} f^{2}-\frac {12}{5} a b f e +\frac {8}{5} b^{2} e^{2}\right ) a c b d}{7}+\frac {48 c^{2} \left (a^{2} f^{2}-\frac {8}{3} a b f e +\frac {8}{3} b^{2} e^{2}\right ) b^{2}}{35}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{128}-\frac {35 \sqrt {b \,x^{2}+a}\, \left (\left (-\frac {32 \left (\frac {1}{2} f^{2} x^{4}+\frac {4}{3} e f \,x^{2}+e^{2}\right ) x^{2} d^{2}}{35}-\frac {128 \left (\frac {1}{3} f^{2} x^{4}+e f \,x^{2}+e^{2}\right ) c d}{35}-\frac {128 \left (\frac {f \,x^{2}}{4}+e \right ) c^{2} f}{35}\right ) b^{\frac {7}{2}}+a \left (\left (\left (\frac {8}{15} f^{2} x^{4}+\frac {32}{21} e f \,x^{2}+\frac {48}{35} e^{2}\right ) d^{2}+\frac {192 c \left (\frac {5 f \,x^{2}}{18}+e \right ) f d}{35}+\frac {48 c^{2} f^{2}}{35}\right ) b^{\frac {5}{2}}+a d \left (\left (\left (-\frac {2 f \,x^{2}}{3}-\frac {16 e}{7}\right ) d -\frac {16 c f}{7}\right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right ) f \right )\right ) x}{128}}{b^{\frac {9}{2}}}\)	\(281\)
risch	\(-\frac {x \left (-48 b^{3} d^{2} f^{2} x^{6}+56 a \,b^{2} d^{2} f^{2} x^{4}-128 b^{3} c d \,f^{2} x^{4}-128 b^{3} d^{2} e f \,x^{4}-70 a^{2} b \,x^{2} d^{2} f^{2}+160 a \,b^{2} c d \,f^{2} x^{2}+160 a \,b^{2} d^{2} e f \,x^{2}-96 b^{3} c^{2} f^{2} x^{2}-384 b^{3} c e \,x^{2} d f -96 b^{3} d^{2} e^{2} x^{2}+105 a^{3} d^{2} f^{2}-240 a^{2} b c d \,f^{2}-240 a^{2} b \,d^{2} e f +144 a \,b^{2} c^{2} f^{2}+576 a \,b^{2} c d e f +144 a \,b^{2} d^{2} e^{2}-384 b^{3} c^{2} e f -384 b^{3} c d \,e^{2}\right ) \sqrt {b \,x^{2}+a}}{384 b^{4}}+\frac {\left (35 a^{4} d^{2} f^{2}-80 a^{3} b c d \,f^{2}-80 a^{3} b \,d^{2} e f +48 a^{2} b^{2} c^{2} f^{2}+192 a^{2} b^{2} c d e f +48 a^{2} b^{2} d^{2} e^{2}-128 a \,b^{3} c^{2} e f -128 a \,b^{3} c d \,e^{2}+128 b^{4} c^{2} e^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {9}{2}}}\)	\(371\)
default	\(\frac {c^{2} e^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+2 d f \left (c f +d e \right ) \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+2 c e \left (c f +d e \right ) \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+\left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}\right ) \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+d^{2} f^{2} \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )\)	\(375\)

Input:

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

Output:

35/128*((a^2*(a^2*f^2+48/35*b^2*e^2-16/7*a*b*f*e)*d^2-16/7*(a^2*f^2-12/5*a 
*b*f*e+8/5*b^2*e^2)*a*c*b*d+48/35*c^2*(a^2*f^2-8/3*a*b*f*e+8/3*b^2*e^2)*b^ 
2)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))-(b*x^2+a)^(1/2)*((-32/35*(1/2*f^2*x^ 
4+4/3*e*f*x^2+e^2)*x^2*d^2-128/35*(1/3*f^2*x^4+e*f*x^2+e^2)*c*d-128/35*(1/ 
4*f*x^2+e)*c^2*f)*b^(7/2)+a*(((8/15*f^2*x^4+32/21*e*f*x^2+48/35*e^2)*d^2+1 
92/35*c*(5/18*f*x^2+e)*f*d+48/35*c^2*f^2)*b^(5/2)+a*d*(((-2/3*f*x^2-16/7*e 
)*d-16/7*c*f)*b^(3/2)+a*d*f*b^(1/2))*f))*x)/b^(9/2)
 

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.13 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/768*(3*(16*(8*b^4*c^2 - 8*a*b^3*c*d + 3*a^2*b^2*d^2)*e^2 - 16*(8*a*b^3* 
c^2 - 12*a^2*b^2*c*d + 5*a^3*b*d^2)*e*f + (48*a^2*b^2*c^2 - 80*a^3*b*c*d + 
 35*a^4*d^2)*f^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) 
+ 2*(48*b^4*d^2*f^2*x^7 + 8*(16*b^4*d^2*e*f + (16*b^4*c*d - 7*a*b^3*d^2)*f 
^2)*x^5 + 2*(48*b^4*d^2*e^2 + 16*(12*b^4*c*d - 5*a*b^3*d^2)*e*f + (48*b^4* 
c^2 - 80*a*b^3*c*d + 35*a^2*b^2*d^2)*f^2)*x^3 + 3*(16*(8*b^4*c*d - 3*a*b^3 
*d^2)*e^2 + 16*(8*b^4*c^2 - 12*a*b^3*c*d + 5*a^2*b^2*d^2)*e*f - (48*a*b^3* 
c^2 - 80*a^2*b^2*c*d + 35*a^3*b*d^2)*f^2)*x)*sqrt(b*x^2 + a))/b^5, -1/384* 
(3*(16*(8*b^4*c^2 - 8*a*b^3*c*d + 3*a^2*b^2*d^2)*e^2 - 16*(8*a*b^3*c^2 - 1 
2*a^2*b^2*c*d + 5*a^3*b*d^2)*e*f + (48*a^2*b^2*c^2 - 80*a^3*b*c*d + 35*a^4 
*d^2)*f^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (48*b^4*d^2*f^2*x 
^7 + 8*(16*b^4*d^2*e*f + (16*b^4*c*d - 7*a*b^3*d^2)*f^2)*x^5 + 2*(48*b^4*d 
^2*e^2 + 16*(12*b^4*c*d - 5*a*b^3*d^2)*e*f + (48*b^4*c^2 - 80*a*b^3*c*d + 
35*a^2*b^2*d^2)*f^2)*x^3 + 3*(16*(8*b^4*c*d - 3*a*b^3*d^2)*e^2 + 16*(8*b^4 
*c^2 - 12*a*b^3*c*d + 5*a^2*b^2*d^2)*e*f - (48*a*b^3*c^2 - 80*a^2*b^2*c*d 
+ 35*a^3*b*d^2)*f^2)*x)*sqrt(b*x^2 + a))/b^5]
 

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.34 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d^{2} f^{2} x^{7}}{8 b} + \frac {x^{5} \left (- \frac {7 a d^{2} f^{2}}{8 b} + 2 c d f^{2} + 2 d^{2} e f\right )}{6 b} + \frac {x^{3} \left (- \frac {5 a \left (- \frac {7 a d^{2} f^{2}}{8 b} + 2 c d f^{2} + 2 d^{2} e f\right )}{6 b} + c^{2} f^{2} + 4 c d e f + d^{2} e^{2}\right )}{4 b} + \frac {x \left (- \frac {3 a \left (- \frac {5 a \left (- \frac {7 a d^{2} f^{2}}{8 b} + 2 c d f^{2} + 2 d^{2} e f\right )}{6 b} + c^{2} f^{2} + 4 c d e f + d^{2} e^{2}\right )}{4 b} + 2 c^{2} e f + 2 c d e^{2}\right )}{2 b}\right ) + \left (- \frac {a \left (- \frac {3 a \left (- \frac {5 a \left (- \frac {7 a d^{2} f^{2}}{8 b} + 2 c d f^{2} + 2 d^{2} e f\right )}{6 b} + c^{2} f^{2} + 4 c d e f + d^{2} e^{2}\right )}{4 b} + 2 c^{2} e f + 2 c d e^{2}\right )}{2 b} + c^{2} e^{2}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\frac {c^{2} e^{2} x + \frac {d^{2} f^{2} x^{9}}{9} + \frac {x^{7} \cdot \left (2 c d f^{2} + 2 d^{2} e f\right )}{7} + \frac {x^{5} \left (c^{2} f^{2} + 4 c d e f + d^{2} e^{2}\right )}{5} + \frac {x^{3} \cdot \left (2 c^{2} e f + 2 c d e^{2}\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(a + b*x**2)*(d**2*f**2*x**7/(8*b) + x**5*(-7*a*d**2*f**2/( 
8*b) + 2*c*d*f**2 + 2*d**2*e*f)/(6*b) + x**3*(-5*a*(-7*a*d**2*f**2/(8*b) + 
 2*c*d*f**2 + 2*d**2*e*f)/(6*b) + c**2*f**2 + 4*c*d*e*f + d**2*e**2)/(4*b) 
 + x*(-3*a*(-5*a*(-7*a*d**2*f**2/(8*b) + 2*c*d*f**2 + 2*d**2*e*f)/(6*b) + 
c**2*f**2 + 4*c*d*e*f + d**2*e**2)/(4*b) + 2*c**2*e*f + 2*c*d*e**2)/(2*b)) 
 + (-a*(-3*a*(-5*a*(-7*a*d**2*f**2/(8*b) + 2*c*d*f**2 + 2*d**2*e*f)/(6*b) 
+ c**2*f**2 + 4*c*d*e*f + d**2*e**2)/(4*b) + 2*c**2*e*f + 2*c*d*e**2)/(2*b 
) + c**2*e**2)*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), 
 Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True)), Ne(b, 0)), ((c**2*e**2*x + d** 
2*f**2*x**9/9 + x**7*(2*c*d*f**2 + 2*d**2*e*f)/7 + x**5*(c**2*f**2 + 4*c*d 
*e*f + d**2*e**2)/5 + x**3*(2*c**2*e*f + 2*c*d*e**2)/3)/sqrt(a), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.28 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} d^{2} f^{2} x^{7}}{8 \, b} - \frac {7 \, \sqrt {b x^{2} + a} a d^{2} f^{2} x^{5}}{48 \, b^{2}} + \frac {35 \, \sqrt {b x^{2} + a} a^{2} d^{2} f^{2} x^{3}}{192 \, b^{3}} + \frac {{\left (d^{2} e f + c d f^{2}\right )} \sqrt {b x^{2} + a} x^{5}}{3 \, b} - \frac {35 \, \sqrt {b x^{2} + a} a^{3} d^{2} f^{2} x}{128 \, b^{4}} + \frac {c^{2} e^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} + \frac {35 \, a^{4} d^{2} f^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {9}{2}}} - \frac {5 \, {\left (d^{2} e f + c d f^{2}\right )} \sqrt {b x^{2} + a} a x^{3}}{12 \, b^{2}} + \frac {{\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} \sqrt {b x^{2} + a} x^{3}}{4 \, b} + \frac {5 \, {\left (d^{2} e f + c d f^{2}\right )} \sqrt {b x^{2} + a} a^{2} x}{8 \, b^{3}} - \frac {3 \, {\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} \sqrt {b x^{2} + a} a x}{8 \, b^{2}} + \frac {{\left (c d e^{2} + c^{2} e f\right )} \sqrt {b x^{2} + a} x}{b} - \frac {5 \, {\left (d^{2} e f + c d f^{2}\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {7}{2}}} + \frac {3 \, {\left (d^{2} e^{2} + 4 \, c d e f + c^{2} f^{2}\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {{\left (c d e^{2} + c^{2} e f\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} \] Input:

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

Output:

1/8*sqrt(b*x^2 + a)*d^2*f^2*x^7/b - 7/48*sqrt(b*x^2 + a)*a*d^2*f^2*x^5/b^2 
 + 35/192*sqrt(b*x^2 + a)*a^2*d^2*f^2*x^3/b^3 + 1/3*(d^2*e*f + c*d*f^2)*sq 
rt(b*x^2 + a)*x^5/b - 35/128*sqrt(b*x^2 + a)*a^3*d^2*f^2*x/b^4 + c^2*e^2*a 
rcsinh(b*x/sqrt(a*b))/sqrt(b) + 35/128*a^4*d^2*f^2*arcsinh(b*x/sqrt(a*b))/ 
b^(9/2) - 5/12*(d^2*e*f + c*d*f^2)*sqrt(b*x^2 + a)*a*x^3/b^2 + 1/4*(d^2*e^ 
2 + 4*c*d*e*f + c^2*f^2)*sqrt(b*x^2 + a)*x^3/b + 5/8*(d^2*e*f + c*d*f^2)*s 
qrt(b*x^2 + a)*a^2*x/b^3 - 3/8*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)*sqrt(b*x^2 
+ a)*a*x/b^2 + (c*d*e^2 + c^2*e*f)*sqrt(b*x^2 + a)*x/b - 5/8*(d^2*e*f + c* 
d*f^2)*a^3*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 3/8*(d^2*e^2 + 4*c*d*e*f + c^2 
*f^2)*a^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) - (c*d*e^2 + c^2*e*f)*a*arcsinh(b 
*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, d^{2} f^{2} x^{2}}{b} + \frac {16 \, b^{6} d^{2} e f + 16 \, b^{6} c d f^{2} - 7 \, a b^{5} d^{2} f^{2}}{b^{7}}\right )} x^{2} + \frac {48 \, b^{6} d^{2} e^{2} + 192 \, b^{6} c d e f - 80 \, a b^{5} d^{2} e f + 48 \, b^{6} c^{2} f^{2} - 80 \, a b^{5} c d f^{2} + 35 \, a^{2} b^{4} d^{2} f^{2}}{b^{7}}\right )} x^{2} + \frac {3 \, {\left (128 \, b^{6} c d e^{2} - 48 \, a b^{5} d^{2} e^{2} + 128 \, b^{6} c^{2} e f - 192 \, a b^{5} c d e f + 80 \, a^{2} b^{4} d^{2} e f - 48 \, a b^{5} c^{2} f^{2} + 80 \, a^{2} b^{4} c d f^{2} - 35 \, a^{3} b^{3} d^{2} f^{2}\right )}}{b^{7}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (128 \, b^{4} c^{2} e^{2} - 128 \, a b^{3} c d e^{2} + 48 \, a^{2} b^{2} d^{2} e^{2} - 128 \, a b^{3} c^{2} e f + 192 \, a^{2} b^{2} c d e f - 80 \, a^{3} b d^{2} e f + 48 \, a^{2} b^{2} c^{2} f^{2} - 80 \, a^{3} b c d f^{2} + 35 \, a^{4} d^{2} f^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {9}{2}}} \] Input:

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

Output:

1/384*(2*(4*(6*d^2*f^2*x^2/b + (16*b^6*d^2*e*f + 16*b^6*c*d*f^2 - 7*a*b^5* 
d^2*f^2)/b^7)*x^2 + (48*b^6*d^2*e^2 + 192*b^6*c*d*e*f - 80*a*b^5*d^2*e*f + 
 48*b^6*c^2*f^2 - 80*a*b^5*c*d*f^2 + 35*a^2*b^4*d^2*f^2)/b^7)*x^2 + 3*(128 
*b^6*c*d*e^2 - 48*a*b^5*d^2*e^2 + 128*b^6*c^2*e*f - 192*a*b^5*c*d*e*f + 80 
*a^2*b^4*d^2*e*f - 48*a*b^5*c^2*f^2 + 80*a^2*b^4*c*d*f^2 - 35*a^3*b^3*d^2* 
f^2)/b^7)*sqrt(b*x^2 + a)*x - 1/128*(128*b^4*c^2*e^2 - 128*a*b^3*c*d*e^2 + 
 48*a^2*b^2*d^2*e^2 - 128*a*b^3*c^2*e*f + 192*a^2*b^2*c*d*e*f - 80*a^3*b*d 
^2*e*f + 48*a^2*b^2*c^2*f^2 - 80*a^3*b*c*d*f^2 + 35*a^4*d^2*f^2)*log(abs(- 
sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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