\(\int \frac {(c+d x^2)^2 (e+f x^2)^3}{\sqrt {a+b x^2}} \, dx\) [323]

Optimal result

Integrand size = 30, antiderivative size = 533 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {\left (63 a^4 d^2 f^3-70 a^3 b d f^2 (3 d e+2 c f)+128 b^4 c e^2 (2 d e+3 c f)+80 a^2 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-96 a b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{256 b^5}-\frac {\left (63 a^3 d^2 f^3-70 a^2 b d f^2 (3 d e+2 c f)+80 a b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )-96 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {f \left (63 a^2 d^2 f^2-70 a b d f (3 d e+2 c f)+80 b^2 \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {d f^2 (30 b d e+20 b c f-9 a d f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {d^2 f^3 x^9 \sqrt {a+b x^2}}{10 b}+\frac {\left (256 b^5 c^2 e^3-63 a^5 d^2 f^3+70 a^4 b d f^2 (3 d e+2 c f)-128 a b^4 c e^2 (2 d e+3 c f)-80 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+96 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{11/2}} \] Output:

1/256*(63*a^4*d^2*f^3-70*a^3*b*d*f^2*(2*c*f+3*d*e)+128*b^4*c*e^2*(3*c*f+2* 
d*e)+80*a^2*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)-96*a*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^5-1/384*(63*a^3*d^2*f^3-70*a^2*b*d*f^2 
*(2*c*f+3*d*e)+80*a*b^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)-96*b^3*e*(3*c^2*f^ 
2+6*c*d*e*f+d^2*e^2))*x^3*(b*x^2+a)^(1/2)/b^4+1/480*f*(63*a^2*d^2*f^2-70*a 
*b*d*f*(2*c*f+3*d*e)+80*b^2*(c^2*f^2+6*c*d*e*f+3*d^2*e^2))*x^5*(b*x^2+a)^( 
1/2)/b^3+1/80*d*f^2*(-9*a*d*f+20*b*c*f+30*b*d*e)*x^7*(b*x^2+a)^(1/2)/b^2+1 
/10*d^2*f^3*x^9*(b*x^2+a)^(1/2)/b+1/256*(256*b^5*c^2*e^3-63*a^5*d^2*f^3+70 
*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)-80*a^3*b^2*f*(c^2 
*f^2+6*c*d*e*f+3*d^2*e^2)+96*a^2*b^3*e*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*arct 
anh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 468, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (945 a^4 d^2 f^3-210 a^3 b d f^2 \left (10 c f+3 d \left (5 e+f x^2\right )\right )+4 a^2 b^2 f \left (300 c^2 f^2+50 c d f \left (36 e+7 f x^2\right )+3 d^2 \left (300 e^2+175 e f x^2+42 f^2 x^4\right )\right )+32 b^4 \left (10 c^2 f \left (18 e^2+9 e f x^2+2 f^2 x^4\right )+30 c d \left (4 e^3+6 e^2 f x^2+4 e f^2 x^4+f^3 x^6\right )+3 d^2 x^2 \left (10 e^3+20 e^2 f x^2+15 e f^2 x^4+4 f^3 x^6\right )\right )-16 a b^3 \left (10 c^2 f^2 \left (27 e+5 f x^2\right )+10 c d f \left (54 e^2+30 e f x^2+7 f^2 x^4\right )+3 d^2 \left (30 e^3+50 e^2 f x^2+35 e f^2 x^4+9 f^3 x^6\right )\right )\right )-15 \left (256 b^5 c^2 e^3-63 a^5 d^2 f^3+70 a^4 b d f^2 (3 d e+2 c f)-128 a b^4 c e^2 (2 d e+3 c f)-80 a^3 b^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+96 a^2 b^3 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{3840 b^{11/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(945*a^4*d^2*f^3 - 210*a^3*b*d*f^2*(10*c*f + 3* 
d*(5*e + f*x^2)) + 4*a^2*b^2*f*(300*c^2*f^2 + 50*c*d*f*(36*e + 7*f*x^2) + 
3*d^2*(300*e^2 + 175*e*f*x^2 + 42*f^2*x^4)) + 32*b^4*(10*c^2*f*(18*e^2 + 9 
*e*f*x^2 + 2*f^2*x^4) + 30*c*d*(4*e^3 + 6*e^2*f*x^2 + 4*e*f^2*x^4 + f^3*x^ 
6) + 3*d^2*x^2*(10*e^3 + 20*e^2*f*x^2 + 15*e*f^2*x^4 + 4*f^3*x^6)) - 16*a* 
b^3*(10*c^2*f^2*(27*e + 5*f*x^2) + 10*c*d*f*(54*e^2 + 30*e*f*x^2 + 7*f^2*x 
^4) + 3*d^2*(30*e^3 + 50*e^2*f*x^2 + 35*e*f^2*x^4 + 9*f^3*x^6))) - 15*(256 
*b^5*c^2*e^3 - 63*a^5*d^2*f^3 + 70*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) - 80*a^3*b^2*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + 
96*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]])/(3840*b^(11/2))
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 800, normalized size of antiderivative = 1.50, 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.

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

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 f^3 \sqrt {b x^2+a} x^9}{10 b}-\frac {9 a d^2 f^3 \sqrt {b x^2+a} x^7}{80 b^2}+\frac {d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^7}{8 b}+\frac {21 a^2 d^2 f^3 \sqrt {b x^2+a} x^5}{160 b^3}-\frac {7 a d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^5}{48 b^2}+\frac {f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \sqrt {b x^2+a} x^5}{6 b}-\frac {21 a^3 d^2 f^3 \sqrt {b x^2+a} x^3}{128 b^4}+\frac {35 a^2 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x^3}{192 b^3}-\frac {5 a f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \sqrt {b x^2+a} x^3}{24 b^2}+\frac {e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \sqrt {b x^2+a} x^3}{4 b}+\frac {63 a^4 d^2 f^3 \sqrt {b x^2+a} x}{256 b^5}-\frac {35 a^3 d f^2 (3 d e+2 c f) \sqrt {b x^2+a} x}{128 b^4}+\frac {c e^2 (2 d e+3 c f) \sqrt {b x^2+a} x}{2 b}+\frac {5 a^2 f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \sqrt {b x^2+a} x}{16 b^3}-\frac {3 a e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \sqrt {b x^2+a} x}{8 b^2}+\frac {c^2 e^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b}}-\frac {63 a^5 d^2 f^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{256 b^{11/2}}+\frac {35 a^4 d f^2 (3 d e+2 c f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{128 b^{9/2}}-\frac {a c e^2 (2 d e+3 c f) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{2 b^{3/2}}-\frac {5 a^3 f \left (3 d^2 e^2+6 c d f e+c^2 f^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{16 b^{7/2}}+\frac {3 a^2 e \left (d^2 e^2+6 c d f e+3 c^2 f^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{8 b^{5/2}}\)

Input:

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

Output:

(63*a^4*d^2*f^3*x*Sqrt[a + b*x^2])/(256*b^5) - (35*a^3*d*f^2*(3*d*e + 2*c* 
f)*x*Sqrt[a + b*x^2])/(128*b^4) + (c*e^2*(2*d*e + 3*c*f)*x*Sqrt[a + b*x^2] 
)/(2*b) + (5*a^2*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x*Sqrt[a + b*x^2])/(1 
6*b^3) - (3*a*e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)*x*Sqrt[a + b*x^2])/(8*b^ 
2) - (21*a^3*d^2*f^3*x^3*Sqrt[a + b*x^2])/(128*b^4) + (35*a^2*d*f^2*(3*d*e 
 + 2*c*f)*x^3*Sqrt[a + b*x^2])/(192*b^3) - (5*a*f*(3*d^2*e^2 + 6*c*d*e*f + 
 c^2*f^2)*x^3*Sqrt[a + b*x^2])/(24*b^2) + (e*(d^2*e^2 + 6*c*d*e*f + 3*c^2* 
f^2)*x^3*Sqrt[a + b*x^2])/(4*b) + (21*a^2*d^2*f^3*x^5*Sqrt[a + b*x^2])/(16 
0*b^3) - (7*a*d*f^2*(3*d*e + 2*c*f)*x^5*Sqrt[a + b*x^2])/(48*b^2) + (f*(3* 
d^2*e^2 + 6*c*d*e*f + c^2*f^2)*x^5*Sqrt[a + b*x^2])/(6*b) - (9*a*d^2*f^3*x 
^7*Sqrt[a + b*x^2])/(80*b^2) + (d*f^2*(3*d*e + 2*c*f)*x^7*Sqrt[a + b*x^2]) 
/(8*b) + (d^2*f^3*x^9*Sqrt[a + b*x^2])/(10*b) + (c^2*e^3*ArcTanh[(Sqrt[b]* 
x)/Sqrt[a + b*x^2]])/Sqrt[b] - (63*a^5*d^2*f^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a 
+ b*x^2]])/(256*b^(11/2)) + (35*a^4*d*f^2*(3*d*e + 2*c*f)*ArcTanh[(Sqrt[b] 
*x)/Sqrt[a + b*x^2]])/(128*b^(9/2)) - (a*c*e^2*(2*d*e + 3*c*f)*ArcTanh[(Sq 
rt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2)) - (5*a^3*f*(3*d^2*e^2 + 6*c*d*e*f + 
 c^2*f^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2)) + (3*a^2*e*(d 
^2*e^2 + 6*c*d*e*f + 3*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 = 0.87 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(-\frac {63 \left (\left (a^{3} \left (a^{2} d^{2}+\frac {80}{63} b^{2} c^{2}-\frac {20}{9} a b c d \right ) f^{3}-\frac {10 a^{2} b \left (a^{2} d^{2}-\frac {16}{7} a b c d +\frac {48}{35} b^{2} c^{2}\right ) e \,f^{2}}{3}+\frac {80 a \,b^{2} e^{2} \left (a^{2} d^{2}-\frac {12}{5} a b c d +\frac {8}{5} b^{2} c^{2}\right ) f}{21}-\frac {32 \left (a^{2} d^{2}-\frac {8}{3} a b c d +\frac {8}{3} b^{2} c^{2}\right ) b^{3} e^{3}}{21}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\sqrt {b \,x^{2}+a}\, \left (\frac {64 \left (\frac {2 x^{4} \left (\frac {3}{5} d^{2} x^{4}+\frac {3}{2} c d \,x^{2}+c^{2}\right ) f^{3}}{9}+\left (\frac {1}{2} d^{2} x^{4}+\frac {4}{3} c d \,x^{2}+c^{2}\right ) x^{2} e \,f^{2}+2 e^{2} \left (\frac {1}{3} d^{2} x^{4}+c d \,x^{2}+c^{2}\right ) f +\frac {4 d \left (\frac {x^{2} d}{4}+c \right ) e^{3}}{3}\right ) b^{\frac {9}{2}}}{21}+a \left (\frac {32 \left (-\frac {5 \left (\frac {27}{50} d^{2} x^{4}+\frac {7}{5} c d \,x^{2}+c^{2}\right ) x^{2} f^{3}}{27}-\left (\frac {7}{18} d^{2} x^{4}+\frac {10}{9} c d \,x^{2}+c^{2}\right ) e \,f^{2}-2 d \left (\frac {5 x^{2} d}{18}+c \right ) e^{2} f -\frac {d^{2} e^{3}}{3}\right ) b^{\frac {7}{2}}}{7}+a f \left (\frac {8 \left (\left (\frac {5}{9} c d \,x^{2}+\frac {10}{21} c^{2}+\frac {1}{5} d^{2} x^{4}\right ) f^{2}+\frac {20 \left (\frac {7 x^{2} d}{24}+c \right ) d e f}{7}+\frac {10 d^{2} e^{2}}{7}\right ) b^{\frac {5}{2}}}{3}+\left (\frac {2 \left (\left (-\frac {10 c}{3}-x^{2} d \right ) f -5 d e \right ) b^{\frac {3}{2}}}{3}+a d f \sqrt {b}\right ) a d f \right )\right )\right ) x \right )}{256 b^{\frac {11}{2}}}\)	\(434\)
default	\(\frac {c^{2} e^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+f^{2} d \left (2 c f +3 d e \right ) \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )+c \,e^{2} \left (3 c f +2 d e \right ) \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+f \left (c^{2} f^{2}+6 c d e f +3 d^{2} e^{2}\right ) \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+e \left (3 c^{2} f^{2}+6 c d e f +d^{2} e^{2}\right ) \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+d^{2} f^{3} \left (\frac {x^{9} \sqrt {b \,x^{2}+a}}{10 b}-\frac {9 a \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )}{10 b}\right )\)	\(541\)
risch	\(\frac {x \left (384 d^{2} f^{3} b^{4} x^{8}-432 a \,b^{3} d^{2} f^{3} x^{6}+960 b^{4} c d \,f^{3} x^{6}+1440 b^{4} d^{2} e \,f^{2} x^{6}+504 a^{2} b^{2} d^{2} f^{3} x^{4}-1120 a \,b^{3} c d \,f^{3} x^{4}-1680 a \,b^{3} d^{2} e \,f^{2} x^{4}+640 b^{4} c^{2} f^{3} x^{4}+3840 b^{4} c d e \,f^{2} x^{4}+1920 b^{4} d^{2} e^{2} f \,x^{4}-630 a^{3} d^{2} f^{3} b \,x^{2}+1400 a^{2} b^{2} c d \,f^{3} x^{2}+2100 a^{2} b^{2} d^{2} e \,f^{2} x^{2}-800 b^{3} a \,c^{2} f^{3} x^{2}-4800 b^{3} a c d e \,f^{2} x^{2}-2400 b^{3} a \,d^{2} e^{2} f \,x^{2}+2880 b^{4} c^{2} e \,f^{2} x^{2}+5760 b^{4} c d \,e^{2} f \,x^{2}+960 b^{4} d^{2} e^{3} x^{2}+945 a^{4} d^{2} f^{3}-2100 a^{3} b c d \,f^{3}-3150 a^{3} b \,d^{2} e \,f^{2}+1200 a^{2} b^{2} c^{2} f^{3}+7200 a^{2} b^{2} c d e \,f^{2}+3600 a^{2} b^{2} d^{2} e^{2} f -4320 a \,b^{3} c^{2} e \,f^{2}-8640 a \,b^{3} c d \,e^{2} f -1440 a \,b^{3} d^{2} e^{3}+5760 b^{4} c^{2} e^{2} f +3840 b^{4} c d \,e^{3}\right ) \sqrt {b \,x^{2}+a}}{3840 b^{5}}-\frac {\left (63 a^{5} d^{2} f^{3}-140 a^{4} b c d \,f^{3}-210 a^{4} b \,d^{2} e \,f^{2}+80 a^{3} b^{2} c^{2} f^{3}+480 a^{3} b^{2} c d e \,f^{2}+240 a^{3} b^{2} d^{2} e^{2} f -288 a^{2} b^{3} c^{2} e \,f^{2}-576 a^{2} b^{3} c d \,e^{2} f -96 a^{2} b^{3} d^{2} e^{3}+384 a \,b^{4} c^{2} e^{2} f +256 a \,b^{4} c d \,e^{3}-256 b^{5} c^{2} e^{3}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {11}{2}}}\)	\(618\)

Input:

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

Output:

-63/256/b^(11/2)*((a^3*(a^2*d^2+80/63*b^2*c^2-20/9*a*b*c*d)*f^3-10/3*a^2*b 
*(a^2*d^2-16/7*a*b*c*d+48/35*b^2*c^2)*e*f^2+80/21*a*b^2*e^2*(a^2*d^2-12/5* 
a*b*c*d+8/5*b^2*c^2)*f-32/21*(a^2*d^2-8/3*a*b*c*d+8/3*b^2*c^2)*b^3*e^3)*ar 
ctanh((b*x^2+a)^(1/2)/x/b^(1/2))-(b*x^2+a)^(1/2)*(64/21*(2/9*x^4*(3/5*d^2* 
x^4+3/2*c*d*x^2+c^2)*f^3+(1/2*d^2*x^4+4/3*c*d*x^2+c^2)*x^2*e*f^2+2*e^2*(1/ 
3*d^2*x^4+c*d*x^2+c^2)*f+4/3*d*(1/4*x^2*d+c)*e^3)*b^(9/2)+a*(32/7*(-5/27*( 
27/50*d^2*x^4+7/5*c*d*x^2+c^2)*x^2*f^3-(7/18*d^2*x^4+10/9*c*d*x^2+c^2)*e*f 
^2-2*d*(5/18*x^2*d+c)*e^2*f-1/3*d^2*e^3)*b^(7/2)+a*f*(8/3*((5/9*c*d*x^2+10 
/21*c^2+1/5*d^2*x^4)*f^2+20/7*(7/24*x^2*d+c)*d*e*f+10/7*d^2*e^2)*b^(5/2)+( 
2/3*((-10/3*c-x^2*d)*f-5*d*e)*b^(3/2)+a*d*f*b^(1/2))*a*d*f)))*x)
 

Fricas [A] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 1110, normalized size of antiderivative = 2.08 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/7680*(15*(32*(8*b^5*c^2 - 8*a*b^4*c*d + 3*a^2*b^3*d^2)*e^3 - 48*(8*a*b 
^4*c^2 - 12*a^2*b^3*c*d + 5*a^3*b^2*d^2)*e^2*f + 6*(48*a^2*b^3*c^2 - 80*a^ 
3*b^2*c*d + 35*a^4*b*d^2)*e*f^2 - (80*a^3*b^2*c^2 - 140*a^4*b*c*d + 63*a^5 
*d^2)*f^3)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(38 
4*b^5*d^2*f^3*x^9 + 48*(30*b^5*d^2*e*f^2 + (20*b^5*c*d - 9*a*b^4*d^2)*f^3) 
*x^7 + 8*(240*b^5*d^2*e^2*f + 30*(16*b^5*c*d - 7*a*b^4*d^2)*e*f^2 + (80*b^ 
5*c^2 - 140*a*b^4*c*d + 63*a^2*b^3*d^2)*f^3)*x^5 + 10*(96*b^5*d^2*e^3 + 48 
*(12*b^5*c*d - 5*a*b^4*d^2)*e^2*f + 6*(48*b^5*c^2 - 80*a*b^4*c*d + 35*a^2* 
b^3*d^2)*e*f^2 - (80*a*b^4*c^2 - 140*a^2*b^3*c*d + 63*a^3*b^2*d^2)*f^3)*x^ 
3 + 15*(32*(8*b^5*c*d - 3*a*b^4*d^2)*e^3 + 48*(8*b^5*c^2 - 12*a*b^4*c*d + 
5*a^2*b^3*d^2)*e^2*f - 6*(48*a*b^4*c^2 - 80*a^2*b^3*c*d + 35*a^3*b^2*d^2)* 
e*f^2 + (80*a^2*b^3*c^2 - 140*a^3*b^2*c*d + 63*a^4*b*d^2)*f^3)*x)*sqrt(b*x 
^2 + a))/b^6, -1/3840*(15*(32*(8*b^5*c^2 - 8*a*b^4*c*d + 3*a^2*b^3*d^2)*e^ 
3 - 48*(8*a*b^4*c^2 - 12*a^2*b^3*c*d + 5*a^3*b^2*d^2)*e^2*f + 6*(48*a^2*b^ 
3*c^2 - 80*a^3*b^2*c*d + 35*a^4*b*d^2)*e*f^2 - (80*a^3*b^2*c^2 - 140*a^4*b 
*c*d + 63*a^5*d^2)*f^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (384 
*b^5*d^2*f^3*x^9 + 48*(30*b^5*d^2*e*f^2 + (20*b^5*c*d - 9*a*b^4*d^2)*f^3)* 
x^7 + 8*(240*b^5*d^2*e^2*f + 30*(16*b^5*c*d - 7*a*b^4*d^2)*e*f^2 + (80*b^5 
*c^2 - 140*a*b^4*c*d + 63*a^2*b^3*d^2)*f^3)*x^5 + 10*(96*b^5*d^2*e^3 + 48* 
(12*b^5*c*d - 5*a*b^4*d^2)*e^2*f + 6*(48*b^5*c^2 - 80*a*b^4*c*d + 35*a^...
 

Sympy [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 690, normalized size of antiderivative = 1.29 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d^{2} f^{3} x^{9}}{10 b} + \frac {x^{7} \left (- \frac {9 a d^{2} f^{3}}{10 b} + 2 c d f^{3} + 3 d^{2} e f^{2}\right )}{8 b} + \frac {x^{5} \left (- \frac {7 a \left (- \frac {9 a d^{2} f^{3}}{10 b} + 2 c d f^{3} + 3 d^{2} e f^{2}\right )}{8 b} + c^{2} f^{3} + 6 c d e f^{2} + 3 d^{2} e^{2} f\right )}{6 b} + \frac {x^{3} \left (- \frac {5 a \left (- \frac {7 a \left (- \frac {9 a d^{2} f^{3}}{10 b} + 2 c d f^{3} + 3 d^{2} e f^{2}\right )}{8 b} + c^{2} f^{3} + 6 c d e f^{2} + 3 d^{2} e^{2} f\right )}{6 b} + 3 c^{2} e f^{2} + 6 c d e^{2} f + d^{2} e^{3}\right )}{4 b} + \frac {x \left (- \frac {3 a \left (- \frac {5 a \left (- \frac {7 a \left (- \frac {9 a d^{2} f^{3}}{10 b} + 2 c d f^{3} + 3 d^{2} e f^{2}\right )}{8 b} + c^{2} f^{3} + 6 c d e f^{2} + 3 d^{2} e^{2} f\right )}{6 b} + 3 c^{2} e f^{2} + 6 c d e^{2} f + d^{2} e^{3}\right )}{4 b} + 3 c^{2} e^{2} f + 2 c d e^{3}\right )}{2 b}\right ) + \left (- \frac {a \left (- \frac {3 a \left (- \frac {5 a \left (- \frac {7 a \left (- \frac {9 a d^{2} f^{3}}{10 b} + 2 c d f^{3} + 3 d^{2} e f^{2}\right )}{8 b} + c^{2} f^{3} + 6 c d e f^{2} + 3 d^{2} e^{2} f\right )}{6 b} + 3 c^{2} e f^{2} + 6 c d e^{2} f + d^{2} e^{3}\right )}{4 b} + 3 c^{2} e^{2} f + 2 c d e^{3}\right )}{2 b} + c^{2} e^{3}\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^{3} x + \frac {d^{2} f^{3} x^{11}}{11} + \frac {x^{9} \cdot \left (2 c d f^{3} + 3 d^{2} e f^{2}\right )}{9} + \frac {x^{7} \left (c^{2} f^{3} + 6 c d e f^{2} + 3 d^{2} e^{2} f\right )}{7} + \frac {x^{5} \cdot \left (3 c^{2} e f^{2} + 6 c d e^{2} f + d^{2} e^{3}\right )}{5} + \frac {x^{3} \cdot \left (3 c^{2} e^{2} f + 2 c d e^{3}\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(a + b*x**2)*(d**2*f**3*x**9/(10*b) + x**7*(-9*a*d**2*f**3/ 
(10*b) + 2*c*d*f**3 + 3*d**2*e*f**2)/(8*b) + x**5*(-7*a*(-9*a*d**2*f**3/(1 
0*b) + 2*c*d*f**3 + 3*d**2*e*f**2)/(8*b) + c**2*f**3 + 6*c*d*e*f**2 + 3*d* 
*2*e**2*f)/(6*b) + x**3*(-5*a*(-7*a*(-9*a*d**2*f**3/(10*b) + 2*c*d*f**3 + 
3*d**2*e*f**2)/(8*b) + c**2*f**3 + 6*c*d*e*f**2 + 3*d**2*e**2*f)/(6*b) + 3 
*c**2*e*f**2 + 6*c*d*e**2*f + d**2*e**3)/(4*b) + x*(-3*a*(-5*a*(-7*a*(-9*a 
*d**2*f**3/(10*b) + 2*c*d*f**3 + 3*d**2*e*f**2)/(8*b) + c**2*f**3 + 6*c*d* 
e*f**2 + 3*d**2*e**2*f)/(6*b) + 3*c**2*e*f**2 + 6*c*d*e**2*f + d**2*e**3)/ 
(4*b) + 3*c**2*e**2*f + 2*c*d*e**3)/(2*b)) + (-a*(-3*a*(-5*a*(-7*a*(-9*a*d 
**2*f**3/(10*b) + 2*c*d*f**3 + 3*d**2*e*f**2)/(8*b) + c**2*f**3 + 6*c*d*e* 
f**2 + 3*d**2*e**2*f)/(6*b) + 3*c**2*e*f**2 + 6*c*d*e**2*f + d**2*e**3)/(4 
*b) + 3*c**2*e**2*f + 2*c*d*e**3)/(2*b) + c**2*e**3)*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**3*x + d**2*f**3*x**11/11 + x**9*(2*c*d*f**3 + 
 3*d**2*e*f**2)/9 + x**7*(c**2*f**3 + 6*c*d*e*f**2 + 3*d**2*e**2*f)/7 + x* 
*5*(3*c**2*e*f**2 + 6*c*d*e**2*f + d**2*e**3)/5 + x**3*(3*c**2*e**2*f + 2* 
c*d*e**3)/3)/sqrt(a), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.31 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/10*sqrt(b*x^2 + a)*d^2*f^3*x^9/b - 9/80*sqrt(b*x^2 + a)*a*d^2*f^3*x^7/b^ 
2 + 21/160*sqrt(b*x^2 + a)*a^2*d^2*f^3*x^5/b^3 - 21/128*sqrt(b*x^2 + a)*a^ 
3*d^2*f^3*x^3/b^4 + 1/8*(3*d^2*e*f^2 + 2*c*d*f^3)*sqrt(b*x^2 + a)*x^7/b + 
63/256*sqrt(b*x^2 + a)*a^4*d^2*f^3*x/b^5 - 7/48*(3*d^2*e*f^2 + 2*c*d*f^3)* 
sqrt(b*x^2 + a)*a*x^5/b^2 + 1/6*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*sqrt 
(b*x^2 + a)*x^5/b + c^2*e^3*arcsinh(b*x/sqrt(a*b))/sqrt(b) - 63/256*a^5*d^ 
2*f^3*arcsinh(b*x/sqrt(a*b))/b^(11/2) + 35/192*(3*d^2*e*f^2 + 2*c*d*f^3)*s 
qrt(b*x^2 + a)*a^2*x^3/b^3 - 5/24*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*sq 
rt(b*x^2 + a)*a*x^3/b^2 + 1/4*(d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*sqrt(b 
*x^2 + a)*x^3/b - 35/128*(3*d^2*e*f^2 + 2*c*d*f^3)*sqrt(b*x^2 + a)*a^3*x/b 
^4 + 5/16*(3*d^2*e^2*f + 6*c*d*e*f^2 + c^2*f^3)*sqrt(b*x^2 + a)*a^2*x/b^3 
- 3/8*(d^2*e^3 + 6*c*d*e^2*f + 3*c^2*e*f^2)*sqrt(b*x^2 + a)*a*x/b^2 + 1/2* 
(2*c*d*e^3 + 3*c^2*e^2*f)*sqrt(b*x^2 + a)*x/b + 35/128*(3*d^2*e*f^2 + 2*c* 
d*f^3)*a^4*arcsinh(b*x/sqrt(a*b))/b^(9/2) - 5/16*(3*d^2*e^2*f + 6*c*d*e*f^ 
2 + c^2*f^3)*a^3*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 3/8*(d^2*e^3 + 6*c*d*e^2 
*f + 3*c^2*e*f^2)*a^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 1/2*(2*c*d*e^3 + 3* 
c^2*e^2*f)*a*arcsinh(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.15 \[ \int \frac {\left (c+d x^2\right )^2 \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, d^{2} f^{3} x^{2}}{b} + \frac {30 \, b^{8} d^{2} e f^{2} + 20 \, b^{8} c d f^{3} - 9 \, a b^{7} d^{2} f^{3}}{b^{9}}\right )} x^{2} + \frac {240 \, b^{8} d^{2} e^{2} f + 480 \, b^{8} c d e f^{2} - 210 \, a b^{7} d^{2} e f^{2} + 80 \, b^{8} c^{2} f^{3} - 140 \, a b^{7} c d f^{3} + 63 \, a^{2} b^{6} d^{2} f^{3}}{b^{9}}\right )} x^{2} + \frac {5 \, {\left (96 \, b^{8} d^{2} e^{3} + 576 \, b^{8} c d e^{2} f - 240 \, a b^{7} d^{2} e^{2} f + 288 \, b^{8} c^{2} e f^{2} - 480 \, a b^{7} c d e f^{2} + 210 \, a^{2} b^{6} d^{2} e f^{2} - 80 \, a b^{7} c^{2} f^{3} + 140 \, a^{2} b^{6} c d f^{3} - 63 \, a^{3} b^{5} d^{2} f^{3}\right )}}{b^{9}}\right )} x^{2} + \frac {15 \, {\left (256 \, b^{8} c d e^{3} - 96 \, a b^{7} d^{2} e^{3} + 384 \, b^{8} c^{2} e^{2} f - 576 \, a b^{7} c d e^{2} f + 240 \, a^{2} b^{6} d^{2} e^{2} f - 288 \, a b^{7} c^{2} e f^{2} + 480 \, a^{2} b^{6} c d e f^{2} - 210 \, a^{3} b^{5} d^{2} e f^{2} + 80 \, a^{2} b^{6} c^{2} f^{3} - 140 \, a^{3} b^{5} c d f^{3} + 63 \, a^{4} b^{4} d^{2} f^{3}\right )}}{b^{9}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (256 \, b^{5} c^{2} e^{3} - 256 \, a b^{4} c d e^{3} + 96 \, a^{2} b^{3} d^{2} e^{3} - 384 \, a b^{4} c^{2} e^{2} f + 576 \, a^{2} b^{3} c d e^{2} f - 240 \, a^{3} b^{2} d^{2} e^{2} f + 288 \, a^{2} b^{3} c^{2} e f^{2} - 480 \, a^{3} b^{2} c d e f^{2} + 210 \, a^{4} b d^{2} e f^{2} - 80 \, a^{3} b^{2} c^{2} f^{3} + 140 \, a^{4} b c d f^{3} - 63 \, a^{5} d^{2} f^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {11}{2}}} \] Input:

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

Output:

1/3840*(2*(4*(6*(8*d^2*f^3*x^2/b + (30*b^8*d^2*e*f^2 + 20*b^8*c*d*f^3 - 9* 
a*b^7*d^2*f^3)/b^9)*x^2 + (240*b^8*d^2*e^2*f + 480*b^8*c*d*e*f^2 - 210*a*b 
^7*d^2*e*f^2 + 80*b^8*c^2*f^3 - 140*a*b^7*c*d*f^3 + 63*a^2*b^6*d^2*f^3)/b^ 
9)*x^2 + 5*(96*b^8*d^2*e^3 + 576*b^8*c*d*e^2*f - 240*a*b^7*d^2*e^2*f + 288 
*b^8*c^2*e*f^2 - 480*a*b^7*c*d*e*f^2 + 210*a^2*b^6*d^2*e*f^2 - 80*a*b^7*c^ 
2*f^3 + 140*a^2*b^6*c*d*f^3 - 63*a^3*b^5*d^2*f^3)/b^9)*x^2 + 15*(256*b^8*c 
*d*e^3 - 96*a*b^7*d^2*e^3 + 384*b^8*c^2*e^2*f - 576*a*b^7*c*d*e^2*f + 240* 
a^2*b^6*d^2*e^2*f - 288*a*b^7*c^2*e*f^2 + 480*a^2*b^6*c*d*e*f^2 - 210*a^3* 
b^5*d^2*e*f^2 + 80*a^2*b^6*c^2*f^3 - 140*a^3*b^5*c*d*f^3 + 63*a^4*b^4*d^2* 
f^3)/b^9)*sqrt(b*x^2 + a)*x - 1/256*(256*b^5*c^2*e^3 - 256*a*b^4*c*d*e^3 + 
 96*a^2*b^3*d^2*e^3 - 384*a*b^4*c^2*e^2*f + 576*a^2*b^3*c*d*e^2*f - 240*a^ 
3*b^2*d^2*e^2*f + 288*a^2*b^3*c^2*e*f^2 - 480*a^3*b^2*c*d*e*f^2 + 210*a^4* 
b*d^2*e*f^2 - 80*a^3*b^2*c^2*f^3 + 140*a^4*b*c*d*f^3 - 63*a^5*d^2*f^3)*log 
(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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