\(\int \sqrt {a+b x^2} (c+d x^2)^3 (e+f x^2) \, dx\) [275]

Optimal result

Integrand size = 28, antiderivative size = 390 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right ) \, dx=\frac {\left (128 b^4 c^3 e+7 a^4 d^3 f+48 a^2 b^2 c d (d e+c f)-32 a b^3 c^2 (3 d e+c f)-10 a^3 b d^2 (d e+3 c f)\right ) x \sqrt {a+b x^2}}{256 b^4}-\frac {\left (7 a^3 d^3 f+48 a b^2 c d (d e+c f)-32 b^3 c^2 (3 d e+c f)-10 a^2 b d^2 (d e+3 c f)\right ) x \left (a+b x^2\right )^{3/2}}{128 b^4}+\frac {d \left (7 a^2 d^2 f+48 b^2 c (d e+c f)-10 a b d (d e+3 c f)\right ) x^3 \left (a+b x^2\right )^{3/2}}{96 b^3}-\frac {d^2 (7 a d f-10 b (d e+3 c f)) x^5 \left (a+b x^2\right )^{3/2}}{80 b^2}+\frac {d^3 f x^7 \left (a+b x^2\right )^{3/2}}{10 b}+\frac {a \left (128 b^4 c^3 e+7 a^4 d^3 f+48 a^2 b^2 c d (d e+c f)-32 a b^3 c^2 (3 d e+c f)-10 a^3 b d^2 (d e+3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{9/2}} \] Output:

1/256*(128*b^4*c^3*e+7*a^4*d^3*f+48*a^2*b^2*c*d*(c*f+d*e)-32*a*b^3*c^2*(c* 
f+3*d*e)-10*a^3*b*d^2*(3*c*f+d*e))*x*(b*x^2+a)^(1/2)/b^4-1/128*(7*a^3*d^3* 
f+48*a*b^2*c*d*(c*f+d*e)-32*b^3*c^2*(c*f+3*d*e)-10*a^2*b*d^2*(3*c*f+d*e))* 
x*(b*x^2+a)^(3/2)/b^4+1/96*d*(7*a^2*d^2*f+48*b^2*c*(c*f+d*e)-10*a*b*d*(3*c 
*f+d*e))*x^3*(b*x^2+a)^(3/2)/b^3-1/80*d^2*(7*a*d*f-10*b*(3*c*f+d*e))*x^5*( 
b*x^2+a)^(3/2)/b^2+1/10*d^3*f*x^7*(b*x^2+a)^(3/2)/b+1/256*a*(128*b^4*c^3*e 
+7*a^4*d^3*f+48*a^2*b^2*c*d*(c*f+d*e)-32*a*b^3*c^2*(c*f+3*d*e)-10*a^3*b*d^ 
2*(3*c*f+d*e))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
 

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.88 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right ) \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^4 d^3 f+10 a^3 b d^2 \left (15 d e+45 c f+7 d f x^2\right )+16 a b^3 \left (30 c^3 f+15 c d^2 x^2 \left (2 e+f x^2\right )+30 c^2 d \left (3 e+f x^2\right )+d^3 x^4 \left (5 e+3 f x^2\right )\right )+96 b^4 \left (10 c^3 \left (2 e+f x^2\right )+10 c^2 d x^2 \left (3 e+2 f x^2\right )+5 c d^2 x^4 \left (4 e+3 f x^2\right )+d^3 x^6 \left (5 e+4 f x^2\right )\right )-4 a^2 b^2 d \left (180 c^2 f+15 c d \left (12 e+5 f x^2\right )+d^2 x^2 \left (25 e+14 f x^2\right )\right )\right )-15 a \left (128 b^4 c^3 e+7 a^4 d^3 f+48 a^2 b^2 c d (d e+c f)-32 a b^3 c^2 (3 d e+c f)-10 a^3 b d^2 (d e+3 c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{3840 b^{9/2}} \] Input:

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

Output:

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^4*d^3*f + 10*a^3*b*d^2*(15*d*e + 45*c*f 
 + 7*d*f*x^2) + 16*a*b^3*(30*c^3*f + 15*c*d^2*x^2*(2*e + f*x^2) + 30*c^2*d 
*(3*e + f*x^2) + d^3*x^4*(5*e + 3*f*x^2)) + 96*b^4*(10*c^3*(2*e + f*x^2) + 
 10*c^2*d*x^2*(3*e + 2*f*x^2) + 5*c*d^2*x^4*(4*e + 3*f*x^2) + d^3*x^6*(5*e 
 + 4*f*x^2)) - 4*a^2*b^2*d*(180*c^2*f + 15*c*d*(12*e + 5*f*x^2) + d^2*x^2* 
(25*e + 14*f*x^2))) - 15*a*(128*b^4*c^3*e + 7*a^4*d^3*f + 48*a^2*b^2*c*d*( 
d*e + c*f) - 32*a*b^3*c^2*(3*d*e + c*f) - 10*a^3*b*d^2*(d*e + 3*c*f))*Log[ 
-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(3840*b^(9/2))
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {403, 403, 403, 299, 211, 224, 219}

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.

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\int \sqrt {b x^2+a} \left (d x^2+c\right )^2 \left ((10 b d e+6 b c f-7 a d f) x^2+c (10 b e-a f)\right )dx}{10 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\int \sqrt {b x^2+a} \left (d x^2+c\right ) \left (\left (24 c (5 d e+c f) b^2-2 a d (25 d e+33 c f) b+35 a^2 d^2 f\right ) x^2+c \left (7 d f a^2-10 b d e a-14 b c f a+80 b^2 c e\right )\right )dx}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (-7 a d f+6 b c f+10 b d e)}{8 b}}{10 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {\frac {\int \sqrt {b x^2+a} \left (c \left (-35 d^2 f a^3+2 b d (25 d e+54 c f) a^2-36 b^2 c (5 d e+3 c f) a+480 b^3 c^2 e\right )-\left (-48 c^2 (15 d e+c f) b^3+8 a c d (65 d e+36 c f) b^2-10 a^2 d^2 (15 d e+31 c f) b+105 a^3 d^3 f\right ) x^2\right )dx}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (35 a^2 d^2 f-2 a b d (33 c f+25 d e)+24 b^2 c (c f+5 d e)\right )}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (-7 a d f+6 b c f+10 b d e)}{8 b}}{10 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {\frac {15 \left (7 a^4 d^3 f-10 a^3 b d^2 (3 c f+d e)+48 a^2 b^2 c d (c f+d e)-32 a b^3 c^2 (c f+3 d e)+128 b^4 c^3 e\right ) \int \sqrt {b x^2+a}dx}{4 b}-\frac {x \left (a+b x^2\right )^{3/2} \left (105 a^3 d^3 f-10 a^2 b d^2 (31 c f+15 d e)+8 a b^2 c d (36 c f+65 d e)-48 b^3 c^2 (c f+15 d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (35 a^2 d^2 f-2 a b d (33 c f+25 d e)+24 b^2 c (c f+5 d e)\right )}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (-7 a d f+6 b c f+10 b d e)}{8 b}}{10 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {15 \left (7 a^4 d^3 f-10 a^3 b d^2 (3 c f+d e)+48 a^2 b^2 c d (c f+d e)-32 a b^3 c^2 (c f+3 d e)+128 b^4 c^3 e\right ) \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}-\frac {x \left (a+b x^2\right )^{3/2} \left (105 a^3 d^3 f-10 a^2 b d^2 (31 c f+15 d e)+8 a b^2 c d (36 c f+65 d e)-48 b^3 c^2 (c f+15 d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (35 a^2 d^2 f-2 a b d (33 c f+25 d e)+24 b^2 c (c f+5 d e)\right )}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (-7 a d f+6 b c f+10 b d e)}{8 b}}{10 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {15 \left (7 a^4 d^3 f-10 a^3 b d^2 (3 c f+d e)+48 a^2 b^2 c d (c f+d e)-32 a b^3 c^2 (c f+3 d e)+128 b^4 c^3 e\right ) \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )}{4 b}-\frac {x \left (a+b x^2\right )^{3/2} \left (105 a^3 d^3 f-10 a^2 b d^2 (31 c f+15 d e)+8 a b^2 c d (36 c f+65 d e)-48 b^3 c^2 (c f+15 d e)\right )}{4 b}}{6 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (35 a^2 d^2 f-2 a b d (33 c f+25 d e)+24 b^2 c (c f+5 d e)\right )}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (-7 a d f+6 b c f+10 b d e)}{8 b}}{10 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{10 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (35 a^2 d^2 f-2 a b d (33 c f+25 d e)+24 b^2 c (c f+5 d e)\right )}{6 b}+\frac {\frac {15 \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right ) \left (7 a^4 d^3 f-10 a^3 b d^2 (3 c f+d e)+48 a^2 b^2 c d (c f+d e)-32 a b^3 c^2 (c f+3 d e)+128 b^4 c^3 e\right )}{4 b}-\frac {x \left (a+b x^2\right )^{3/2} \left (105 a^3 d^3 f-10 a^2 b d^2 (31 c f+15 d e)+8 a b^2 c d (36 c f+65 d e)-48 b^3 c^2 (c f+15 d e)\right )}{4 b}}{6 b}}{8 b}+\frac {x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 (-7 a d f+6 b c f+10 b d e)}{8 b}}{10 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3}{10 b}\)

Input:

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

Output:

(f*x*(a + b*x^2)^(3/2)*(c + d*x^2)^3)/(10*b) + (((10*b*d*e + 6*b*c*f - 7*a 
*d*f)*x*(a + b*x^2)^(3/2)*(c + d*x^2)^2)/(8*b) + (((35*a^2*d^2*f + 24*b^2* 
c*(5*d*e + c*f) - 2*a*b*d*(25*d*e + 33*c*f))*x*(a + b*x^2)^(3/2)*(c + d*x^ 
2))/(6*b) + (-1/4*((105*a^3*d^3*f - 48*b^3*c^2*(15*d*e + c*f) - 10*a^2*b*d 
^2*(15*d*e + 31*c*f) + 8*a*b^2*c*d*(65*d*e + 36*c*f))*x*(a + b*x^2)^(3/2)) 
/b + (15*(128*b^4*c^3*e + 7*a^4*d^3*f + 48*a^2*b^2*c*d*(d*e + c*f) - 32*a* 
b^3*c^2*(3*d*e + c*f) - 10*a^3*b*d^2*(d*e + 3*c*f))*((x*Sqrt[a + b*x^2])/2 
 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/(4*b))/(6*b))/(8 
*b))/(10*b)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(\frac {\frac {7 a \left (a^{3} \left (a f -\frac {10 b e}{7}\right ) d^{3}-\frac {30 a^{2} c b \left (a f -\frac {8 b e}{5}\right ) d^{2}}{7}+\frac {48 a \,b^{2} c^{2} \left (a f -2 b e \right ) d}{7}-\frac {32 b^{3} c^{3} \left (a f -4 b e \right )}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{256}-\frac {7 \left (\left (-\frac {32 \left (\frac {4 f \,x^{2}}{5}+e \right ) x^{6} d^{3}}{7}-\frac {128 c \left (\frac {3 f \,x^{2}}{4}+e \right ) x^{4} d^{2}}{7}-\frac {192 c^{2} \left (\frac {2 f \,x^{2}}{3}+e \right ) x^{2} d}{7}-\frac {128 c^{3} \left (\frac {f \,x^{2}}{2}+e \right )}{7}\right ) b^{\frac {9}{2}}+a \left (\left (-\frac {16 \left (\frac {3 f \,x^{2}}{5}+e \right ) x^{4} d^{3}}{21}-\frac {32 c \,x^{2} \left (\frac {f \,x^{2}}{2}+e \right ) d^{2}}{7}-\frac {96 c^{2} \left (\frac {f \,x^{2}}{3}+e \right ) d}{7}-\frac {32 f \,c^{3}}{7}\right ) b^{\frac {7}{2}}+a d \left (\left (\left (\frac {8}{15} f \,x^{4}+\frac {20}{21} e \,x^{2}\right ) d^{2}+\frac {48 \left (\frac {5 f \,x^{2}}{12}+e \right ) c d}{7}+\frac {48 c^{2} f}{7}\right ) b^{\frac {5}{2}}+a \left (\left (\left (-\frac {2 f \,x^{2}}{3}-\frac {10 e}{7}\right ) d -\frac {30 c f}{7}\right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right ) d \right )\right )\right ) \sqrt {b \,x^{2}+a}\, x}{256}}{b^{\frac {9}{2}}}\)	\(312\)
risch	\(-\frac {x \left (-384 f \,d^{3} b^{4} x^{8}-48 a \,b^{3} d^{3} f \,x^{6}-1440 b^{4} c \,d^{2} f \,x^{6}-480 b^{4} d^{3} e \,x^{6}+56 a^{2} b^{2} d^{3} f \,x^{4}-240 a \,b^{3} c \,d^{2} f \,x^{4}-80 a \,b^{3} d^{3} e \,x^{4}-1920 b^{4} c^{2} d f \,x^{4}-1920 b^{4} c \,d^{2} e \,x^{4}-70 a^{3} b \,d^{3} f \,x^{2}+300 a^{2} b^{2} c \,d^{2} f \,x^{2}+100 a^{2} b^{2} d^{3} e \,x^{2}-480 a \,b^{3} c^{2} d f \,x^{2}-480 a \,b^{3} c \,d^{2} e \,x^{2}-960 b^{4} c^{3} f \,x^{2}-2880 b^{4} c^{2} d e \,x^{2}+105 a^{4} d^{3} f -450 a^{3} b c \,d^{2} f -150 a^{3} b \,d^{3} e +720 a^{2} b^{2} c^{2} d f +720 a^{2} b^{2} c \,d^{2} e -480 a \,b^{3} c^{3} f -1440 a \,b^{3} c^{2} d e -1920 b^{4} c^{3} e \right ) \sqrt {b \,x^{2}+a}}{3840 b^{4}}+\frac {a \left (7 a^{4} d^{3} f -30 a^{3} b c \,d^{2} f -10 a^{3} b \,d^{3} e +48 a^{2} b^{2} c^{2} d f +48 a^{2} b^{2} c \,d^{2} e -32 a \,b^{3} c^{3} f -96 a \,b^{3} c^{2} d e +128 b^{4} c^{3} e \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {9}{2}}}\)	\(428\)
default	\(e \,c^{3} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+d^{2} \left (3 c f +d e \right ) \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{8 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )}{8 b}\right )+3 c d \left (c f +d e \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )+c^{2} \left (c f +3 d e \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )+f \,d^{3} \left (\frac {x^{7} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{10 b}-\frac {7 a \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{8 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )}{2 b}\right )}{8 b}\right )}{10 b}\right )\)	\(454\)

Input:

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

Output:

7/256/b^(9/2)*(a*(a^3*(a*f-10/7*b*e)*d^3-30/7*a^2*c*b*(a*f-8/5*b*e)*d^2+48 
/7*a*b^2*c^2*(a*f-2*b*e)*d-32/7*b^3*c^3*(a*f-4*b*e))*arctanh((b*x^2+a)^(1/ 
2)/x/b^(1/2))-((-32/7*(4/5*f*x^2+e)*x^6*d^3-128/7*c*(3/4*f*x^2+e)*x^4*d^2- 
192/7*c^2*(2/3*f*x^2+e)*x^2*d-128/7*c^3*(1/2*f*x^2+e))*b^(9/2)+a*((-16/21* 
(3/5*f*x^2+e)*x^4*d^3-32/7*c*x^2*(1/2*f*x^2+e)*d^2-96/7*c^2*(1/3*f*x^2+e)* 
d-32/7*f*c^3)*b^(7/2)+a*d*(((8/15*f*x^4+20/21*e*x^2)*d^2+48/7*(5/12*f*x^2+ 
e)*c*d+48/7*c^2*f)*b^(5/2)+a*(((-2/3*f*x^2-10/7*e)*d-30/7*c*f)*b^(3/2)+a*d 
*f*b^(1/2))*d)))*(b*x^2+a)^(1/2)*x)
 

Fricas [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.17 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

[1/7680*(15*(2*(64*a*b^4*c^3 - 48*a^2*b^3*c^2*d + 24*a^3*b^2*c*d^2 - 5*a^4 
*b*d^3)*e - (32*a^2*b^3*c^3 - 48*a^3*b^2*c^2*d + 30*a^4*b*c*d^2 - 7*a^5*d^ 
3)*f)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(384*b^5 
*d^3*f*x^9 + 48*(10*b^5*d^3*e + (30*b^5*c*d^2 + a*b^4*d^3)*f)*x^7 + 8*(10* 
(24*b^5*c*d^2 + a*b^4*d^3)*e + (240*b^5*c^2*d + 30*a*b^4*c*d^2 - 7*a^2*b^3 
*d^3)*f)*x^5 + 10*(2*(144*b^5*c^2*d + 24*a*b^4*c*d^2 - 5*a^2*b^3*d^3)*e + 
(96*b^5*c^3 + 48*a*b^4*c^2*d - 30*a^2*b^3*c*d^2 + 7*a^3*b^2*d^3)*f)*x^3 + 
15*(2*(64*b^5*c^3 + 48*a*b^4*c^2*d - 24*a^2*b^3*c*d^2 + 5*a^3*b^2*d^3)*e + 
 (32*a*b^4*c^3 - 48*a^2*b^3*c^2*d + 30*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*f)*x)* 
sqrt(b*x^2 + a))/b^5, -1/3840*(15*(2*(64*a*b^4*c^3 - 48*a^2*b^3*c^2*d + 24 
*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*e - (32*a^2*b^3*c^3 - 48*a^3*b^2*c^2*d + 30* 
a^4*b*c*d^2 - 7*a^5*d^3)*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 
(384*b^5*d^3*f*x^9 + 48*(10*b^5*d^3*e + (30*b^5*c*d^2 + a*b^4*d^3)*f)*x^7 
+ 8*(10*(24*b^5*c*d^2 + a*b^4*d^3)*e + (240*b^5*c^2*d + 30*a*b^4*c*d^2 - 7 
*a^2*b^3*d^3)*f)*x^5 + 10*(2*(144*b^5*c^2*d + 24*a*b^4*c*d^2 - 5*a^2*b^3*d 
^3)*e + (96*b^5*c^3 + 48*a*b^4*c^2*d - 30*a^2*b^3*c*d^2 + 7*a^3*b^2*d^3)*f 
)*x^3 + 15*(2*(64*b^5*c^3 + 48*a*b^4*c^2*d - 24*a^2*b^3*c*d^2 + 5*a^3*b^2* 
d^3)*e + (32*a*b^4*c^3 - 48*a^2*b^3*c^2*d + 30*a^3*b^2*c*d^2 - 7*a^4*b*d^3 
)*f)*x)*sqrt(b*x^2 + a))/b^5]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.74 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right ) \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d^{3} f x^{9}}{10} + \frac {x^{7} \left (\frac {a d^{3} f}{10} + 3 b c d^{2} f + b d^{3} e\right )}{8 b} + \frac {x^{5} \cdot \left (3 a c d^{2} f + a d^{3} e - \frac {7 a \left (\frac {a d^{3} f}{10} + 3 b c d^{2} f + b d^{3} e\right )}{8 b} + 3 b c^{2} d f + 3 b c d^{2} e\right )}{6 b} + \frac {x^{3} \cdot \left (3 a c^{2} d f + 3 a c d^{2} e - \frac {5 a \left (3 a c d^{2} f + a d^{3} e - \frac {7 a \left (\frac {a d^{3} f}{10} + 3 b c d^{2} f + b d^{3} e\right )}{8 b} + 3 b c^{2} d f + 3 b c d^{2} e\right )}{6 b} + b c^{3} f + 3 b c^{2} d e\right )}{4 b} + \frac {x \left (a c^{3} f + 3 a c^{2} d e - \frac {3 a \left (3 a c^{2} d f + 3 a c d^{2} e - \frac {5 a \left (3 a c d^{2} f + a d^{3} e - \frac {7 a \left (\frac {a d^{3} f}{10} + 3 b c d^{2} f + b d^{3} e\right )}{8 b} + 3 b c^{2} d f + 3 b c d^{2} e\right )}{6 b} + b c^{3} f + 3 b c^{2} d e\right )}{4 b} + b c^{3} e\right )}{2 b}\right ) + \left (a c^{3} e - \frac {a \left (a c^{3} f + 3 a c^{2} d e - \frac {3 a \left (3 a c^{2} d f + 3 a c d^{2} e - \frac {5 a \left (3 a c d^{2} f + a d^{3} e - \frac {7 a \left (\frac {a d^{3} f}{10} + 3 b c d^{2} f + b d^{3} e\right )}{8 b} + 3 b c^{2} d f + 3 b c d^{2} e\right )}{6 b} + b c^{3} f + 3 b c^{2} d e\right )}{4 b} + b c^{3} e\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\sqrt {a} \left (c^{3} e x + \frac {d^{3} f x^{9}}{9} + \frac {x^{7} \cdot \left (3 c d^{2} f + d^{3} e\right )}{7} + \frac {x^{5} \cdot \left (3 c^{2} d f + 3 c d^{2} e\right )}{5} + \frac {x^{3} \left (c^{3} f + 3 c^{2} d e\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.35 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} d^{3} f x^{7}}{10 \, b} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a d^{3} f x^{5}}{80 \, b^{2}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{3} f x^{3}}{96 \, b^{3}} + \frac {{\left (d^{3} e + 3 \, c d^{2} f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{5}}{8 \, b} + \frac {1}{2} \, \sqrt {b x^{2} + a} c^{3} e x - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} d^{3} f x}{128 \, b^{4}} + \frac {7 \, \sqrt {b x^{2} + a} a^{4} d^{3} f x}{256 \, b^{4}} + \frac {a c^{3} e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} + \frac {7 \, a^{5} d^{3} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {9}{2}}} - \frac {5 \, {\left (d^{3} e + 3 \, c d^{2} f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{3}}{48 \, b^{2}} + \frac {{\left (c d^{2} e + c^{2} d f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{3}}{2 \, b} + \frac {5 \, {\left (d^{3} e + 3 \, c d^{2} f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x}{64 \, b^{3}} - \frac {5 \, {\left (d^{3} e + 3 \, c d^{2} f\right )} \sqrt {b x^{2} + a} a^{3} x}{128 \, b^{3}} - \frac {3 \, {\left (c d^{2} e + c^{2} d f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x}{8 \, b^{2}} + \frac {3 \, {\left (c d^{2} e + c^{2} d f\right )} \sqrt {b x^{2} + a} a^{2} x}{16 \, b^{2}} + \frac {{\left (3 \, c^{2} d e + c^{3} f\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} x}{4 \, b} - \frac {{\left (3 \, c^{2} d e + c^{3} f\right )} \sqrt {b x^{2} + a} a x}{8 \, b} - \frac {5 \, {\left (d^{3} e + 3 \, c d^{2} f\right )} a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {7}{2}}} + \frac {3 \, {\left (c d^{2} e + c^{2} d f\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {5}{2}}} - \frac {{\left (3 \, c^{2} d e + c^{3} f\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} \] Input:

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

Output:

1/10*(b*x^2 + a)^(3/2)*d^3*f*x^7/b - 7/80*(b*x^2 + a)^(3/2)*a*d^3*f*x^5/b^ 
2 + 7/96*(b*x^2 + a)^(3/2)*a^2*d^3*f*x^3/b^3 + 1/8*(d^3*e + 3*c*d^2*f)*(b* 
x^2 + a)^(3/2)*x^5/b + 1/2*sqrt(b*x^2 + a)*c^3*e*x - 7/128*(b*x^2 + a)^(3/ 
2)*a^3*d^3*f*x/b^4 + 7/256*sqrt(b*x^2 + a)*a^4*d^3*f*x/b^4 + 1/2*a*c^3*e*a 
rcsinh(b*x/sqrt(a*b))/sqrt(b) + 7/256*a^5*d^3*f*arcsinh(b*x/sqrt(a*b))/b^( 
9/2) - 5/48*(d^3*e + 3*c*d^2*f)*(b*x^2 + a)^(3/2)*a*x^3/b^2 + 1/2*(c*d^2*e 
 + c^2*d*f)*(b*x^2 + a)^(3/2)*x^3/b + 5/64*(d^3*e + 3*c*d^2*f)*(b*x^2 + a) 
^(3/2)*a^2*x/b^3 - 5/128*(d^3*e + 3*c*d^2*f)*sqrt(b*x^2 + a)*a^3*x/b^3 - 3 
/8*(c*d^2*e + c^2*d*f)*(b*x^2 + a)^(3/2)*a*x/b^2 + 3/16*(c*d^2*e + c^2*d*f 
)*sqrt(b*x^2 + a)*a^2*x/b^2 + 1/4*(3*c^2*d*e + c^3*f)*(b*x^2 + a)^(3/2)*x/ 
b - 1/8*(3*c^2*d*e + c^3*f)*sqrt(b*x^2 + a)*a*x/b - 5/128*(d^3*e + 3*c*d^2 
*f)*a^4*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 3/16*(c*d^2*e + c^2*d*f)*a^3*arcs 
inh(b*x/sqrt(a*b))/b^(5/2) - 1/8*(3*c^2*d*e + c^3*f)*a^2*arcsinh(b*x/sqrt( 
a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.10 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^3 \left (e+f x^2\right ) \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, d^{3} f x^{2} + \frac {10 \, b^{8} d^{3} e + 30 \, b^{8} c d^{2} f + a b^{7} d^{3} f}{b^{8}}\right )} x^{2} + \frac {240 \, b^{8} c d^{2} e + 10 \, a b^{7} d^{3} e + 240 \, b^{8} c^{2} d f + 30 \, a b^{7} c d^{2} f - 7 \, a^{2} b^{6} d^{3} f}{b^{8}}\right )} x^{2} + \frac {5 \, {\left (288 \, b^{8} c^{2} d e + 48 \, a b^{7} c d^{2} e - 10 \, a^{2} b^{6} d^{3} e + 96 \, b^{8} c^{3} f + 48 \, a b^{7} c^{2} d f - 30 \, a^{2} b^{6} c d^{2} f + 7 \, a^{3} b^{5} d^{3} f\right )}}{b^{8}}\right )} x^{2} + \frac {15 \, {\left (128 \, b^{8} c^{3} e + 96 \, a b^{7} c^{2} d e - 48 \, a^{2} b^{6} c d^{2} e + 10 \, a^{3} b^{5} d^{3} e + 32 \, a b^{7} c^{3} f - 48 \, a^{2} b^{6} c^{2} d f + 30 \, a^{3} b^{5} c d^{2} f - 7 \, a^{4} b^{4} d^{3} f\right )}}{b^{8}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (128 \, a b^{4} c^{3} e - 96 \, a^{2} b^{3} c^{2} d e + 48 \, a^{3} b^{2} c d^{2} e - 10 \, a^{4} b d^{3} e - 32 \, a^{2} b^{3} c^{3} f + 48 \, a^{3} b^{2} c^{2} d f - 30 \, a^{4} b c d^{2} f + 7 \, a^{5} d^{3} f\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {9}{2}}} \] Input:

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

Output:

1/3840*(2*(4*(6*(8*d^3*f*x^2 + (10*b^8*d^3*e + 30*b^8*c*d^2*f + a*b^7*d^3* 
f)/b^8)*x^2 + (240*b^8*c*d^2*e + 10*a*b^7*d^3*e + 240*b^8*c^2*d*f + 30*a*b 
^7*c*d^2*f - 7*a^2*b^6*d^3*f)/b^8)*x^2 + 5*(288*b^8*c^2*d*e + 48*a*b^7*c*d 
^2*e - 10*a^2*b^6*d^3*e + 96*b^8*c^3*f + 48*a*b^7*c^2*d*f - 30*a^2*b^6*c*d 
^2*f + 7*a^3*b^5*d^3*f)/b^8)*x^2 + 15*(128*b^8*c^3*e + 96*a*b^7*c^2*d*e - 
48*a^2*b^6*c*d^2*e + 10*a^3*b^5*d^3*e + 32*a*b^7*c^3*f - 48*a^2*b^6*c^2*d* 
f + 30*a^3*b^5*c*d^2*f - 7*a^4*b^4*d^3*f)/b^8)*sqrt(b*x^2 + a)*x - 1/256*( 
128*a*b^4*c^3*e - 96*a^2*b^3*c^2*d*e + 48*a^3*b^2*c*d^2*e - 10*a^4*b*d^3*e 
 - 32*a^2*b^3*c^3*f + 48*a^3*b^2*c^2*d*f - 30*a^4*b*c*d^2*f + 7*a^5*d^3*f) 
*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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