\(\int (g+h x)^m (a+b x+c x^2) (d+f x^2)^2 \, dx\) [7]

Optimal result

Integrand size = 27, antiderivative size = 362 \[ \int (g+h x)^m \left (a+b x+c x^2\right ) \left (d+f x^2\right )^2 \, dx=\frac {\left (c g^2-b g h+a h^2\right ) \left (f g^2+d h^2\right )^2 (g+h x)^{1+m}}{h^7 (1+m)}-\frac {\left (f g^2+d h^2\right ) \left (6 c f g^3-5 b f g^2 h+2 c d g h^2+4 a f g h^2-b d h^3\right ) (g+h x)^{2+m}}{h^7 (2+m)}-\frac {\left (2 f h \left (5 b f g^3-3 a f g^2 h+3 b d g h^2-a d h^3\right )-c \left (15 f^2 g^4+12 d f g^2 h^2+d^2 h^4\right )\right ) (g+h x)^{3+m}}{h^7 (3+m)}-\frac {2 f \left (10 c f g^3-5 b f g^2 h+4 c d g h^2+2 a f g h^2-b d h^3\right ) (g+h x)^{4+m}}{h^7 (4+m)}-\frac {f \left (f h (5 b g-a h)-c \left (15 f g^2+2 d h^2\right )\right ) (g+h x)^{5+m}}{h^7 (5+m)}-\frac {f^2 (6 c g-b h) (g+h x)^{6+m}}{h^7 (6+m)}+\frac {c f^2 (g+h x)^{7+m}}{h^7 (7+m)} \] Output:

(a*h^2-b*g*h+c*g^2)*(d*h^2+f*g^2)^2*(h*x+g)^(1+m)/h^7/(1+m)-(d*h^2+f*g^2)* 
(4*a*f*g*h^2-b*d*h^3-5*b*f*g^2*h+2*c*d*g*h^2+6*c*f*g^3)*(h*x+g)^(2+m)/h^7/ 
(2+m)-(2*f*h*(-a*d*h^3-3*a*f*g^2*h+3*b*d*g*h^2+5*b*f*g^3)-c*(d^2*h^4+12*d* 
f*g^2*h^2+15*f^2*g^4))*(h*x+g)^(3+m)/h^7/(3+m)-2*f*(2*a*f*g*h^2-b*d*h^3-5* 
b*f*g^2*h+4*c*d*g*h^2+10*c*f*g^3)*(h*x+g)^(4+m)/h^7/(4+m)-f*(f*h*(-a*h+5*b 
*g)-c*(2*d*h^2+15*f*g^2))*(h*x+g)^(5+m)/h^7/(5+m)-f^2*(-b*h+6*c*g)*(h*x+g) 
^(6+m)/h^7/(6+m)+c*f^2*(h*x+g)^(7+m)/h^7/(7+m)
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.90 \[ \int (g+h x)^m \left (a+b x+c x^2\right ) \left (d+f x^2\right )^2 \, dx=\frac {(g+h x)^{1+m} \left (\frac {\left (f g^2+d h^2\right )^2 \left (c g^2+h (-b g+a h)\right )}{1+m}-\frac {\left (f g^2+d h^2\right ) \left (6 c f g^3-5 b f g^2 h+2 c d g h^2+4 a f g h^2-b d h^3\right ) (g+h x)}{2+m}+\frac {\left (2 f h \left (-5 b f g^3+3 a f g^2 h-3 b d g h^2+a d h^3\right )+c \left (15 f^2 g^4+12 d f g^2 h^2+d^2 h^4\right )\right ) (g+h x)^2}{3+m}-\frac {2 f \left (10 c f g^3-5 b f g^2 h+4 c d g h^2+2 a f g h^2-b d h^3\right ) (g+h x)^3}{4+m}+\frac {f \left (f h (-5 b g+a h)+c \left (15 f g^2+2 d h^2\right )\right ) (g+h x)^4}{5+m}-\frac {f^2 (6 c g-b h) (g+h x)^5}{6+m}+\frac {c f^2 (g+h x)^6}{7+m}\right )}{h^7} \] Input:

Integrate[(g + h*x)^m*(a + b*x + c*x^2)*(d + f*x^2)^2,x]
 

Output:

((g + h*x)^(1 + m)*(((f*g^2 + d*h^2)^2*(c*g^2 + h*(-(b*g) + a*h)))/(1 + m) 
 - ((f*g^2 + d*h^2)*(6*c*f*g^3 - 5*b*f*g^2*h + 2*c*d*g*h^2 + 4*a*f*g*h^2 - 
 b*d*h^3)*(g + h*x))/(2 + m) + ((2*f*h*(-5*b*f*g^3 + 3*a*f*g^2*h - 3*b*d*g 
*h^2 + a*d*h^3) + c*(15*f^2*g^4 + 12*d*f*g^2*h^2 + d^2*h^4))*(g + h*x)^2)/ 
(3 + m) - (2*f*(10*c*f*g^3 - 5*b*f*g^2*h + 4*c*d*g*h^2 + 2*a*f*g*h^2 - b*d 
*h^3)*(g + h*x)^3)/(4 + m) + (f*(f*h*(-5*b*g + a*h) + c*(15*f*g^2 + 2*d*h^ 
2))*(g + h*x)^4)/(5 + m) - (f^2*(6*c*g - b*h)*(g + h*x)^5)/(6 + m) + (c*f^ 
2*(g + h*x)^6)/(7 + m)))/h^7
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2159, 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.

\(\displaystyle \int \left (d+f x^2\right )^2 \left (a+b x+c x^2\right ) (g+h x)^m \, dx\)

\(\Big \downarrow \) 2159

\(\displaystyle \int \left (\frac {(g+h x)^{m+2} \left (c \left (d^2 h^4+12 d f g^2 h^2+15 f^2 g^4\right )-2 f h \left (-a d h^3-3 a f g^2 h+3 b d g h^2+5 b f g^3\right )\right )}{h^6}+\frac {\left (d h^2+f g^2\right )^2 (g+h x)^m \left (a h^2-b g h+c g^2\right )}{h^6}+\frac {f (g+h x)^{m+4} \left (c \left (2 d h^2+15 f g^2\right )-f h (5 b g-a h)\right )}{h^6}+\frac {\left (d h^2+f g^2\right ) (g+h x)^{m+1} \left (-4 a f g h^2+b d h^3+5 b f g^2 h-2 c d g h^2-6 c f g^3\right )}{h^6}-\frac {2 f (g+h x)^{m+3} \left (2 a f g h^2-b d h^3-5 b f g^2 h+4 c d g h^2+10 c f g^3\right )}{h^6}+\frac {f^2 (b h-6 c g) (g+h x)^{m+5}}{h^6}+\frac {c f^2 (g+h x)^{m+6}}{h^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(g+h x)^{m+3} \left (2 f h \left (-a d h^3-3 a f g^2 h+3 b d g h^2+5 b f g^3\right )-c \left (d^2 h^4+12 d f g^2 h^2+15 f^2 g^4\right )\right )}{h^7 (m+3)}+\frac {\left (d h^2+f g^2\right )^2 (g+h x)^{m+1} \left (a h^2-b g h+c g^2\right )}{h^7 (m+1)}-\frac {f (g+h x)^{m+5} \left (f h (5 b g-a h)-c \left (2 d h^2+15 f g^2\right )\right )}{h^7 (m+5)}-\frac {\left (d h^2+f g^2\right ) (g+h x)^{m+2} \left (4 a f g h^2-b d h^3-5 b f g^2 h+2 c d g h^2+6 c f g^3\right )}{h^7 (m+2)}-\frac {2 f (g+h x)^{m+4} \left (2 a f g h^2-b d h^3-5 b f g^2 h+4 c d g h^2+10 c f g^3\right )}{h^7 (m+4)}-\frac {f^2 (6 c g-b h) (g+h x)^{m+6}}{h^7 (m+6)}+\frac {c f^2 (g+h x)^{m+7}}{h^7 (m+7)}\)

Input:

Int[(g + h*x)^m*(a + b*x + c*x^2)*(d + f*x^2)^2,x]
 

Output:

((c*g^2 - b*g*h + a*h^2)*(f*g^2 + d*h^2)^2*(g + h*x)^(1 + m))/(h^7*(1 + m) 
) - ((f*g^2 + d*h^2)*(6*c*f*g^3 - 5*b*f*g^2*h + 2*c*d*g*h^2 + 4*a*f*g*h^2 
- b*d*h^3)*(g + h*x)^(2 + m))/(h^7*(2 + m)) - ((2*f*h*(5*b*f*g^3 - 3*a*f*g 
^2*h + 3*b*d*g*h^2 - a*d*h^3) - c*(15*f^2*g^4 + 12*d*f*g^2*h^2 + d^2*h^4)) 
*(g + h*x)^(3 + m))/(h^7*(3 + m)) - (2*f*(10*c*f*g^3 - 5*b*f*g^2*h + 4*c*d 
*g*h^2 + 2*a*f*g*h^2 - b*d*h^3)*(g + h*x)^(4 + m))/(h^7*(4 + m)) - (f*(f*h 
*(5*b*g - a*h) - c*(15*f*g^2 + 2*d*h^2))*(g + h*x)^(5 + m))/(h^7*(5 + m)) 
- (f^2*(6*c*g - b*h)*(g + h*x)^(6 + m))/(h^7*(6 + m)) + (c*f^2*(g + h*x)^( 
7 + m))/(h^7*(7 + m))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2159

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2054\) vs. \(2(362)=724\).

Time = 1.28 (sec) , antiderivative size = 2055, normalized size of antiderivative = 5.68

method	result	size
norman	\(\text {Expression too large to display}\)	\(2055\)
gosper	\(\text {Expression too large to display}\)	\(2617\)
orering	\(\text {Expression too large to display}\)	\(2620\)
risch	\(\text {Expression too large to display}\)	\(3271\)
parallelrisch	\(\text {Expression too large to display}\)	\(4797\)

Input:

int((h*x+g)^m*(c*x^2+b*x+a)*(f*x^2+d)^2,x,method=_RETURNVERBOSE)
 

Output:

c*f^2/(7+m)*x^7*exp(m*ln(h*x+g))+g*(a*d^2*h^6*m^6+27*a*d^2*h^6*m^5-b*d^2*g 
*h^5*m^5+295*a*d^2*h^6*m^4+4*a*d*f*g^2*h^4*m^4-25*b*d^2*g*h^5*m^4+2*c*d^2* 
g^2*h^4*m^4+1665*a*d^2*h^6*m^3+88*a*d*f*g^2*h^4*m^3-245*b*d^2*g*h^5*m^3-12 
*b*d*f*g^3*h^3*m^3+44*c*d^2*g^2*h^4*m^3+5104*a*d^2*h^6*m^2+716*a*d*f*g^2*h 
^4*m^2+24*a*f^2*g^4*h^2*m^2-1175*b*d^2*g*h^5*m^2-216*b*d*f*g^3*h^3*m^2+358 
*c*d^2*g^2*h^4*m^2+48*c*d*f*g^4*h^2*m^2+8028*a*d^2*h^6*m+2552*a*d*f*g^2*h^ 
4*m+312*a*f^2*g^4*h^2*m-2754*b*d^2*g*h^5*m-1284*b*d*f*g^3*h^3*m-120*b*f^2* 
g^5*h*m+1276*c*d^2*g^2*h^4*m+624*c*d*f*g^4*h^2*m+5040*a*d^2*h^6+3360*a*d*f 
*g^2*h^4+1008*a*f^2*g^4*h^2-2520*b*d^2*g*h^5-2520*b*d*f*g^3*h^3-840*b*f^2* 
g^5*h+1680*c*d^2*g^2*h^4+2016*c*d*f*g^4*h^2+720*c*f^2*g^6)/h^7/(m^7+28*m^6 
+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)*exp(m*ln(h*x+g))+(2*a*d 
*f*h^4*m^4+2*b*d*f*g*h^3*m^4+c*d^2*h^4*m^4+44*a*d*f*h^4*m^3-4*a*f^2*g^2*h^ 
2*m^3+36*b*d*f*g*h^3*m^3+22*c*d^2*h^4*m^3-8*c*d*f*g^2*h^2*m^3+358*a*d*f*h^ 
4*m^2-52*a*f^2*g^2*h^2*m^2+214*b*d*f*g*h^3*m^2+20*b*f^2*g^3*h*m^2+179*c*d^ 
2*h^4*m^2-104*c*d*f*g^2*h^2*m^2+1276*a*d*f*h^4*m-168*a*f^2*g^2*h^2*m+420*b 
*d*f*g*h^3*m+140*b*f^2*g^3*h*m+638*c*d^2*h^4*m-336*c*d*f*g^2*h^2*m-120*c*f 
^2*g^4*m+1680*a*d*f*h^4+840*c*d^2*h^4)/h^4/(m^5+25*m^4+245*m^3+1175*m^2+27 
54*m+2520)*x^3*exp(m*ln(h*x+g))+(2*a*d*f*g*h^4*m^5+b*d^2*h^5*m^5+c*d^2*g*h 
^4*m^5+44*a*d*f*g*h^4*m^4+25*b*d^2*h^5*m^4-6*b*d*f*g^2*h^3*m^4+22*c*d^2*g* 
h^4*m^4+358*a*d*f*g*h^4*m^3+12*a*f^2*g^3*h^2*m^3+245*b*d^2*h^5*m^3-108*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2430 vs. \(2 (362) = 724\).

Time = 0.12 (sec) , antiderivative size = 2430, normalized size of antiderivative = 6.71 \[ \int (g+h x)^m \left (a+b x+c x^2\right ) \left (d+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)^m*(c*x^2+b*x+a)*(f*x^2+d)^2,x, algorithm="fricas")
 

Output:

(a*d^2*g*h^6*m^6 + 720*c*f^2*g^7 - 840*b*f^2*g^6*h - 2520*b*d*f*g^4*h^3 - 
2520*b*d^2*g^2*h^5 + 5040*a*d^2*g*h^6 + 1008*(2*c*d*f + a*f^2)*g^5*h^2 + 1 
680*(c*d^2 + 2*a*d*f)*g^3*h^4 + (c*f^2*h^7*m^6 + 21*c*f^2*h^7*m^5 + 175*c* 
f^2*h^7*m^4 + 735*c*f^2*h^7*m^3 + 1624*c*f^2*h^7*m^2 + 1764*c*f^2*h^7*m + 
720*c*f^2*h^7)*x^7 + (840*b*f^2*h^7 + (c*f^2*g*h^6 + b*f^2*h^7)*m^6 + (15* 
c*f^2*g*h^6 + 22*b*f^2*h^7)*m^5 + 5*(17*c*f^2*g*h^6 + 38*b*f^2*h^7)*m^4 + 
5*(45*c*f^2*g*h^6 + 164*b*f^2*h^7)*m^3 + (274*c*f^2*g*h^6 + 1849*b*f^2*h^7 
)*m^2 + 2*(60*c*f^2*g*h^6 + 1019*b*f^2*h^7)*m)*x^6 - (b*d^2*g^2*h^5 - 27*a 
*d^2*g*h^6)*m^5 + (1008*(2*c*d*f + a*f^2)*h^7 + (b*f^2*g*h^6 + (2*c*d*f + 
a*f^2)*h^7)*m^6 - (6*c*f^2*g^2*h^5 - 17*b*f^2*g*h^6 - 23*(2*c*d*f + a*f^2) 
*h^7)*m^5 - 3*(20*c*f^2*g^2*h^5 - 35*b*f^2*g*h^6 - 69*(2*c*d*f + a*f^2)*h^ 
7)*m^4 - 5*(42*c*f^2*g^2*h^5 - 59*b*f^2*g*h^6 - 185*(2*c*d*f + a*f^2)*h^7) 
*m^3 - 2*(150*c*f^2*g^2*h^5 - 187*b*f^2*g*h^6 - 1072*(2*c*d*f + a*f^2)*h^7 
)*m^2 - 12*(12*c*f^2*g^2*h^5 - 14*b*f^2*g*h^6 - 201*(2*c*d*f + a*f^2)*h^7) 
*m)*x^5 - (25*b*d^2*g^2*h^5 - 295*a*d^2*g*h^6 - 2*(c*d^2 + 2*a*d*f)*g^3*h^ 
4)*m^4 + (2520*b*d*f*h^7 + (2*b*d*f*h^7 + (2*c*d*f + a*f^2)*g*h^6)*m^6 - ( 
5*b*f^2*g^2*h^5 - 48*b*d*f*h^7 - 19*(2*c*d*f + a*f^2)*g*h^6)*m^5 + (30*c*f 
^2*g^3*h^4 - 65*b*f^2*g^2*h^5 + 452*b*d*f*h^7 + 131*(2*c*d*f + a*f^2)*g*h^ 
6)*m^4 + (180*c*f^2*g^3*h^4 - 265*b*f^2*g^2*h^5 + 2112*b*d*f*h^7 + 401*(2* 
c*d*f + a*f^2)*g*h^6)*m^3 + 5*(66*c*f^2*g^3*h^4 - 83*b*f^2*g^2*h^5 + 10...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36319 vs. \(2 (354) = 708\).

Time = 7.09 (sec) , antiderivative size = 36319, normalized size of antiderivative = 100.33 \[ \int (g+h x)^m \left (a+b x+c x^2\right ) \left (d+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)**m*(c*x**2+b*x+a)*(f*x**2+d)**2,x)
 

Output:

Piecewise((g**m*(a*d**2*x + 2*a*d*f*x**3/3 + a*f**2*x**5/5 + b*d**2*x**2/2 
 + b*d*f*x**4/2 + b*f**2*x**6/6 + c*d**2*x**3/3 + 2*c*d*f*x**5/5 + c*f**2* 
x**7/7), Eq(h, 0)), (-10*a*d**2*h**6/(60*g**6*h**7 + 360*g**5*h**8*x + 900 
*g**4*h**9*x**2 + 1200*g**3*h**10*x**3 + 900*g**2*h**11*x**4 + 360*g*h**12 
*x**5 + 60*h**13*x**6) - 2*a*d*f*g**2*h**4/(60*g**6*h**7 + 360*g**5*h**8*x 
 + 900*g**4*h**9*x**2 + 1200*g**3*h**10*x**3 + 900*g**2*h**11*x**4 + 360*g 
*h**12*x**5 + 60*h**13*x**6) - 12*a*d*f*g*h**5*x/(60*g**6*h**7 + 360*g**5* 
h**8*x + 900*g**4*h**9*x**2 + 1200*g**3*h**10*x**3 + 900*g**2*h**11*x**4 + 
 360*g*h**12*x**5 + 60*h**13*x**6) - 30*a*d*f*h**6*x**2/(60*g**6*h**7 + 36 
0*g**5*h**8*x + 900*g**4*h**9*x**2 + 1200*g**3*h**10*x**3 + 900*g**2*h**11 
*x**4 + 360*g*h**12*x**5 + 60*h**13*x**6) - 2*a*f**2*g**4*h**2/(60*g**6*h* 
*7 + 360*g**5*h**8*x + 900*g**4*h**9*x**2 + 1200*g**3*h**10*x**3 + 900*g** 
2*h**11*x**4 + 360*g*h**12*x**5 + 60*h**13*x**6) - 12*a*f**2*g**3*h**3*x/( 
60*g**6*h**7 + 360*g**5*h**8*x + 900*g**4*h**9*x**2 + 1200*g**3*h**10*x**3 
 + 900*g**2*h**11*x**4 + 360*g*h**12*x**5 + 60*h**13*x**6) - 30*a*f**2*g** 
2*h**4*x**2/(60*g**6*h**7 + 360*g**5*h**8*x + 900*g**4*h**9*x**2 + 1200*g* 
*3*h**10*x**3 + 900*g**2*h**11*x**4 + 360*g*h**12*x**5 + 60*h**13*x**6) - 
40*a*f**2*g*h**5*x**3/(60*g**6*h**7 + 360*g**5*h**8*x + 900*g**4*h**9*x**2 
 + 1200*g**3*h**10*x**3 + 900*g**2*h**11*x**4 + 360*g*h**12*x**5 + 60*h**1 
3*x**6) - 30*a*f**2*h**6*x**4/(60*g**6*h**7 + 360*g**5*h**8*x + 900*g**...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1024 vs. \(2 (362) = 724\).

Time = 0.07 (sec) , antiderivative size = 1024, normalized size of antiderivative = 2.83 \[ \int (g+h x)^m \left (a+b x+c x^2\right ) \left (d+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)^m*(c*x^2+b*x+a)*(f*x^2+d)^2,x, algorithm="maxima")
 

Output:

(h^2*(m + 1)*x^2 + g*h*m*x - g^2)*(h*x + g)^m*b*d^2/((m^2 + 3*m + 2)*h^2) 
+ (h*x + g)^(m + 1)*a*d^2/(h*(m + 1)) + ((m^2 + 3*m + 2)*h^3*x^3 + (m^2 + 
m)*g*h^2*x^2 - 2*g^2*h*m*x + 2*g^3)*(h*x + g)^m*c*d^2/((m^3 + 6*m^2 + 11*m 
 + 6)*h^3) + 2*((m^2 + 3*m + 2)*h^3*x^3 + (m^2 + m)*g*h^2*x^2 - 2*g^2*h*m* 
x + 2*g^3)*(h*x + g)^m*a*d*f/((m^3 + 6*m^2 + 11*m + 6)*h^3) + 2*((m^3 + 6* 
m^2 + 11*m + 6)*h^4*x^4 + (m^3 + 3*m^2 + 2*m)*g*h^3*x^3 - 3*(m^2 + m)*g^2* 
h^2*x^2 + 6*g^3*h*m*x - 6*g^4)*(h*x + g)^m*b*d*f/((m^4 + 10*m^3 + 35*m^2 + 
 50*m + 24)*h^4) + 2*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*h^5*x^5 + (m^4 + 
 6*m^3 + 11*m^2 + 6*m)*g*h^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*g^2*h^3*x^3 + 12* 
(m^2 + m)*g^3*h^2*x^2 - 24*g^4*h*m*x + 24*g^5)*(h*x + g)^m*c*d*f/((m^5 + 1 
5*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*h^5) + ((m^4 + 10*m^3 + 35*m^2 + 5 
0*m + 24)*h^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*g*h^4*x^4 - 4*(m^3 + 3*m^ 
2 + 2*m)*g^2*h^3*x^3 + 12*(m^2 + m)*g^3*h^2*x^2 - 24*g^4*h*m*x + 24*g^5)*( 
h*x + g)^m*a*f^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*h^5) + ( 
(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*h^6*x^6 + (m^5 + 10*m^4 + 
35*m^3 + 50*m^2 + 24*m)*g*h^5*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*g^2*h^4 
*x^4 + 20*(m^3 + 3*m^2 + 2*m)*g^3*h^3*x^3 - 60*(m^2 + m)*g^4*h^2*x^2 + 120 
*g^5*h*m*x - 120*g^6)*(h*x + g)^m*b*f^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 
 + 1624*m^2 + 1764*m + 720)*h^6) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 16 
24*m^2 + 1764*m + 720)*h^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4819 vs. \(2 (362) = 724\).

Time = 0.17 (sec) , antiderivative size = 4819, normalized size of antiderivative = 13.31 \[ \int (g+h x)^m \left (a+b x+c x^2\right ) \left (d+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)^m*(c*x^2+b*x+a)*(f*x^2+d)^2,x, algorithm="giac")
 

Output:

((h*x + g)^m*c*f^2*h^7*m^6*x^7 + (h*x + g)^m*c*f^2*g*h^6*m^6*x^6 + (h*x + 
g)^m*b*f^2*h^7*m^6*x^6 + 21*(h*x + g)^m*c*f^2*h^7*m^5*x^7 + (h*x + g)^m*b* 
f^2*g*h^6*m^6*x^5 + 2*(h*x + g)^m*c*d*f*h^7*m^6*x^5 + (h*x + g)^m*a*f^2*h^ 
7*m^6*x^5 + 15*(h*x + g)^m*c*f^2*g*h^6*m^5*x^6 + 22*(h*x + g)^m*b*f^2*h^7* 
m^5*x^6 + 175*(h*x + g)^m*c*f^2*h^7*m^4*x^7 + 2*(h*x + g)^m*c*d*f*g*h^6*m^ 
6*x^4 + (h*x + g)^m*a*f^2*g*h^6*m^6*x^4 + 2*(h*x + g)^m*b*d*f*h^7*m^6*x^4 
- 6*(h*x + g)^m*c*f^2*g^2*h^5*m^5*x^5 + 17*(h*x + g)^m*b*f^2*g*h^6*m^5*x^5 
 + 46*(h*x + g)^m*c*d*f*h^7*m^5*x^5 + 23*(h*x + g)^m*a*f^2*h^7*m^5*x^5 + 8 
5*(h*x + g)^m*c*f^2*g*h^6*m^4*x^6 + 190*(h*x + g)^m*b*f^2*h^7*m^4*x^6 + 73 
5*(h*x + g)^m*c*f^2*h^7*m^3*x^7 + 2*(h*x + g)^m*b*d*f*g*h^6*m^6*x^3 + (h*x 
 + g)^m*c*d^2*h^7*m^6*x^3 + 2*(h*x + g)^m*a*d*f*h^7*m^6*x^3 - 5*(h*x + g)^ 
m*b*f^2*g^2*h^5*m^5*x^4 + 38*(h*x + g)^m*c*d*f*g*h^6*m^5*x^4 + 19*(h*x + g 
)^m*a*f^2*g*h^6*m^5*x^4 + 48*(h*x + g)^m*b*d*f*h^7*m^5*x^4 - 60*(h*x + g)^ 
m*c*f^2*g^2*h^5*m^4*x^5 + 105*(h*x + g)^m*b*f^2*g*h^6*m^4*x^5 + 414*(h*x + 
 g)^m*c*d*f*h^7*m^4*x^5 + 207*(h*x + g)^m*a*f^2*h^7*m^4*x^5 + 225*(h*x + g 
)^m*c*f^2*g*h^6*m^3*x^6 + 820*(h*x + g)^m*b*f^2*h^7*m^3*x^6 + 1624*(h*x + 
g)^m*c*f^2*h^7*m^2*x^7 + (h*x + g)^m*c*d^2*g*h^6*m^6*x^2 + 2*(h*x + g)^m*a 
*d*f*g*h^6*m^6*x^2 + (h*x + g)^m*b*d^2*h^7*m^6*x^2 - 8*(h*x + g)^m*c*d*f*g 
^2*h^5*m^5*x^3 - 4*(h*x + g)^m*a*f^2*g^2*h^5*m^5*x^3 + 42*(h*x + g)^m*b*d* 
f*g*h^6*m^5*x^3 + 25*(h*x + g)^m*c*d^2*h^7*m^5*x^3 + 50*(h*x + g)^m*a*d...
 

Mupad [B] (verification not implemented)

Time = 17.61 (sec) , antiderivative size = 2259, normalized size of antiderivative = 6.24 \[ \int (g+h x)^m \left (a+b x+c x^2\right ) \left (d+f x^2\right )^2 \, dx=\text {Too large to display} \] Input:

int((g + h*x)^m*(d + f*x^2)^2*(a + b*x + c*x^2),x)
 

Output:

((g + h*x)^m*(720*c*f^2*g^7 - 2520*b*d^2*g^2*h^5 + 1008*a*f^2*g^5*h^2 + 16 
80*c*d^2*g^3*h^4 + 5040*a*d^2*g*h^6 - 840*b*f^2*g^6*h + 3360*a*d*f*g^3*h^4 
 - 2520*b*d*f*g^4*h^3 + 2016*c*d*f*g^5*h^2 + 8028*a*d^2*g*h^6*m - 120*b*f^ 
2*g^6*h*m + 5104*a*d^2*g*h^6*m^2 + 1665*a*d^2*g*h^6*m^3 + 295*a*d^2*g*h^6* 
m^4 + 27*a*d^2*g*h^6*m^5 + a*d^2*g*h^6*m^6 - 2754*b*d^2*g^2*h^5*m + 312*a* 
f^2*g^5*h^2*m + 1276*c*d^2*g^3*h^4*m - 1175*b*d^2*g^2*h^5*m^2 - 245*b*d^2* 
g^2*h^5*m^3 - 25*b*d^2*g^2*h^5*m^4 - b*d^2*g^2*h^5*m^5 + 24*a*f^2*g^5*h^2* 
m^2 + 358*c*d^2*g^3*h^4*m^2 + 44*c*d^2*g^3*h^4*m^3 + 2*c*d^2*g^3*h^4*m^4 + 
 2552*a*d*f*g^3*h^4*m - 1284*b*d*f*g^4*h^3*m + 624*c*d*f*g^5*h^2*m + 716*a 
*d*f*g^3*h^4*m^2 + 88*a*d*f*g^3*h^4*m^3 + 4*a*d*f*g^3*h^4*m^4 - 216*b*d*f* 
g^4*h^3*m^2 - 12*b*d*f*g^4*h^3*m^3 + 48*c*d*f*g^5*h^2*m^2))/(h^7*(13068*m 
+ 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x*( 
g + h*x)^m*(5040*a*d^2*h^7 + 5104*a*d^2*h^7*m^2 + 1665*a*d^2*h^7*m^3 + 295 
*a*d^2*h^7*m^4 + 27*a*d^2*h^7*m^5 + a*d^2*h^7*m^6 + 8028*a*d^2*h^7*m + 252 
0*b*d^2*g*h^6*m - 720*c*f^2*g^6*h*m + 2754*b*d^2*g*h^6*m^2 + 1175*b*d^2*g* 
h^6*m^3 + 245*b*d^2*g*h^6*m^4 + 25*b*d^2*g*h^6*m^5 + b*d^2*g*h^6*m^6 - 100 
8*a*f^2*g^4*h^3*m - 1680*c*d^2*g^2*h^5*m + 840*b*f^2*g^5*h^2*m - 312*a*f^2 
*g^4*h^3*m^2 - 1276*c*d^2*g^2*h^5*m^2 - 24*a*f^2*g^4*h^3*m^3 - 358*c*d^2*g 
^2*h^5*m^3 - 44*c*d^2*g^2*h^5*m^4 - 2*c*d^2*g^2*h^5*m^5 + 120*b*f^2*g^5*h^ 
2*m^2 - 3360*a*d*f*g^2*h^5*m + 2520*b*d*f*g^3*h^4*m - 2016*c*d*f*g^4*h^...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 3270, normalized size of antiderivative = 9.03 \[ \int (g+h x)^m \left (a+b x+c x^2\right ) \left (d+f x^2\right )^2 \, dx =\text {Too large to display} \] Input:

int((h*x+g)^m*(c*x^2+b*x+a)*(f*x^2+d)^2,x)
 

Output:

((g + h*x)**m*(a*d**2*g*h**6*m**6 + 27*a*d**2*g*h**6*m**5 + 295*a*d**2*g*h 
**6*m**4 + 1665*a*d**2*g*h**6*m**3 + 5104*a*d**2*g*h**6*m**2 + 8028*a*d**2 
*g*h**6*m + 5040*a*d**2*g*h**6 + a*d**2*h**7*m**6*x + 27*a*d**2*h**7*m**5* 
x + 295*a*d**2*h**7*m**4*x + 1665*a*d**2*h**7*m**3*x + 5104*a*d**2*h**7*m* 
*2*x + 8028*a*d**2*h**7*m*x + 5040*a*d**2*h**7*x + 4*a*d*f*g**3*h**4*m**4 
+ 88*a*d*f*g**3*h**4*m**3 + 716*a*d*f*g**3*h**4*m**2 + 2552*a*d*f*g**3*h** 
4*m + 3360*a*d*f*g**3*h**4 - 4*a*d*f*g**2*h**5*m**5*x - 88*a*d*f*g**2*h**5 
*m**4*x - 716*a*d*f*g**2*h**5*m**3*x - 2552*a*d*f*g**2*h**5*m**2*x - 3360* 
a*d*f*g**2*h**5*m*x + 2*a*d*f*g*h**6*m**6*x**2 + 46*a*d*f*g*h**6*m**5*x**2 
 + 402*a*d*f*g*h**6*m**4*x**2 + 1634*a*d*f*g*h**6*m**3*x**2 + 2956*a*d*f*g 
*h**6*m**2*x**2 + 1680*a*d*f*g*h**6*m*x**2 + 2*a*d*f*h**7*m**6*x**3 + 50*a 
*d*f*h**7*m**5*x**3 + 494*a*d*f*h**7*m**4*x**3 + 2438*a*d*f*h**7*m**3*x**3 
 + 6224*a*d*f*h**7*m**2*x**3 + 7592*a*d*f*h**7*m*x**3 + 3360*a*d*f*h**7*x* 
*3 + 24*a*f**2*g**5*h**2*m**2 + 312*a*f**2*g**5*h**2*m + 1008*a*f**2*g**5* 
h**2 - 24*a*f**2*g**4*h**3*m**3*x - 312*a*f**2*g**4*h**3*m**2*x - 1008*a*f 
**2*g**4*h**3*m*x + 12*a*f**2*g**3*h**4*m**4*x**2 + 168*a*f**2*g**3*h**4*m 
**3*x**2 + 660*a*f**2*g**3*h**4*m**2*x**2 + 504*a*f**2*g**3*h**4*m*x**2 - 
4*a*f**2*g**2*h**5*m**5*x**3 - 64*a*f**2*g**2*h**5*m**4*x**3 - 332*a*f**2* 
g**2*h**5*m**3*x**3 - 608*a*f**2*g**2*h**5*m**2*x**3 - 336*a*f**2*g**2*h** 
5*m*x**3 + a*f**2*g*h**6*m**6*x**4 + 19*a*f**2*g*h**6*m**5*x**4 + 131*a...