\(\int (g+h x)^2 \sqrt {a+b x+c x^2} (d+e x+f x^2) \, dx\) [24]

Optimal result

Integrand size = 32, antiderivative size = 575 \[ \int (g+h x)^2 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {\left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (4 c e-3 b f-\frac {2 c f g}{h}\right ) (g+h x)^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\left (105 b^3 f h^2+\frac {64 c^3 \left (f g^3-2 g h (e g+5 d h)\right )}{h}-28 b c h (7 a f h+5 b (2 f g+e h))+8 c^2 \left (7 b f g^2+25 b h (2 e g+d h)+16 a h (2 f g+e h)\right )-6 c \left (21 b^2 f h^2-4 c h (2 b f g+7 b e h+5 a f h)-8 c^2 \left (f g^2-h (2 e g+5 d h)\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}-\frac {\left (b^2-4 a c\right ) \left (128 c^4 d g^2+21 b^4 f h^2-28 b^2 c h (2 b f g+b e h+2 a f h)-32 c^3 \left (a f g^2+a h (2 e g+d h)+2 b g (e g+2 d h)\right )+8 c^2 \left (2 a^2 f h^2+6 a b h (2 f g+e h)+5 b^2 \left (f g^2+h (2 e g+d h)\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}} \] Output:

1/512*(128*c^4*d*g^2+21*b^4*f*h^2-28*b^2*c*h*(2*a*f*h+b*e*h+2*b*f*g)-32*c^ 
3*(a*f*g^2+a*h*(d*h+2*e*g)+2*b*g*(2*d*h+e*g))+8*c^2*(2*a^2*f*h^2+6*a*b*h*( 
e*h+2*f*g)+5*b^2*(f*g^2+h*(d*h+2*e*g))))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^5 
+1/20*(4*c*e-3*b*f-2*c*f*g/h)*(h*x+g)^2*(c*x^2+b*x+a)^(3/2)/c^2+1/6*f*(h*x 
+g)^3*(c*x^2+b*x+a)^(3/2)/c/h-1/960*(105*b^3*f*h^2+64*c^3*(f*g^3-2*g*h*(5* 
d*h+e*g))/h-28*b*c*h*(7*a*f*h+5*b*(e*h+2*f*g))+8*c^2*(7*b*f*g^2+25*b*h*(d* 
h+2*e*g)+16*a*h*(e*h+2*f*g))-6*c*(21*b^2*f*h^2-4*c*h*(5*a*f*h+7*b*e*h+2*b* 
f*g)-8*c^2*(f*g^2-h*(5*d*h+2*e*g)))*x)*(c*x^2+b*x+a)^(3/2)/c^4-1/1024*(-4* 
a*c+b^2)*(128*c^4*d*g^2+21*b^4*f*h^2-28*b^2*c*h*(2*a*f*h+b*e*h+2*b*f*g)-32 
*c^3*(a*f*g^2+a*h*(d*h+2*e*g)+2*b*g*(2*d*h+e*g))+8*c^2*(2*a^2*f*h^2+6*a*b* 
h*(e*h+2*f*g)+5*b^2*(f*g^2+h*(d*h+2*e*g))))*arctanh(1/2*(2*c*x+b)/c^(1/2)/ 
(c*x^2+b*x+a)^(1/2))/c^(11/2)
 

Mathematica [A] (verified)

Time = 11.26 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.23 \[ \int (g+h x)^2 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\frac {3840 c^{9/2} d g^2 (b+2 c x) \sqrt {a+x (b+c x)}+5120 c^{9/2} g (e g+2 d h) (a+x (b+c x))^{3/2}+3840 c^{9/2} \left (f g^2+h (2 e g+d h)\right ) x (a+x (b+c x))^{3/2}+3072 c^{9/2} h (2 f g+e h) x^2 (a+x (b+c x))^{3/2}+2560 c^{9/2} f h^2 x^3 (a+x (b+c x))^{3/2}-1920 c^4 \left (b^2-4 a c\right ) d g^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-960 b c^3 g (e g+2 d h) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )+4 c h (2 f g+e h) \left (-16 c^{3/2} \left (-35 b^2+32 a c+42 b c x\right ) (a+x (b+c x))^{3/2}-15 b \left (7 b^2-12 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )-40 c^2 \left (f g^2+h (2 e g+d h)\right ) \left (80 b c^{3/2} (a+x (b+c x))^{3/2}-3 \left (5 b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )-f h^2 \left (2304 b c^{7/2} x^2 (a+x (b+c x))^{3/2}+16 c^{3/2} \left (105 b^3-196 a b c-126 b^2 c x+120 a c^2 x\right ) (a+x (b+c x))^{3/2}-15 \left (21 b^4-56 a b^2 c+16 a^2 c^2\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )}{15360 c^{11/2}} \] Input:

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

Output:

(3840*c^(9/2)*d*g^2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + 5120*c^(9/2)*g*(e* 
g + 2*d*h)*(a + x*(b + c*x))^(3/2) + 3840*c^(9/2)*(f*g^2 + h*(2*e*g + d*h) 
)*x*(a + x*(b + c*x))^(3/2) + 3072*c^(9/2)*h*(2*f*g + e*h)*x^2*(a + x*(b + 
 c*x))^(3/2) + 2560*c^(9/2)*f*h^2*x^3*(a + x*(b + c*x))^(3/2) - 1920*c^4*( 
b^2 - 4*a*c)*d*g^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] 
- 960*b*c^3*g*(e*g + 2*d*h)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - 
 (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]) + 4 
*c*h*(2*f*g + e*h)*(-16*c^(3/2)*(-35*b^2 + 32*a*c + 42*b*c*x)*(a + x*(b + 
c*x))^(3/2) - 15*b*(7*b^2 - 12*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + 
 c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x) 
])])) - 40*c^2*(f*g^2 + h*(2*e*g + d*h))*(80*b*c^(3/2)*(a + x*(b + c*x))^( 
3/2) - 3*(5*b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b 
^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])) - f*h 
^2*(2304*b*c^(7/2)*x^2*(a + x*(b + c*x))^(3/2) + 16*c^(3/2)*(105*b^3 - 196 
*a*b*c - 126*b^2*c*x + 120*a*c^2*x)*(a + x*(b + c*x))^(3/2) - 15*(21*b^4 - 
 56*a*b^2*c + 16*a^2*c^2)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - ( 
b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/(15 
360*c^(11/2))
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 478, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2184, 27, 1236, 27, 1225, 1087, 1092, 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.

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

\(\Big \downarrow \) 2184

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\frac {\int -\frac {1}{2} (g+h x) \left (9 f g h b^2+12 a f h^2 b-4 c g (f g+3 e h) b+4 c h (10 c d g-3 a f g-4 a e h)+\left (-8 \left (f g^2-h (2 e g+5 d h)\right ) c^2-4 h (2 b f g+7 b e h+5 a f h) c+21 b^2 f h^2\right ) x\right ) \sqrt {c x^2+b x+a}dx}{5 c}+\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{5 c}}{4 c h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{5 c}-\frac {\int (g+h x) \left (9 f g h b^2+12 a f h^2 b-4 c g (f g+3 e h) b+4 c h (10 c d g-3 a f g-4 a e h)+\left (-8 \left (f g^2-h (2 e g+5 d h)\right ) c^2-4 h (2 b f g+7 b e h+5 a f h) c+21 b^2 f h^2\right ) x\right ) \sqrt {c x^2+b x+a}dx}{10 c}}{4 c h}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{5 c}-\frac {\frac {5 h \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a h (d h+2 e g)+a f g^2+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right ) \int \sqrt {c x^2+b x+a}dx}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c h x \left (-4 c h (5 a f h+7 b e h+2 b f g)+21 b^2 f h^2-8 c^2 \left (f g^2-h (5 d h+2 e g)\right )\right )+8 c^2 h \left (16 a h (e h+2 f g)+25 b h (d h+2 e g)+7 b f g^2\right )-28 b c h^2 (7 a f h+5 b (e h+2 f g))+105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (5 d h+e g)\right )\right )}{24 c^2}}{10 c}}{4 c h}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{5 c}-\frac {\frac {5 h \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a h (d h+2 e g)+a f g^2+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c h x \left (-4 c h (5 a f h+7 b e h+2 b f g)+21 b^2 f h^2-8 c^2 \left (f g^2-h (5 d h+2 e g)\right )\right )+8 c^2 h \left (16 a h (e h+2 f g)+25 b h (d h+2 e g)+7 b f g^2\right )-28 b c h^2 (7 a f h+5 b (e h+2 f g))+105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (5 d h+e g)\right )\right )}{24 c^2}}{10 c}}{4 c h}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{5 c}-\frac {\frac {5 h \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a h (d h+2 e g)+a f g^2+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c h x \left (-4 c h (5 a f h+7 b e h+2 b f g)+21 b^2 f h^2-8 c^2 \left (f g^2-h (5 d h+2 e g)\right )\right )+8 c^2 h \left (16 a h (e h+2 f g)+25 b h (d h+2 e g)+7 b f g^2\right )-28 b c h^2 (7 a f h+5 b (e h+2 f g))+105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (5 d h+e g)\right )\right )}{24 c^2}}{10 c}}{4 c h}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {f (g+h x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c h}-\frac {\frac {(g+h x)^2 \left (a+b x+c x^2\right )^{3/2} (3 b f h-4 c e h+2 c f g)}{5 c}-\frac {\frac {5 h \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right ) \left (8 c^2 \left (2 a^2 f h^2+6 a b h (e h+2 f g)+5 b^2 \left (h (d h+2 e g)+f g^2\right )\right )-28 b^2 c h (2 a f h+b e h+2 b f g)-32 c^3 \left (a h (d h+2 e g)+a f g^2+2 b g (2 d h+e g)\right )+21 b^4 f h^2+128 c^4 d g^2\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c h x \left (-4 c h (5 a f h+7 b e h+2 b f g)+21 b^2 f h^2-8 c^2 \left (f g^2-h (5 d h+2 e g)\right )\right )+8 c^2 h \left (16 a h (e h+2 f g)+25 b h (d h+2 e g)+7 b f g^2\right )-28 b c h^2 (7 a f h+5 b (e h+2 f g))+105 b^3 f h^3+64 c^3 \left (f g^3-2 g h (5 d h+e g)\right )\right )}{24 c^2}}{10 c}}{4 c h}\)

Input:

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

Output:

(f*(g + h*x)^3*(a + b*x + c*x^2)^(3/2))/(6*c*h) - (((2*c*f*g - 4*c*e*h + 3 
*b*f*h)*(g + h*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) - (-1/24*((105*b^3*f*h^ 
3 + 64*c^3*(f*g^3 - 2*g*h*(e*g + 5*d*h)) - 28*b*c*h^2*(7*a*f*h + 5*b*(2*f* 
g + e*h)) + 8*c^2*h*(7*b*f*g^2 + 25*b*h*(2*e*g + d*h) + 16*a*h*(2*f*g + e* 
h)) - 6*c*h*(21*b^2*f*h^2 - 4*c*h*(2*b*f*g + 7*b*e*h + 5*a*f*h) - 8*c^2*(f 
*g^2 - h*(2*e*g + 5*d*h)))*x)*(a + b*x + c*x^2)^(3/2))/c^2 + (5*h*(128*c^4 
*d*g^2 + 21*b^4*f*h^2 - 28*b^2*c*h*(2*b*f*g + b*e*h + 2*a*f*h) - 32*c^3*(a 
*f*g^2 + a*h*(2*e*g + d*h) + 2*b*g*(e*g + 2*d*h)) + 8*c^2*(2*a^2*f*h^2 + 6 
*a*b*h*(2*f*g + e*h) + 5*b^2*(f*g^2 + h*(2*e*g + d*h))))*(((b + 2*c*x)*Sqr 
t[a + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]* 
Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))))/(16*c^2))/(10*c))/(4*c*h)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 1027, normalized size of antiderivative = 1.79

method	result	size
risch	\(\text {Expression too large to display}\)	\(1027\)
default	\(\text {Expression too large to display}\)	\(1208\)

Input:

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

Output:

1/7680*(1280*c^5*f*h^2*x^5+128*b*c^4*f*h^2*x^4+1536*c^5*e*h^2*x^4+3072*c^5 
*f*g*h*x^4+320*a*c^4*f*h^2*x^3-144*b^2*c^3*f*h^2*x^3+192*b*c^4*e*h^2*x^3+3 
84*b*c^4*f*g*h*x^3+1920*c^5*d*h^2*x^3+3840*c^5*e*g*h*x^3+1920*c^5*f*g^2*x^ 
3-544*a*b*c^3*f*h^2*x^2+512*a*c^4*e*h^2*x^2+1024*a*c^4*f*g*h*x^2+168*b^3*c 
^2*f*h^2*x^2-224*b^2*c^3*e*h^2*x^2-448*b^2*c^3*f*g*h*x^2+320*b*c^4*d*h^2*x 
^2+640*b*c^4*e*g*h*x^2+320*b*c^4*f*g^2*x^2+5120*c^5*d*g*h*x^2+2560*c^5*e*g 
^2*x^2-480*a^2*c^3*f*h^2*x+896*a*b^2*c^2*f*h^2*x-928*a*b*c^3*e*h^2*x-1856* 
a*b*c^3*f*g*h*x+960*a*c^4*d*h^2*x+1920*a*c^4*e*g*h*x+960*a*c^4*f*g^2*x-210 
*b^4*c*f*h^2*x+280*b^3*c^2*e*h^2*x+560*b^3*c^2*f*g*h*x-400*b^2*c^3*d*h^2*x 
-800*b^2*c^3*e*g*h*x-400*b^2*c^3*f*g^2*x+1280*b*c^4*d*g*h*x+640*b*c^4*e*g^ 
2*x+3840*c^5*d*g^2*x+1808*a^2*b*c^2*f*h^2-1024*a^2*c^3*e*h^2-2048*a^2*c^3* 
f*g*h-1680*a*b^3*c*f*h^2+1840*a*b^2*c^2*e*h^2+3680*a*b^2*c^2*f*g*h-2080*a* 
b*c^3*d*h^2-4160*a*b*c^3*e*g*h-2080*a*b*c^3*f*g^2+5120*a*c^4*d*g*h+2560*a* 
c^4*e*g^2+315*b^5*f*h^2-420*b^4*c*e*h^2-840*b^4*c*f*g*h+600*b^3*c^2*d*h^2+ 
1200*b^3*c^2*e*g*h+600*b^3*c^2*f*g^2-1920*b^2*c^3*d*g*h-960*b^2*c^3*e*g^2+ 
1920*b*c^4*d*g^2)*(c*x^2+b*x+a)^(1/2)/c^5+1/1024*(64*a^3*c^3*f*h^2-240*a^2 
*b^2*c^2*f*h^2+192*a^2*b*c^3*e*h^2+384*a^2*b*c^3*f*g*h-128*a^2*c^4*d*h^2-2 
56*a^2*c^4*e*g*h-128*a^2*c^4*f*g^2+140*a*b^4*c*f*h^2-160*a*b^3*c^2*e*h^2-3 
20*a*b^3*c^2*f*g*h+192*a*b^2*c^3*d*h^2+384*a*b^2*c^3*e*g*h+192*a*b^2*c^3*f 
*g^2-512*a*b*c^4*d*g*h-256*a*b*c^4*e*g^2+512*a*c^5*d*g^2-21*b^6*f*h^2+2...
 

Fricas [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 1791, normalized size of antiderivative = 3.11 \[ \int (g+h x)^2 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/30720*(15*(8*(16*(b^2*c^4 - 4*a*c^5)*d - 8*(b^3*c^3 - 4*a*b*c^4)*e + ( 
5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*f)*g^2 - 8*(16*(b^3*c^3 - 4*a*b*c^4 
)*d - 2*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*e + (7*b^5*c - 40*a*b^3*c^ 
2 + 48*a^2*b*c^3)*f)*g*h + (8*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d - 
4*(7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*e + (21*b^6 - 140*a*b^4*c + 240* 
a^2*b^2*c^2 - 64*a^3*c^3)*f)*h^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 
 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(1280*c^6*f*h^2* 
x^5 + 128*(24*c^6*f*g*h + (12*c^6*e + b*c^5*f)*h^2)*x^4 + 16*(120*c^6*f*g^ 
2 + 24*(10*c^6*e + b*c^5*f)*g*h + (120*c^6*d + 12*b*c^5*e - (9*b^2*c^4 - 2 
0*a*c^5)*f)*h^2)*x^3 + 40*(48*b*c^5*d - 8*(3*b^2*c^4 - 8*a*c^5)*e + (15*b^ 
3*c^3 - 52*a*b*c^4)*f)*g^2 - 8*(80*(3*b^2*c^4 - 8*a*c^5)*d - 10*(15*b^3*c^ 
3 - 52*a*b*c^4)*e + (105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2*c^4)*f)*g*h + ( 
40*(15*b^3*c^3 - 52*a*b*c^4)*d - 4*(105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2* 
c^4)*e + (315*b^5*c - 1680*a*b^3*c^2 + 1808*a^2*b*c^3)*f)*h^2 + 8*(40*(8*c 
^6*e + b*c^5*f)*g^2 + 8*(80*c^6*d + 10*b*c^5*e - (7*b^2*c^4 - 16*a*c^5)*f) 
*g*h + (40*b*c^5*d - 4*(7*b^2*c^4 - 16*a*c^5)*e + (21*b^3*c^3 - 68*a*b*c^4 
)*f)*h^2)*x^2 + 2*(40*(48*c^6*d + 8*b*c^5*e - (5*b^2*c^4 - 12*a*c^5)*f)*g^ 
2 + 8*(80*b*c^5*d - 10*(5*b^2*c^4 - 12*a*c^5)*e + (35*b^3*c^3 - 116*a*b*c^ 
4)*f)*g*h - (40*(5*b^2*c^4 - 12*a*c^5)*d - 4*(35*b^3*c^3 - 116*a*b*c^4)*e 
+ (105*b^4*c^2 - 448*a*b^2*c^3 + 240*a^2*c^4)*f)*h^2)*x)*sqrt(c*x^2 + b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2440 vs. \(2 (597) = 1194\).

Time = 1.22 (sec) , antiderivative size = 2440, normalized size of antiderivative = 4.24 \[ \int (g+h x)^2 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int (g+h x)^2 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.71 \[ \int (g+h x)^2 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*f*h^2*x + (24*c^5*f*g*h + 12* 
c^5*e*h^2 + b*c^4*f*h^2)/c^5)*x + (120*c^5*f*g^2 + 240*c^5*e*g*h + 24*b*c^ 
4*f*g*h + 120*c^5*d*h^2 + 12*b*c^4*e*h^2 - 9*b^2*c^3*f*h^2 + 20*a*c^4*f*h^ 
2)/c^5)*x + (320*c^5*e*g^2 + 40*b*c^4*f*g^2 + 640*c^5*d*g*h + 80*b*c^4*e*g 
*h - 56*b^2*c^3*f*g*h + 128*a*c^4*f*g*h + 40*b*c^4*d*h^2 - 28*b^2*c^3*e*h^ 
2 + 64*a*c^4*e*h^2 + 21*b^3*c^2*f*h^2 - 68*a*b*c^3*f*h^2)/c^5)*x + (1920*c 
^5*d*g^2 + 320*b*c^4*e*g^2 - 200*b^2*c^3*f*g^2 + 480*a*c^4*f*g^2 + 640*b*c 
^4*d*g*h - 400*b^2*c^3*e*g*h + 960*a*c^4*e*g*h + 280*b^3*c^2*f*g*h - 928*a 
*b*c^3*f*g*h - 200*b^2*c^3*d*h^2 + 480*a*c^4*d*h^2 + 140*b^3*c^2*e*h^2 - 4 
64*a*b*c^3*e*h^2 - 105*b^4*c*f*h^2 + 448*a*b^2*c^2*f*h^2 - 240*a^2*c^3*f*h 
^2)/c^5)*x + (1920*b*c^4*d*g^2 - 960*b^2*c^3*e*g^2 + 2560*a*c^4*e*g^2 + 60 
0*b^3*c^2*f*g^2 - 2080*a*b*c^3*f*g^2 - 1920*b^2*c^3*d*g*h + 5120*a*c^4*d*g 
*h + 1200*b^3*c^2*e*g*h - 4160*a*b*c^3*e*g*h - 840*b^4*c*f*g*h + 3680*a*b^ 
2*c^2*f*g*h - 2048*a^2*c^3*f*g*h + 600*b^3*c^2*d*h^2 - 2080*a*b*c^3*d*h^2 
- 420*b^4*c*e*h^2 + 1840*a*b^2*c^2*e*h^2 - 1024*a^2*c^3*e*h^2 + 315*b^5*f* 
h^2 - 1680*a*b^3*c*f*h^2 + 1808*a^2*b*c^2*f*h^2)/c^5) + 1/1024*(128*b^2*c^ 
4*d*g^2 - 512*a*c^5*d*g^2 - 64*b^3*c^3*e*g^2 + 256*a*b*c^4*e*g^2 + 40*b^4* 
c^2*f*g^2 - 192*a*b^2*c^3*f*g^2 + 128*a^2*c^4*f*g^2 - 128*b^3*c^3*d*g*h + 
512*a*b*c^4*d*g*h + 80*b^4*c^2*e*g*h - 384*a*b^2*c^3*e*g*h + 256*a^2*c^4*e 
*g*h - 56*b^5*c*f*g*h + 320*a*b^3*c^2*f*g*h - 384*a^2*b*c^3*f*g*h + 40*...
 

Mupad [B] (verification not implemented)

Time = 21.22 (sec) , antiderivative size = 1881, normalized size of antiderivative = 3.27 \[ \int (g+h x)^2 \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

d*g^2*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (e*h^2*x^2*(a + b*x + c*x^ 
2)^(3/2))/(5*c) + (f*h^2*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) - (a*d*h^2*((x 
/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b* 
x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c) - (a*f*g^2*((x/2 + b/ 
(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x 
^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c) + (d*g^2*log((b/2 + c*x)/c^( 
1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2)) + (e*g^2*log((b 
 + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2 
)) - (2*a*e*h^2*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^ 
3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x 
 + c*x^2)^(1/2))/(24*c^2)))/(5*c) - (5*b*d*h^2*((log((b + 2*c*x)/c^(1/2) + 
 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x 
^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (5*b*f* 
g^2*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c) 
)/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1 
/2))/(24*c^2)))/(8*c) + (e*g^2*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b* 
x + c*x^2)^(1/2))/(24*c^2) + (d*h^2*x*(a + b*x + c*x^2)^(3/2))/(4*c) + (f* 
g^2*x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*f*h^2*((5*b*((log((b + 2*c*x)/c^ 
(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*( 
a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c)...
 

Reduce [F]

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

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

Output:

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