\(\int \frac {\sqrt {a+b x+c x^2} (d+e x+f x^2)}{(g+h x)^3} \, dx\) [29]

Optimal result

Integrand size = 32, antiderivative size = 446 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^3} \, dx=-\frac {\left (11 b f g^2-b h (3 e g+d h)-\frac {4 c g^2 (3 f g-e h)}{h}-4 a h (3 f g-e h)+2 h \left (c e g+2 b f g-\frac {3 c f g^2}{h}-c d h-2 a f h\right ) x\right ) \sqrt {a+b x+c x^2}}{4 h^2 \left (c g^2-b g h+a h^2\right ) (g+h x)}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{2 h \left (c g^2-b g h+a h^2\right ) (g+h x)^2}-\frac {(6 c f g-2 c e h-b f h) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} h^4}+\frac {\left (8 c^2 g^3 (3 f g-e h)-4 c h \left (b g^2 (10 f g-3 e h)-a h \left (9 f g^2-3 e g h+d h^2\right )\right )+h^2 \left (8 a^2 f h^2-4 a b h (6 f g-e h)+b^2 \left (15 f g^2-h (3 e g+d h)\right )\right )\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{8 h^4 \left (c g^2-b g h+a h^2\right )^{3/2}} \] Output:

-1/4*(11*b*f*g^2-b*h*(d*h+3*e*g)-4*c*g^2*(-e*h+3*f*g)/h-4*a*h*(-e*h+3*f*g) 
+2*h*(c*e*g+2*b*f*g-3*c*f*g^2/h-c*d*h-2*a*f*h)*x)*(c*x^2+b*x+a)^(1/2)/h^2/ 
(a*h^2-b*g*h+c*g^2)/(h*x+g)-1/2*(f*g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(3/2)/h 
/(a*h^2-b*g*h+c*g^2)/(h*x+g)^2-1/2*(-b*f*h-2*c*e*h+6*c*f*g)*arctanh(1/2*(2 
*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(1/2)/h^4+1/8*(8*c^2*g^3*(-e*h+3*f* 
g)-4*c*h*(b*g^2*(-3*e*h+10*f*g)-a*h*(d*h^2-3*e*g*h+9*f*g^2))+h^2*(8*a^2*f* 
h^2-4*a*b*h*(-e*h+6*f*g)+b^2*(15*f*g^2-h*(d*h+3*e*g))))*arctanh(1/2*(b*g-2 
*a*h+(-b*h+2*c*g)*x)/(a*h^2-b*g*h+c*g^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/h^4/(a 
*h^2-b*g*h+c*g^2)^(3/2)
 

Mathematica [A] (verified)

Time = 11.47 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^3} \, dx=\frac {8 f \sqrt {a+x (b+c x)}+\frac {8 (2 f g-e h) \sqrt {a+x (b+c x)}}{g+h x}+\frac {2 h \left (f g^2+h (-e g+d h)\right ) \sqrt {a+x (b+c x)} (-2 a h+2 c g x+b (g-h x))}{\left (c g^2+h (-b g+a h)\right ) (g+h x)^2}+\frac {\left (-b^2+4 a c\right ) h \left (f g^2+h (-e g+d h)\right ) \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{\left (c g^2+h (-b g+a h)\right )^{3/2}}-\frac {4 (2 f g-e h) \left (2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-\frac {(2 c g-b h) \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c g^2+h (-b g+a h)}}\right )}{h}+\frac {4 f \left (\frac {(-2 c g+b h) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+2 \sqrt {c g^2+h (-b g+a h)} \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )\right )}{h}}{8 h^3} \] Input:

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

Output:

(8*f*Sqrt[a + x*(b + c*x)] + (8*(2*f*g - e*h)*Sqrt[a + x*(b + c*x)])/(g + 
h*x) + (2*h*(f*g^2 + h*(-(e*g) + d*h))*Sqrt[a + x*(b + c*x)]*(-2*a*h + 2*c 
*g*x + b*(g - h*x)))/((c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^2) + ((-b^2 + 4 
*a*c)*h*(f*g^2 + h*(-(e*g) + d*h))*ArcTanh[(-2*a*h + 2*c*g*x + b*(g - h*x) 
)/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/(c*g^2 + h*(- 
(b*g) + a*h))^(3/2) - (4*(2*f*g - e*h)*(2*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*S 
qrt[c]*Sqrt[a + x*(b + c*x)])] - ((2*c*g - b*h)*ArcTanh[(-2*a*h + 2*c*g*x 
+ b*(g - h*x))/(2*Sqrt[c*g^2 + h*(-(b*g) + a*h)]*Sqrt[a + x*(b + c*x)])])/ 
Sqrt[c*g^2 + h*(-(b*g) + a*h)]))/h + (4*f*(((-2*c*g + b*h)*ArcTanh[(b + 2* 
c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + 2*Sqrt[c*g^2 + h*(-(b*g 
) + a*h)]*ArcTanh[(-2*a*h + 2*c*g*x + b*(g - h*x))/(2*Sqrt[c*g^2 + h*(-(b* 
g) + a*h)]*Sqrt[a + x*(b + c*x)])]))/h)/(8*h^3)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2181, 27, 1230, 25, 27, 1269, 1092, 219, 1154, 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.

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1230

\(\displaystyle \frac {-\frac {\int -\frac {h \left (2 (2 b g-2 a h) \left (-\frac {3 c f g^2}{h}+c e g+2 b f g-c d h-2 a f h\right )+b \left (3 b f g^2-b h (3 e g+d h)+4 h (c d g-a f g+a e h)\right )\right )-4 (6 c f g-2 c e h-b f h) \left (c g^2-b h g+a h^2\right ) x}{h (g+h x) \sqrt {c x^2+b x+a}}dx}{2 h^2}-\frac {\sqrt {a+b x+c x^2} \left (2 h x \left (-2 a f h+2 b f g-c d h+c e g-\frac {3 c f g^2}{h}\right )-4 a h (3 f g-e h)-b h (d h+3 e g)+11 b f g^2-\frac {4 c g^2 (3 f g-e h)}{h}\right )}{h^2 (g+h x)}}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {h \left (2 (2 b g-2 a h) \left (-\frac {3 c f g^2}{h}+c e g+2 b f g-c d h-2 a f h\right )+b \left (3 b f g^2-b h (3 e g+d h)+4 h (c d g-a f g+a e h)\right )\right )-4 (6 c f g-2 c e h-b f h) \left (c g^2-b h g+a h^2\right ) x}{h (g+h x) \sqrt {c x^2+b x+a}}dx}{2 h^2}-\frac {\sqrt {a+b x+c x^2} \left (2 h x \left (-2 a f h+2 b f g-c d h+c e g-\frac {3 c f g^2}{h}\right )-4 a h (3 f g-e h)-b h (d h+3 e g)+11 b f g^2-\frac {4 c g^2 (3 f g-e h)}{h}\right )}{h^2 (g+h x)}}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {h \left (2 (2 b g-2 a h) \left (-\frac {3 c f g^2}{h}+c e g+2 b f g-c d h-2 a f h\right )+b \left (3 b f g^2-b h (3 e g+d h)+4 h (c d g-a f g+a e h)\right )\right )-4 (6 c f g-2 c e h-b f h) \left (c g^2-b h g+a h^2\right ) x}{(g+h x) \sqrt {c x^2+b x+a}}dx}{2 h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 h x \left (-2 a f h+2 b f g-c d h+c e g-\frac {3 c f g^2}{h}\right )-4 a h (3 f g-e h)-b h (d h+3 e g)+11 b f g^2-\frac {4 c g^2 (3 f g-e h)}{h}\right )}{h^2 (g+h x)}}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {\left (h^2 \left (8 a^2 f h^2-4 a b h (6 f g-e h)+b^2 \left (15 f g^2-h (d h+3 e g)\right )\right )-4 c h \left (b g^2 (10 f g-3 e h)-a h \left (d h^2-3 e g h+9 f g^2\right )\right )+8 c^2 g^3 (3 f g-e h)\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}-\frac {4 \left (a h^2-b g h+c g^2\right ) (-b f h-2 c e h+6 c f g) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{h}}{2 h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 h x \left (-2 a f h+2 b f g-c d h+c e g-\frac {3 c f g^2}{h}\right )-4 a h (3 f g-e h)-b h (d h+3 e g)+11 b f g^2-\frac {4 c g^2 (3 f g-e h)}{h}\right )}{h^2 (g+h x)}}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\left (h^2 \left (8 a^2 f h^2-4 a b h (6 f g-e h)+b^2 \left (15 f g^2-h (d h+3 e g)\right )\right )-4 c h \left (b g^2 (10 f g-3 e h)-a h \left (d h^2-3 e g h+9 f g^2\right )\right )+8 c^2 g^3 (3 f g-e h)\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}-\frac {8 \left (a h^2-b g h+c g^2\right ) (-b f h-2 c e h+6 c f g) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{h}}{2 h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 h x \left (-2 a f h+2 b f g-c d h+c e g-\frac {3 c f g^2}{h}\right )-4 a h (3 f g-e h)-b h (d h+3 e g)+11 b f g^2-\frac {4 c g^2 (3 f g-e h)}{h}\right )}{h^2 (g+h x)}}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\left (h^2 \left (8 a^2 f h^2-4 a b h (6 f g-e h)+b^2 \left (15 f g^2-h (d h+3 e g)\right )\right )-4 c h \left (b g^2 (10 f g-3 e h)-a h \left (d h^2-3 e g h+9 f g^2\right )\right )+8 c^2 g^3 (3 f g-e h)\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}-\frac {4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a h^2-b g h+c g^2\right ) (-b f h-2 c e h+6 c f g)}{\sqrt {c} h}}{2 h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 h x \left (-2 a f h+2 b f g-c d h+c e g-\frac {3 c f g^2}{h}\right )-4 a h (3 f g-e h)-b h (d h+3 e g)+11 b f g^2-\frac {4 c g^2 (3 f g-e h)}{h}\right )}{h^2 (g+h x)}}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {-\frac {2 \left (h^2 \left (8 a^2 f h^2-4 a b h (6 f g-e h)+b^2 \left (15 f g^2-h (d h+3 e g)\right )\right )-4 c h \left (b g^2 (10 f g-3 e h)-a h \left (d h^2-3 e g h+9 f g^2\right )\right )+8 c^2 g^3 (3 f g-e h)\right ) \int \frac {1}{4 \left (c g^2-b h g+a h^2\right )-\frac {(b g-2 a h+(2 c g-b h) x)^2}{c x^2+b x+a}}d\left (-\frac {b g-2 a h+(2 c g-b h) x}{\sqrt {c x^2+b x+a}}\right )}{h}-\frac {4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a h^2-b g h+c g^2\right ) (-b f h-2 c e h+6 c f g)}{\sqrt {c} h}}{2 h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 h x \left (-2 a f h+2 b f g-c d h+c e g-\frac {3 c f g^2}{h}\right )-4 a h (3 f g-e h)-b h (d h+3 e g)+11 b f g^2-\frac {4 c g^2 (3 f g-e h)}{h}\right )}{h^2 (g+h x)}}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right ) \left (h^2 \left (8 a^2 f h^2-4 a b h (6 f g-e h)+b^2 \left (15 f g^2-h (d h+3 e g)\right )\right )-4 c h \left (b g^2 (10 f g-3 e h)-a h \left (d h^2-3 e g h+9 f g^2\right )\right )+8 c^2 g^3 (3 f g-e h)\right )}{h \sqrt {a h^2-b g h+c g^2}}-\frac {4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a h^2-b g h+c g^2\right ) (-b f h-2 c e h+6 c f g)}{\sqrt {c} h}}{2 h^3}-\frac {\sqrt {a+b x+c x^2} \left (2 h x \left (-2 a f h+2 b f g-c d h+c e g-\frac {3 c f g^2}{h}\right )-4 a h (3 f g-e h)-b h (d h+3 e g)+11 b f g^2-\frac {4 c g^2 (3 f g-e h)}{h}\right )}{h^2 (g+h x)}}{4 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{2 h (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\)

Input:

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

Output:

-1/2*((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(3/2))/(h*(c*g^2 - b*g*h + 
 a*h^2)*(g + h*x)^2) + (-(((11*b*f*g^2 - b*h*(3*e*g + d*h) - (4*c*g^2*(3*f 
*g - e*h))/h - 4*a*h*(3*f*g - e*h) + 2*h*(c*e*g + 2*b*f*g - (3*c*f*g^2)/h 
- c*d*h - 2*a*f*h)*x)*Sqrt[a + b*x + c*x^2])/(h^2*(g + h*x))) + ((-4*(6*c* 
f*g - 2*c*e*h - b*f*h)*(c*g^2 - b*g*h + a*h^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt 
[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*h) + ((8*c^2*g^3*(3*f*g - e*h) - 4*c 
*h*(b*g^2*(10*f*g - 3*e*h) - a*h*(9*f*g^2 - 3*e*g*h + d*h^2)) + h^2*(8*a^2 
*f*h^2 - 4*a*b*h*(6*f*g - e*h) + b^2*(15*f*g^2 - h*(3*e*g + d*h))))*ArcTan 
h[(b*g - 2*a*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + 
b*x + c*x^2])])/(h*Sqrt[c*g^2 - b*g*h + a*h^2]))/(2*h^3))/(4*(c*g^2 - b*g* 
h + a*h^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

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

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) 
^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - 
b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1229\) vs. \(2(418)=836\).

Time = 0.46 (sec) , antiderivative size = 1230, normalized size of antiderivative = 2.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(1230\)
default	\(\text {Expression too large to display}\)	\(2166\)

Input:

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

Output:

f/h^3*(c*x^2+b*x+a)^(1/2)+1/2/h^3*((b*f*h+2*c*e*h-6*c*f*g)/h*ln((1/2*b+c*x 
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-2/h^2*(a*f*h^2+b*e*h^2-3*b*f*g*h+c* 
d*h^2-3*c*e*g*h+6*c*f*g^2)/((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b* 
g*h+c*g^2)/h^2+(b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x 
+g/h)^2*c+(b*h-2*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h))+2 
/h^3*(a*e*h^3-2*a*f*g*h^2+b*d*h^3-2*b*e*g*h^2+3*b*f*g^2*h-2*c*d*g*h^2+3*c* 
e*g^2*h-4*c*f*g^3)*(-1/(a*h^2-b*g*h+c*g^2)*h^2/(x+g/h)*((x+g/h)^2*c+(b*h-2 
*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)+1/2*(b*h-2*c*g)*h/(a*h^2-b* 
g*h+c*g^2)/((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+( 
b*h-2*c*g)/h*(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c+(b*h-2 
*c*g)/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h)))+2*(a*d*h^4-a*e*g 
*h^3+a*f*g^2*h^2-b*d*g*h^3+b*e*g^2*h^2-b*f*g^3*h+c*d*g^2*h^2-c*e*g^3*h+c*f 
*g^4)/h^4*(-1/2/(a*h^2-b*g*h+c*g^2)*h^2/(x+g/h)^2*((x+g/h)^2*c+(b*h-2*c*g) 
/h*(x+g/h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)-3/4*(b*h-2*c*g)*h/(a*h^2-b*g*h+c 
*g^2)*(-1/(a*h^2-b*g*h+c*g^2)*h^2/(x+g/h)*((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/ 
h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2)+1/2*(b*h-2*c*g)*h/(a*h^2-b*g*h+c*g^2)/(( 
a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c*g)/h* 
(x+g/h)+2*((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*((x+g/h)^2*c+(b*h-2*c*g)/h*(x+g/ 
h)+(a*h^2-b*g*h+c*g^2)/h^2)^(1/2))/(x+g/h)))+1/2*c/(a*h^2-b*g*h+c*g^2)*h^2 
/((a*h^2-b*g*h+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2-b*g*h+c*g^2)/h^2+(b*h-2*c...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*x + c*x**2)*(d + e*x + f*x**2)/(g + h*x)**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)/(h*x+g)^3,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(a*h^2-b*g*h>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2364 vs. \(2 (418) = 836\).

Time = 0.46 (sec) , antiderivative size = 2364, normalized size of antiderivative = 5.30 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(24*c^2*f*g^4 - 8*c^2*e*g^3*h - 40*b*c*f*g^3*h + 12*b*c*e*g^2*h^2 + 15 
*b^2*f*g^2*h^2 + 36*a*c*f*g^2*h^2 - 3*b^2*e*g*h^3 - 12*a*c*e*g*h^3 - 24*a* 
b*f*g*h^3 - b^2*d*h^4 + 4*a*c*d*h^4 + 4*a*b*e*h^4 + 8*a^2*f*h^4)*arctan(-( 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*h + sqrt(c)*g)/sqrt(-c*g^2 + b*g*h - a 
*h^2))/((c*g^2*h^4 - b*g*h^5 + a*h^6)*sqrt(-c*g^2 + b*g*h - a*h^2)) + sqrt 
(c*x^2 + b*x + a)*f/h^3 + 1/4*(24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^ 
2*f*g^4*h - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*e*g^3*h^2 - 32*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*f*g^3*h^2 + 8*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^3*c^2*d*g^2*h^3 + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3 
*b*c*e*g^2*h^3 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*f*g^2*h^3 + 2 
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*f*g^2*h^3 - 8*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^3*b*c*d*g*h^4 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
3*b^2*e*g*h^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*e*g*h^4 - 8*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*f*g*h^4 + (sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^3*b^2*d*h^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*d*h 
^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*e*h^5 + 40*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))^2*c^(5/2)*f*g^5 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^2*c^(5/2)*e*g^4*h - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2 
)*f*g^4*h + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d*g^3*h^2 + 20 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*e*g^3*h^2 + 3*(sqrt(c)...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 7427, normalized size of antiderivative = 16.65 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*sqrt(a*h**2 - b*g*h + c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 
- b*g*h + c*g**2) - 2*a*h + b*g - b*h*x + 2*c*g*x)*a**2*c*f*g**2*h**4 + 16 
*sqrt(a*h**2 - b*g*h + c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - 
b*g*h + c*g**2) - 2*a*h + b*g - b*h*x + 2*c*g*x)*a**2*c*f*g*h**5*x + 8*sqr 
t(a*h**2 - b*g*h + c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - b*g* 
h + c*g**2) - 2*a*h + b*g - b*h*x + 2*c*g*x)*a**2*c*f*h**6*x**2 + 4*sqrt(a 
*h**2 - b*g*h + c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - b*g*h + 
 c*g**2) - 2*a*h + b*g - b*h*x + 2*c*g*x)*a*b*c*e*g**2*h**4 + 8*sqrt(a*h** 
2 - b*g*h + c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - b*g*h + c*g 
**2) - 2*a*h + b*g - b*h*x + 2*c*g*x)*a*b*c*e*g*h**5*x + 4*sqrt(a*h**2 - b 
*g*h + c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - b*g*h + c*g**2) 
- 2*a*h + b*g - b*h*x + 2*c*g*x)*a*b*c*e*h**6*x**2 - 24*sqrt(a*h**2 - b*g* 
h + c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - b*g*h + c*g**2) - 2 
*a*h + b*g - b*h*x + 2*c*g*x)*a*b*c*f*g**3*h**3 - 48*sqrt(a*h**2 - b*g*h + 
 c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - b*g*h + c*g**2) - 2*a* 
h + b*g - b*h*x + 2*c*g*x)*a*b*c*f*g**2*h**4*x - 24*sqrt(a*h**2 - b*g*h + 
c*g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - b*g*h + c*g**2) - 2*a*h 
 + b*g - b*h*x + 2*c*g*x)*a*b*c*f*g*h**5*x**2 + 4*sqrt(a*h**2 - b*g*h + c* 
g**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*h**2 - b*g*h + c*g**2) - 2*a*h + 
 b*g - b*h*x + 2*c*g*x)*a*c**2*d*g**2*h**4 + 8*sqrt(a*h**2 - b*g*h + c*...