\(\int \frac {\sqrt {a+b x+c x^2} (d+e x+f x^2)}{(g+h x)^4} \, dx\) [30]

Optimal result

Integrand size = 32, antiderivative size = 485 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx=-\frac {f \sqrt {a+b x+c x^2}}{h^3 (g+h x)}-\frac {\left (2 c \left (f g^3-d g h^2\right )-h \left (3 b f g^2-b h (e g+d h)-2 a h (2 f g-e h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{8 h^2 \left (c g^2-b g h+a h^2\right )^2 (g+h x)^2}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 h \left (c g^2-b g h+a h^2\right ) (g+h x)^3}+\frac {\sqrt {c} f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{h^4}-\frac {\left (16 c^3 f g^5-8 c^2 g h \left (5 b f g^3-5 a f g^2 h+a d h^3\right )-b h^3 \left (8 a^2 f h^2-2 a b h (6 f g+e h)+b^2 \left (5 f g^2+e g h+d h^2\right )\right )+2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (15 f g^2-e g h-d h^2\right )+b^2 \left (15 f g^3+d g h^2\right )\right )\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{16 h^4 \left (c g^2-b g h+a h^2\right )^{5/2}} \] Output:

-f*(c*x^2+b*x+a)^(1/2)/h^3/(h*x+g)-1/8*(2*c*(-d*g*h^2+f*g^3)-h*(3*b*f*g^2- 
b*h*(d*h+e*g)-2*a*h*(-e*h+2*f*g)))*(b*g-2*a*h+(-b*h+2*c*g)*x)*(c*x^2+b*x+a 
)^(1/2)/h^2/(a*h^2-b*g*h+c*g^2)^2/(h*x+g)^2-1/3*(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)^3+c^(1/2)*f*arctanh(1/2*(2*c* 
x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/h^4-1/16*(16*c^3*f*g^5-8*c^2*g*h*(a*d*h^ 
3-5*a*f*g^2*h+5*b*f*g^3)-b*h^3*(8*a^2*f*h^2-2*a*b*h*(e*h+6*f*g)+b^2*(d*h^2 
+e*g*h+5*f*g^2))+2*c*h^2*(4*a^2*h^2*(-e*h+4*f*g)-2*a*b*h*(-d*h^2-e*g*h+15* 
f*g^2)+b^2*(d*g*h^2+15*f*g^3)))*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)^(5/2)
 

Mathematica [A] (verified)

Time = 12.27 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx=\frac {-\frac {48 f \sqrt {a+x (b+c x)}}{g+h x}-\frac {16 h^2 \left (f g^2+h (-e g+d h)\right ) (a+x (b+c x))^{3/2}}{\left (c g^2+h (-b g+a h)\right ) (g+h x)^3}+\frac {12 h (-2 f g+e h) \sqrt {a+x (b+c x)} (-2 a h+2 c g x+b (g-h x))}{\left (c g^2+h (-b g+a h)\right ) (g+h x)^2}-\frac {6 \left (b^2-4 a c\right ) h (-2 f g+e h) \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{\left (c g^2+h (-b g+a h)\right )^{3/2}}+\frac {3 h (2 c g-b h) \left (f g^2+h (-e g+d h)\right ) \left (\frac {2 \sqrt {a+x (b+c x)} (-2 a h+2 c g x+b (g-h x))}{\left (c g^2+h (-b g+a h)\right ) (g+h x)^2}+\frac {\left (-b^2+4 a c\right ) \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{\left (c g^2+h (-b g+a h)\right )^{3/2}}\right )}{c g^2+h (-b g+a h)}+\frac {24 f \left (2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-\frac {(2 c g-b h) \text {arctanh}\left (\frac {-2 a h+2 c g x+b (g-h x)}{2 \sqrt {c g^2+h (-b g+a h)} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c g^2+h (-b g+a h)}}\right )}{h}}{48 h^3} \] Input:

Integrate[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^4,x]
 

Output:

((-48*f*Sqrt[a + x*(b + c*x)])/(g + h*x) - (16*h^2*(f*g^2 + h*(-(e*g) + d* 
h))*(a + x*(b + c*x))^(3/2))/((c*g^2 + h*(-(b*g) + a*h))*(g + h*x)^3) + (1 
2*h*(-2*f*g + e*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) - (6*(b^2 - 4*a*c)*h*(-2*f*g + e 
*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) + (3*h*(2*c 
*g - b*h)*(f*g^2 + h*(-(e*g) + d*h))*((2*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)*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)))/(c*g^ 
2 + h*(-(b*g) + a*h)) + (24*f*(2*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sq 
rt[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)/(48*h^3)
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 645, normalized size of antiderivative = 1.33, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2181, 27, 1229, 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.

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1229

\(\displaystyle \frac {-\frac {\int -\frac {16 c f x \left (c g^2-b h g+a h^2\right )^2+h \left (\frac {8 b c^2 f g^4}{h}+8 a^2 b f h^3+4 a b c h \left (7 f g^2-h (e g+d h)\right )+b^3 h \left (5 f g^2+h (e g+d h)\right )-8 a c \left (a (2 f g-e h) h^2+c \left (f g^3-d g h^2\right )\right )-2 b^2 \left (a (6 f g+e h) h^2+c \left (7 f g^3+d h^2 g\right )\right )\right )}{2 h (g+h x) \sqrt {c x^2+b x+a}}dx}{4 h^2 \left (a h^2-b g h+c g^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 a^2 e h^4-2 b \left (a h^2 \left (d h^2+2 e g h+3 f g^2\right )+c \left (d g^2 h^2+7 f g^4\right )\right )+x \left ((2 c g-b h) \left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right )+8 f \left (c g^2-h (b g-a h)\right )^2\right )+4 a c g h \left (d h^2+3 f g^2\right )+b^2 g h \left (h (d h+e g)+5 f g^2\right )+\frac {8 c^2 f g^5}{h}\right )}{4 h^2 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {16 c f x \left (c g^2-b h g+a h^2\right )^2+h \left (\frac {8 b c^2 f g^4}{h}+8 a^2 b f h^3+4 a b c h \left (7 f g^2-h (e g+d h)\right )+b^3 h \left (5 f g^2+h (e g+d h)\right )-8 a c \left (a (2 f g-e h) h^2+c \left (f g^3-d g h^2\right )\right )-2 b^2 \left (a (6 f g+e h) h^2+c \left (7 f g^3+d h^2 g\right )\right )\right )}{(g+h x) \sqrt {c x^2+b x+a}}dx}{8 h^3 \left (a h^2-b g h+c g^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 a^2 e h^4-2 b \left (a h^2 \left (d h^2+2 e g h+3 f g^2\right )+c \left (d g^2 h^2+7 f g^4\right )\right )+x \left ((2 c g-b h) \left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right )+8 f \left (c g^2-h (b g-a h)\right )^2\right )+4 a c g h \left (d h^2+3 f g^2\right )+b^2 g h \left (h (d h+e g)+5 f g^2\right )+\frac {8 c^2 f g^5}{h}\right )}{4 h^2 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {16 c f \left (a h^2-b g h+c g^2\right )^2 \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{h}-\frac {\left (2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (-d h^2-e g h+15 f g^2\right )+b^2 \left (d g h^2+15 f g^3\right )\right )-b h^3 \left (8 a^2 f h^2-2 a b h (e h+6 f g)+b^2 \left (d h^2+e g h+5 f g^2\right )\right )-8 c^2 g h \left (a d h^3-5 a f g^2 h+5 b f g^3\right )+16 c^3 f g^5\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}}{8 h^3 \left (a h^2-b g h+c g^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 a^2 e h^4-2 b \left (a h^2 \left (d h^2+2 e g h+3 f g^2\right )+c \left (d g^2 h^2+7 f g^4\right )\right )+x \left ((2 c g-b h) \left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right )+8 f \left (c g^2-h (b g-a h)\right )^2\right )+4 a c g h \left (d h^2+3 f g^2\right )+b^2 g h \left (h (d h+e g)+5 f g^2\right )+\frac {8 c^2 f g^5}{h}\right )}{4 h^2 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {32 c f \left (a h^2-b g h+c g^2\right )^2 \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{h}-\frac {\left (2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (-d h^2-e g h+15 f g^2\right )+b^2 \left (d g h^2+15 f g^3\right )\right )-b h^3 \left (8 a^2 f h^2-2 a b h (e h+6 f g)+b^2 \left (d h^2+e g h+5 f g^2\right )\right )-8 c^2 g h \left (a d h^3-5 a f g^2 h+5 b f g^3\right )+16 c^3 f g^5\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}}{8 h^3 \left (a h^2-b g h+c g^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 a^2 e h^4-2 b \left (a h^2 \left (d h^2+2 e g h+3 f g^2\right )+c \left (d g^2 h^2+7 f g^4\right )\right )+x \left ((2 c g-b h) \left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right )+8 f \left (c g^2-h (b g-a h)\right )^2\right )+4 a c g h \left (d h^2+3 f g^2\right )+b^2 g h \left (h (d h+e g)+5 f g^2\right )+\frac {8 c^2 f g^5}{h}\right )}{4 h^2 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {16 \sqrt {c} f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a h^2-b g h+c g^2\right )^2}{h}-\frac {\left (2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (-d h^2-e g h+15 f g^2\right )+b^2 \left (d g h^2+15 f g^3\right )\right )-b h^3 \left (8 a^2 f h^2-2 a b h (e h+6 f g)+b^2 \left (d h^2+e g h+5 f g^2\right )\right )-8 c^2 g h \left (a d h^3-5 a f g^2 h+5 b f g^3\right )+16 c^3 f g^5\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{h}}{8 h^3 \left (a h^2-b g h+c g^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 a^2 e h^4-2 b \left (a h^2 \left (d h^2+2 e g h+3 f g^2\right )+c \left (d g^2 h^2+7 f g^4\right )\right )+x \left ((2 c g-b h) \left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right )+8 f \left (c g^2-h (b g-a h)\right )^2\right )+4 a c g h \left (d h^2+3 f g^2\right )+b^2 g h \left (h (d h+e g)+5 f g^2\right )+\frac {8 c^2 f g^5}{h}\right )}{4 h^2 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {2 \left (2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (-d h^2-e g h+15 f g^2\right )+b^2 \left (d g h^2+15 f g^3\right )\right )-b h^3 \left (8 a^2 f h^2-2 a b h (e h+6 f g)+b^2 \left (d h^2+e g h+5 f g^2\right )\right )-8 c^2 g h \left (a d h^3-5 a f g^2 h+5 b f g^3\right )+16 c^3 f g^5\right ) \int \frac {1}{4 \left (c g^2-b h g+a h^2\right )-\frac {(b g-2 a h+(2 c g-b h) x)^2}{c x^2+b x+a}}d\left (-\frac {b g-2 a h+(2 c g-b h) x}{\sqrt {c x^2+b x+a}}\right )}{h}+\frac {16 \sqrt {c} f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a h^2-b g h+c g^2\right )^2}{h}}{8 h^3 \left (a h^2-b g h+c g^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 a^2 e h^4-2 b \left (a h^2 \left (d h^2+2 e g h+3 f g^2\right )+c \left (d g^2 h^2+7 f g^4\right )\right )+x \left ((2 c g-b h) \left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right )+8 f \left (c g^2-h (b g-a h)\right )^2\right )+4 a c g h \left (d h^2+3 f g^2\right )+b^2 g h \left (h (d h+e g)+5 f g^2\right )+\frac {8 c^2 f g^5}{h}\right )}{4 h^2 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {16 \sqrt {c} f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (a h^2-b g h+c g^2\right )^2}{h}-\frac {\text {arctanh}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right ) \left (2 c h^2 \left (4 a^2 h^2 (4 f g-e h)-2 a b h \left (-d h^2-e g h+15 f g^2\right )+b^2 \left (d g h^2+15 f g^3\right )\right )-b h^3 \left (8 a^2 f h^2-2 a b h (e h+6 f g)+b^2 \left (d h^2+e g h+5 f g^2\right )\right )-8 c^2 g h \left (a d h^3-5 a f g^2 h+5 b f g^3\right )+16 c^3 f g^5\right )}{h \sqrt {a h^2-b g h+c g^2}}}{8 h^3 \left (a h^2-b g h+c g^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 a^2 e h^4-2 b \left (a h^2 \left (d h^2+2 e g h+3 f g^2\right )+c \left (d g^2 h^2+7 f g^4\right )\right )+x \left ((2 c g-b h) \left (2 c \left (f g^3-d g h^2\right )-h \left (-2 a h (2 f g-e h)-b h (d h+e g)+3 b f g^2\right )\right )+8 f \left (c g^2-h (b g-a h)\right )^2\right )+4 a c g h \left (d h^2+3 f g^2\right )+b^2 g h \left (h (d h+e g)+5 f g^2\right )+\frac {8 c^2 f g^5}{h}\right )}{4 h^2 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}}{2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (f g^2-h (e g-d h)\right )}{3 h (g+h x)^3 \left (a h^2-b g h+c g^2\right )}\)

Input:

Int[(Sqrt[a + b*x + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^4,x]
 

Output:

-1/3*((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)^3) + (-1/4*(((8*c^2*f*g^5)/h + 4*a^2*e*h^4 + 4*a*c*g*h*( 
3*f*g^2 + d*h^2) + b^2*g*h*(5*f*g^2 + h*(e*g + d*h)) - 2*b*(a*h^2*(3*f*g^2 
 + 2*e*g*h + d*h^2) + c*(7*f*g^4 + d*g^2*h^2)) + (8*f*(c*g^2 - h*(b*g - a* 
h))^2 + (2*c*g - b*h)*(2*c*(f*g^3 - d*g*h^2) - h*(3*b*f*g^2 - b*h*(e*g + d 
*h) - 2*a*h*(2*f*g - e*h))))*x)*Sqrt[a + b*x + c*x^2])/(h^2*(c*g^2 - b*g*h 
 + a*h^2)*(g + h*x)^2) + ((16*Sqrt[c]*f*(c*g^2 - b*g*h + a*h^2)^2*ArcTanh[ 
(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/h - ((16*c^3*f*g^5 - 8*c^2 
*g*h*(5*b*f*g^3 - 5*a*f*g^2*h + a*d*h^3) - b*h^3*(8*a^2*f*h^2 - 2*a*b*h*(6 
*f*g + e*h) + b^2*(5*f*g^2 + e*g*h + d*h^2)) + 2*c*h^2*(4*a^2*h^2*(4*f*g - 
 e*h) - 2*a*b*h*(15*f*g^2 - e*g*h - d*h^2) + b^2*(15*f*g^3 + d*g*h^2)))*Ar 
cTanh[(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]))/(8*h^3*(c*g^2 - b*g*h 
 + a*h^2)))/(2*(c*g^2 - b*g*h + a*h^2))
 

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 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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

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. \(3089\) vs. \(2(457)=914\).

Time = 0.52 (sec) , antiderivative size = 3090, normalized size of antiderivative = 6.37

method	result	size
default	\(\text {Expression too large to display}\)	\(3090\)

Input:

int((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)/(h*x+g)^4,x,method=_RETURNVERBOSE)
 

Output:

f/h^4*(-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)^(3/2)+1/2*(b*h-2*c*g)*h/(a*h^2-b*g*h+c*g^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)+1/2*(b*h- 
2*c*g)/h*ln((1/2*(b*h-2*c*g)/h+c*(x+g/h))/c^(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))/c^(1/2)-(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*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*c/(a*h^2-b*g*h+c*g^2)*h^ 
2*(1/4*(2*c*(x+g/h)+(b*h-2*c*g)/h)/c*((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)+1/8*(4*c*(a*h^2-b*g*h+c*g^2)/h^2-(b*h-2*c*g)^ 
2/h^2)/c^(3/2)*ln((1/2*(b*h-2*c*g)/h+c*(x+g/h))/c^(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))))+(e*h-2*f*g)/h^5*(-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)^(3/2)-1/4*(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)^(3/2)+1/2*(b*h-2*c*g)*h/(a*h^2-b*g*h+c*g^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)+1/2*(b*h-2*c*g)/h*ln((1/2* 
(b*h-2*c*g)/h+c*(x+g/h))/c^(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))/c^(1/2)-(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*g)/h*(x+g/h)+2*(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)/(h*x+g)^4,x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx=\int \frac {\sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{4}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(1/2)*(f*x**2+e*x+d)/(h*x+g)**4,x)
 

Output:

Integral(sqrt(a + b*x + c*x**2)*(d + e*x + f*x**2)/(g + h*x)**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)/(h*x+g)^4,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. 6846 vs. \(2 (457) = 914\).

Time = 4.09 (sec) , antiderivative size = 6846, normalized size of antiderivative = 14.12 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)/(h*x+g)^4,x, algorithm="giac")
 

Output:

-1/8*(16*c^3*f*g^5 - 40*b*c^2*f*g^4*h + 30*b^2*c*f*g^3*h^2 + 40*a*c^2*f*g^ 
3*h^2 - 5*b^3*f*g^2*h^3 - 60*a*b*c*f*g^2*h^3 + 2*b^2*c*d*g*h^4 - 8*a*c^2*d 
*g*h^4 - b^3*e*g*h^4 + 4*a*b*c*e*g*h^4 + 12*a*b^2*f*g*h^4 + 32*a^2*c*f*g*h 
^4 - b^3*d*h^5 + 4*a*b*c*d*h^5 + 2*a*b^2*e*h^5 - 8*a^2*c*e*h^5 - 8*a^2*b*f 
*h^5)*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^2*g^4*h^4 - 2*b*c*g^3*h^5 + b^2*g^2*h^6 + 2*a*c* 
g^2*h^6 - 2*a*b*g*h^7 + a^2*h^8)*sqrt(-c*g^2 + b*g*h - a*h^2)) - sqrt(c)*f 
*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/h^4 - 1/24*( 
144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^3*f*g^5*h^2 - 48*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^5*c^3*e*g^4*h^3 - 312*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^5*b*c^2*f*g^4*h^3 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^2* 
e*g^3*h^4 + 198*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*f*g^3*h^4 + 26 
4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^2*f*g^3*h^4 - 48*(sqrt(c)*x - 
sqrt(c*x^2 + b*x + a))^5*b^2*c*e*g^2*h^5 - 96*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^5*a*c^2*e*g^2*h^5 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*f 
*g^2*h^5 - 300*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*f*g^2*h^5 - 6*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c*d*g*h^6 + 24*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^5*a*c^2*d*g*h^6 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
5*b^3*e*g*h^6 + 84*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c*e*g*h^6 + 6 
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*f*g*h^6 + 96*(sqrt(c)*x -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^4} \,d x \] Input:

int(((a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2))/(g + h*x)^4,x)
 

Output:

int(((a + b*x + c*x^2)^(1/2)*(d + e*x + f*x^2))/(g + h*x)^4, x)
 

Reduce [B] (verification not implemented)

Time = 1.20 (sec) , antiderivative size = 11991, normalized size of antiderivative = 24.72 \[ \int \frac {\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(1/2)*(f*x^2+e*x+d)/(h*x+g)^4,x)
 

Output:

(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**2*b*f*g**3*h**5 + 7 
2*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*b*f*g**2*h**6*x + 7 
2*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*b*f*g*h**7*x**2 + 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**2*b*f*h**8*x**3 + 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**2*c*e*g**3*h**5 + 72*sq 
rt(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*e*g**2*h**6*x + 72*sq 
rt(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*e*g*h**7*x**2 + 24*sq 
rt(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*e*h**8*x**3 - 96*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**4*h**4 - 288*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**3*h**5*x - 288*s...