\(\int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx\) [879]

Optimal result

Integrand size = 27, antiderivative size = 608 \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\frac {\left (32 c^3 d^3 f-b^2 e^2 (7 b e f+3 b d g-10 a e g)-8 c^2 d (3 a e (e f-2 d g)+2 b d (3 e f+d g))-2 c e \left (4 a^2 e^2 g-6 a b e (e f-3 d g)-3 b^2 d (5 e f+2 d g)\right )\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}-\frac {(e f-d g) \left (a+b x+c x^2\right )^{3/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {(2 c d (7 e f-2 d g)-e (7 b e f+3 b d g-10 a e g)) \left (a+b x+c x^2\right )^{3/2}}{40 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac {\left (4 c^2 d^2 (27 e f-2 d g)+5 b e^2 (7 b e f+3 b d g-10 a e g)-2 c e (2 a e (8 e f-33 d g)+3 b d (18 e f+7 d g))\right ) \left (a+b x+c x^2\right )^{3/2}}{240 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) \left (32 c^3 d^3 f-b^2 e^2 (7 b e f+3 b d g-10 a e g)-8 c^2 d (3 a e (e f-2 d g)+2 b d (3 e f+d g))-2 c e \left (4 a^2 e^2 g-6 a b e (e f-3 d g)-3 b^2 d (5 e f+2 d g)\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{256 \left (c d^2-b d e+a e^2\right )^{9/2}} \] Output:

1/128*(32*c^3*d^3*f-b^2*e^2*(-10*a*e*g+3*b*d*g+7*b*e*f)-8*c^2*d*(3*a*e*(-2 
*d*g+e*f)+2*b*d*(d*g+3*e*f))-2*c*e*(4*a^2*e^2*g-6*a*b*e*(-3*d*g+e*f)-3*b^2 
*d*(2*d*g+5*e*f)))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b 
*d*e+c*d^2)^4/(e*x+d)^2-1/5*(-d*g+e*f)*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c* 
d^2)/(e*x+d)^5-1/40*(2*c*d*(-2*d*g+7*e*f)-e*(-10*a*e*g+3*b*d*g+7*b*e*f))*( 
c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^4-1/240*(4*c^2*d^2*(-2*d* 
g+27*e*f)+5*b*e^2*(-10*a*e*g+3*b*d*g+7*b*e*f)-2*c*e*(2*a*e*(-33*d*g+8*e*f) 
+3*b*d*(7*d*g+18*e*f)))*(c*x^2+b*x+a)^(3/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^ 
3-1/256*(-4*a*c+b^2)*(32*c^3*d^3*f-b^2*e^2*(-10*a*e*g+3*b*d*g+7*b*e*f)-8*c 
^2*d*(3*a*e*(-2*d*g+e*f)+2*b*d*(d*g+3*e*f))-2*c*e*(4*a^2*e^2*g-6*a*b*e*(-3 
*d*g+e*f)-3*b^2*d*(2*d*g+5*e*f)))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/( 
a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(9/2)
 

Mathematica [A] (verified)

Time = 16.18 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.28 \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=-\frac {(e f-d g) \left (a+b x+c x^2\right ) \sqrt {a+x (b+c x)}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac {\sqrt {a+x (b+c x)} \left (-\frac {\left (-2 c d (e f-d g)+\frac {1}{2} e (-10 c d f-10 a e g+b (7 e f+3 d g))\right ) \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {\frac {\left (-\frac {1}{2} c d (2 c d (7 e f-2 d g)-e (7 b e f+3 b d g-10 a e g))-\frac {1}{4} e \left (80 c^2 d^2 f+5 b e (7 b e f+3 b d g-10 a e g)-2 c (8 a e (2 e f-7 d g)+b d (47 e f+18 d g))\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\left (-2 \left (-\frac {1}{2} a c e (2 c d (7 e f-2 d g)-e (7 b e f+3 b d g-10 a e g))+\frac {1}{4} c d \left (80 c^2 d^2 f+5 b e (7 b e f+3 b d g-10 a e g)-2 c (8 a e (2 e f-7 d g)+b d (47 e f+18 d g))\right )\right )+b \left (-\frac {1}{2} c d (2 c d (7 e f-2 d g)-e (7 b e f+3 b d g-10 a e g))+\frac {1}{4} e \left (80 c^2 d^2 f+5 b e (7 b e f+3 b d g-10 a e g)-2 c (8 a e (2 e f-7 d g)+b d (47 e f+18 d g))\right )\right )\right ) \left (\frac {(b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c d^2-b d e+a e^2} \left (4 c d^2-4 b d e+4 a e^2\right )}\right )}{2 \left (c d^2-b d e+a e^2\right )}}{4 \left (c d^2-b d e+a e^2\right )}\right )}{5 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}} \] Input:

Integrate[((f + g*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^6,x]
 

Output:

-1/5*((e*f - d*g)*(a + b*x + c*x^2)*Sqrt[a + x*(b + c*x)])/((c*d^2 - b*d*e 
 + a*e^2)*(d + e*x)^5) - (Sqrt[a + x*(b + c*x)]*(-1/4*((-2*c*d*(e*f - d*g) 
 + (e*(-10*c*d*f - 10*a*e*g + b*(7*e*f + 3*d*g)))/2)*(a + b*x + c*x^2)^(3/ 
2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (((-1/2*(c*d*(2*c*d*(7*e*f - 2 
*d*g) - e*(7*b*e*f + 3*b*d*g - 10*a*e*g))) - (e*(80*c^2*d^2*f + 5*b*e*(7*b 
*e*f + 3*b*d*g - 10*a*e*g) - 2*c*(8*a*e*(2*e*f - 7*d*g) + b*d*(47*e*f + 18 
*d*g))))/4)*(a + b*x + c*x^2)^(3/2))/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^ 
3) - ((-2*(-1/2*(a*c*e*(2*c*d*(7*e*f - 2*d*g) - e*(7*b*e*f + 3*b*d*g - 10* 
a*e*g))) + (c*d*(80*c^2*d^2*f + 5*b*e*(7*b*e*f + 3*b*d*g - 10*a*e*g) - 2*c 
*(8*a*e*(2*e*f - 7*d*g) + b*d*(47*e*f + 18*d*g))))/4) + b*(-1/2*(c*d*(2*c* 
d*(7*e*f - 2*d*g) - e*(7*b*e*f + 3*b*d*g - 10*a*e*g))) + (e*(80*c^2*d^2*f 
+ 5*b*e*(7*b*e*f + 3*b*d*g - 10*a*e*g) - 2*c*(8*a*e*(2*e*f - 7*d*g) + b*d* 
(47*e*f + 18*d*g))))/4))*(((b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + 
c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[( 
-(b*d) + 2*a*e - (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + 
b*x + c*x^2])])/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*(4*c*d^2 - 4*b*d*e + 4*a*e^ 
2))))/(2*(c*d^2 - b*d*e + a*e^2)))/(4*(c*d^2 - b*d*e + a*e^2))))/(5*(c*d^2 
 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2])
 

Rubi [A] (verified)

Time = 1.70 (sec) , antiderivative size = 570, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1237, 27, 1237, 27, 1228, 1152, 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.

\(\displaystyle \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx\)

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1237

\(\displaystyle \frac {-\frac {\int -\frac {\left (80 c^2 f d^2-2 b c (47 e f+18 d g) d-16 a c e (2 e f-7 d g)+5 b e (7 b e f+3 b d g-10 a e g)-2 c (2 c d (7 e f-2 d g)-e (7 b e f+3 b d g-10 a e g)) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^4}dx}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (7 e f-2 d g)-e (-10 a e g+3 b d g+7 b e f))}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (80 c^2 f d^2+5 b e (7 b e f+3 b d g-10 a e g)-2 c (8 a e (2 e f-7 d g)+b d (47 e f+18 d g))-2 c (2 c d (7 e f-2 d g)-e (7 b e f+3 b d g-10 a e g)) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^4}dx}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (7 e f-2 d g)-e (-10 a e g+3 b d g+7 b e f))}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {\frac {5 \left (-2 c e \left (4 a^2 e^2 g-6 a b e (e f-3 d g)-3 b^2 d (2 d g+5 e f)\right )-b^2 e^2 (-10 a e g+3 b d g+7 b e f)-8 c^2 d (3 a e (e f-2 d g)+2 b d (d g+3 e f))+32 c^3 d^3 f\right ) \int \frac {\sqrt {c x^2+b x+a}}{(d+e x)^3}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (2 a e (8 e f-33 d g)+3 b d (7 d g+18 e f))+5 b e^2 (-10 a e g+3 b d g+7 b e f)+4 c^2 d^2 (27 e f-2 d g)\right )}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (7 e f-2 d g)-e (-10 a e g+3 b d g+7 b e f))}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\frac {\frac {5 \left (-2 c e \left (4 a^2 e^2 g-6 a b e (e f-3 d g)-3 b^2 d (2 d g+5 e f)\right )-b^2 e^2 (-10 a e g+3 b d g+7 b e f)-8 c^2 d (3 a e (e f-2 d g)+2 b d (d g+3 e f))+32 c^3 d^3 f\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (2 a e (8 e f-33 d g)+3 b d (7 d g+18 e f))+5 b e^2 (-10 a e g+3 b d g+7 b e f)+4 c^2 d^2 (27 e f-2 d g)\right )}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (7 e f-2 d g)-e (-10 a e g+3 b d g+7 b e f))}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {5 \left (-2 c e \left (4 a^2 e^2 g-6 a b e (e f-3 d g)-3 b^2 d (2 d g+5 e f)\right )-b^2 e^2 (-10 a e g+3 b d g+7 b e f)-8 c^2 d (3 a e (e f-2 d g)+2 b d (d g+3 e f))+32 c^3 d^3 f\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (2 a e (8 e f-33 d g)+3 b d (7 d g+18 e f))+5 b e^2 (-10 a e g+3 b d g+7 b e f)+4 c^2 d^2 (27 e f-2 d g)\right )}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (7 e f-2 d g)-e (-10 a e g+3 b d g+7 b e f))}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {5 \left (\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}}\right ) \left (-2 c e \left (4 a^2 e^2 g-6 a b e (e f-3 d g)-3 b^2 d (2 d g+5 e f)\right )-b^2 e^2 (-10 a e g+3 b d g+7 b e f)-8 c^2 d (3 a e (e f-2 d g)+2 b d (d g+3 e f))+32 c^3 d^3 f\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (2 a e (8 e f-33 d g)+3 b d (7 d g+18 e f))+5 b e^2 (-10 a e g+3 b d g+7 b e f)+4 c^2 d^2 (27 e f-2 d g)\right )}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}}{8 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (2 c d (7 e f-2 d g)-e (-10 a e g+3 b d g+7 b e f))}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )}}{10 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{3/2} (e f-d g)}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[((f + g*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^6,x]
 

Output:

-1/5*((e*f - d*g)*(a + b*x + c*x^2)^(3/2))/((c*d^2 - b*d*e + a*e^2)*(d + e 
*x)^5) + (-1/4*((2*c*d*(7*e*f - 2*d*g) - e*(7*b*e*f + 3*b*d*g - 10*a*e*g)) 
*(a + b*x + c*x^2)^(3/2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + (-1/3*(( 
4*c^2*d^2*(27*e*f - 2*d*g) + 5*b*e^2*(7*b*e*f + 3*b*d*g - 10*a*e*g) - 2*c* 
e*(2*a*e*(8*e*f - 33*d*g) + 3*b*d*(18*e*f + 7*d*g)))*(a + b*x + c*x^2)^(3/ 
2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + (5*(32*c^3*d^3*f - b^2*e^2*(7* 
b*e*f + 3*b*d*g - 10*a*e*g) - 8*c^2*d*(3*a*e*(e*f - 2*d*g) + 2*b*d*(3*e*f 
+ d*g)) - 2*c*e*(4*a^2*e^2*g - 6*a*b*e*(e*f - 3*d*g) - 3*b^2*d*(5*e*f + 2* 
d*g)))*(((b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*d^2 
- b*d*e + a*e^2)*(d + e*x)^2) - ((b^2 - 4*a*c)*ArcTanh[(b*d - 2*a*e + (2*c 
*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c 
*d^2 - b*d*e + a*e^2)^(3/2))))/(2*(c*d^2 - b*d*e + a*e^2)))/(8*(c*d^2 - b* 
d*e + a*e^2)))/(10*(c*d^2 - b*d*e + a*e^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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

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 1228

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

rule 1237

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6446\) vs. \(2(582)=1164\).

Time = 3.83 (sec) , antiderivative size = 6447, normalized size of antiderivative = 10.60

method	result	size
default	\(\text {Expression too large to display}\)	\(6447\)

Input:

int((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\int \frac {\left (f + g x\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{6}}\, dx \] Input:

integrate((g*x+f)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**6,x)
 

Output:

Integral((f + g*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,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*e^2-b*d*e>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. 18289 vs. \(2 (582) = 1164\).

Time = 5.67 (sec) , antiderivative size = 18289, normalized size of antiderivative = 30.08 \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x, algorithm="giac")
 

Output:

-1/128*(32*b^2*c^3*d^3*f - 128*a*c^4*d^3*f - 48*b^3*c^2*d^2*e*f + 192*a*b* 
c^3*d^2*e*f + 30*b^4*c*d*e^2*f - 144*a*b^2*c^2*d*e^2*f + 96*a^2*c^3*d*e^2* 
f - 7*b^5*e^3*f + 40*a*b^3*c*e^3*f - 48*a^2*b*c^2*e^3*f - 16*b^3*c^2*d^3*g 
 + 64*a*b*c^3*d^3*g + 12*b^4*c*d^2*e*g - 192*a^2*c^3*d^2*e*g - 3*b^5*d*e^2 
*g - 24*a*b^3*c*d*e^2*g + 144*a^2*b*c^2*d*e^2*g + 10*a*b^4*e^3*g - 48*a^2* 
b^2*c*e^3*g + 32*a^3*c^2*e^3*g)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a 
))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 - 4*b*c^3*d^7*e 
+ 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e 
^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^ 
5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e 
^7 + a^4*e^8)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/1920*(480*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^9*b^2*c^3*d^3*e^7*f - 1920*(sqrt(c)*x - sqrt(c*x^2 + b 
*x + a))^9*a*c^4*d^3*e^7*f - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3 
*c^2*d^2*e^8*f + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b*c^3*d^2*e^ 
8*f + 450*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^4*c*d*e^9*f - 2160*(sqrt 
(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^2*c^2*d*e^9*f + 1440*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^9*a^2*c^3*d*e^9*f - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))^9*b^5*e^10*f + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c*e^1 
0*f - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b*c^2*e^10*f - 240*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^2*d^3*e^7*g + 960*(sqrt(c)*x -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^6} \,d x \] Input:

int(((f + g*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^6,x)
 

Output:

int(((f + g*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^6, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 65.98 (sec) , antiderivative size = 21307, normalized size of antiderivative = 35.04 \[ \int \frac {(f+g x) \sqrt {a+b x+c x^2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^6,x)
 

Output:

(480*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a* 
e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d**5*e** 
3*g + 2400*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*s 
qrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d* 
*4*e**4*g*x + 4800*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c 
*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3 
*c**2*d**3*e**5*g*x**2 + 4800*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt( 
a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2* 
c*d*x)*a**3*c**2*d**2*e**6*g*x**3 + 2400*sqrt(a*e**2 - b*d*e + c*d**2)*log 
( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - 
 b*e*x + 2*c*d*x)*a**3*c**2*d*e**7*g*x**4 + 480*sqrt(a*e**2 - b*d*e + c*d* 
*2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e 
+ b*d - b*e*x + 2*c*d*x)*a**3*c**2*e**8*g*x**5 - 720*sqrt(a*e**2 - b*d*e + 
 c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2 
*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b**2*c*d**5*e**3*g - 3600*sqrt(a*e**2 - 
 b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d 
**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b**2*c*d**4*e**4*g*x - 7200*sqr 
t(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b 
*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*b**2*c*d**3*e**5*g*x* 
*2 - 7200*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)...