\(\int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx\) [204]

Optimal result

Integrand size = 29, antiderivative size = 330 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx=-\frac {2 (b e-a f) (b g-a h) (c+d x)^{3/2}}{9 b^2 (b c-a d) (a+b x)^{9/2}}-\frac {2 \left (4 a^2 d f h-a b (d f g+d e h+6 c f h)-b^2 (2 d e g-3 c (f g+e h))\right ) (c+d x)^{3/2}}{21 b^2 (b c-a d)^2 (a+b x)^{7/2}}-\frac {2 \left (5 a^2 d^2 f h-2 a b d (9 c f h-2 d (f g+e h))+b^2 \left (8 d^2 e g+21 c^2 f h-12 c d (f g+e h)\right )\right ) (c+d x)^{3/2}}{105 b^2 (b c-a d)^3 (a+b x)^{5/2}}+\frac {4 d \left (5 a^2 d^2 f h-2 a b d (9 c f h-2 d (f g+e h))+b^2 \left (8 d^2 e g+21 c^2 f h-12 c d (f g+e h)\right )\right ) (c+d x)^{3/2}}{315 b^2 (b c-a d)^4 (a+b x)^{3/2}} \] Output:

-2/9*(-a*f+b*e)*(-a*h+b*g)*(d*x+c)^(3/2)/b^2/(-a*d+b*c)/(b*x+a)^(9/2)-2/21 
*(4*a^2*d*f*h-a*b*(6*c*f*h+d*e*h+d*f*g)-b^2*(2*d*e*g-3*c*(e*h+f*g)))*(d*x+ 
c)^(3/2)/b^2/(-a*d+b*c)^2/(b*x+a)^(7/2)-2/105*(5*a^2*d^2*f*h-2*a*b*d*(9*c* 
f*h-2*d*(e*h+f*g))+b^2*(8*d^2*e*g+21*c^2*f*h-12*c*d*(e*h+f*g)))*(d*x+c)^(3 
/2)/b^2/(-a*d+b*c)^3/(b*x+a)^(5/2)+4/315*d*(5*a^2*d^2*f*h-2*a*b*d*(9*c*f*h 
-2*d*(e*h+f*g))+b^2*(8*d^2*e*g+21*c^2*f*h-12*c*d*(e*h+f*g)))*(d*x+c)^(3/2) 
/b^2/(-a*d+b*c)^4/(b*x+a)^(3/2)
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx=\frac {2 (c+d x)^{3/2} \left (a b^2 \left (8 d^3 x^2 (9 e g+f g x+e h x)-2 c^3 (5 f g+5 e h+18 f h x)-12 c d^2 x \left (9 e g+10 f g x+10 e h x+3 f h x^2\right )+3 c^2 d \left (45 e g+59 f g x+59 e h x+81 f h x^2\right )\right )+3 a^3 d \left (8 c^2 f h-2 c d (7 f g+7 e h+6 f h x)+d^2 (7 e (5 g+3 h x)+3 f x (7 g+5 h x))\right )-b^3 \left (-16 d^3 e g x^3+24 c d^2 x^2 (f g x+e (g+h x))-6 c^2 d x (e (5 g+6 h x)+f x (6 g+7 h x))+c^3 (9 f x (5 g+7 h x)+5 e (7 g+9 h x))\right )+a^2 b \left (-8 c^3 f h+12 c^2 d (3 f g+3 e h+10 f h x)+2 d^3 x (9 e (7 g+2 h x)+f x (18 g+5 h x))-3 c d^2 (9 e (7 g+9 h x)+f x (81 g+59 h x))\right )\right )}{315 (b c-a d)^4 (a+b x)^{9/2}} \] Input:

Integrate[(Sqrt[c + d*x]*(e + f*x)*(g + h*x))/(a + b*x)^(11/2),x]
 

Output:

(2*(c + d*x)^(3/2)*(a*b^2*(8*d^3*x^2*(9*e*g + f*g*x + e*h*x) - 2*c^3*(5*f* 
g + 5*e*h + 18*f*h*x) - 12*c*d^2*x*(9*e*g + 10*f*g*x + 10*e*h*x + 3*f*h*x^ 
2) + 3*c^2*d*(45*e*g + 59*f*g*x + 59*e*h*x + 81*f*h*x^2)) + 3*a^3*d*(8*c^2 
*f*h - 2*c*d*(7*f*g + 7*e*h + 6*f*h*x) + d^2*(7*e*(5*g + 3*h*x) + 3*f*x*(7 
*g + 5*h*x))) - b^3*(-16*d^3*e*g*x^3 + 24*c*d^2*x^2*(f*g*x + e*(g + h*x)) 
- 6*c^2*d*x*(e*(5*g + 6*h*x) + f*x*(6*g + 7*h*x)) + c^3*(9*f*x*(5*g + 7*h* 
x) + 5*e*(7*g + 9*h*x))) + a^2*b*(-8*c^3*f*h + 12*c^2*d*(3*f*g + 3*e*h + 1 
0*f*h*x) + 2*d^3*x*(9*e*(7*g + 2*h*x) + f*x*(18*g + 5*h*x)) - 3*c*d^2*(9*e 
*(7*g + 9*h*x) + f*x*(81*g + 59*h*x)))))/(315*(b*c - a*d)^4*(a + b*x)^(9/2 
))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {162, 55, 48}

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.

\(\displaystyle \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx\)

\(\Big \downarrow \) 162

\(\displaystyle \frac {\left (5 a^2 d^2 f h-2 a b d (9 c f h-2 d (e h+f g))+b^2 \left (21 c^2 f h-12 c d (e h+f g)+8 d^2 e g\right )\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}}dx}{21 b^2 (b c-a d)^2}-\frac {2 (c+d x)^{3/2} \left (5 a^3 d f h+3 b x \left (4 a^2 d f h-a b (6 c f h+d e h+d f g)-b^2 (2 d e g-3 c (e h+f g))\right )-a^2 b (11 c f h-4 d (e h+f g))-a b^2 (13 d e g-2 c (e h+f g))+7 b^3 c e g\right )}{63 b^2 (a+b x)^{9/2} (b c-a d)^2}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {\left (5 a^2 d^2 f h-2 a b d (9 c f h-2 d (e h+f g))+b^2 \left (21 c^2 f h-12 c d (e h+f g)+8 d^2 e g\right )\right ) \left (-\frac {2 d \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}}dx}{5 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{21 b^2 (b c-a d)^2}-\frac {2 (c+d x)^{3/2} \left (5 a^3 d f h+3 b x \left (4 a^2 d f h-a b (6 c f h+d e h+d f g)-b^2 (2 d e g-3 c (e h+f g))\right )-a^2 b (11 c f h-4 d (e h+f g))-a b^2 (13 d e g-2 c (e h+f g))+7 b^3 c e g\right )}{63 b^2 (a+b x)^{9/2} (b c-a d)^2}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (\frac {4 d (c+d x)^{3/2}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)}\right ) \left (5 a^2 d^2 f h-2 a b d (9 c f h-2 d (e h+f g))+b^2 \left (21 c^2 f h-12 c d (e h+f g)+8 d^2 e g\right )\right )}{21 b^2 (b c-a d)^2}-\frac {2 (c+d x)^{3/2} \left (5 a^3 d f h+3 b x \left (4 a^2 d f h-a b (6 c f h+d e h+d f g)-b^2 (2 d e g-3 c (e h+f g))\right )-a^2 b (11 c f h-4 d (e h+f g))-a b^2 (13 d e g-2 c (e h+f g))+7 b^3 c e g\right )}{63 b^2 (a+b x)^{9/2} (b c-a d)^2}\)

Input:

Int[(Sqrt[c + d*x]*(e + f*x)*(g + h*x))/(a + b*x)^(11/2),x]
 

Output:

(-2*(c + d*x)^(3/2)*(7*b^3*c*e*g + 5*a^3*d*f*h - a*b^2*(13*d*e*g - 2*c*(f* 
g + e*h)) - a^2*b*(11*c*f*h - 4*d*(f*g + e*h)) + 3*b*(4*a^2*d*f*h - a*b*(d 
*f*g + d*e*h + 6*c*f*h) - b^2*(2*d*e*g - 3*c*(f*g + e*h)))*x))/(63*b^2*(b* 
c - a*d)^2*(a + b*x)^(9/2)) + ((5*a^2*d^2*f*h - 2*a*b*d*(9*c*f*h - 2*d*(f* 
g + e*h)) + b^2*(8*d^2*e*g + 21*c^2*f*h - 12*c*d*(f*g + e*h)))*((-2*(c + d 
*x)^(3/2))/(5*(b*c - a*d)*(a + b*x)^(5/2)) + (4*d*(c + d*x)^(3/2))/(15*(b* 
c - a*d)^2*(a + b*x)^(3/2))))/(21*b^2*(b*c - a*d)^2)
 

Defintions of rubi rules used

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(613\) vs. \(2(306)=612\).

Time = 0.35 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.86

method	result	size
default	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (10 a^{2} b \,d^{3} f h \,x^{3}-36 a \,b^{2} c \,d^{2} f h \,x^{3}+8 a \,b^{2} d^{3} e h \,x^{3}+8 a \,b^{2} d^{3} f g \,x^{3}+42 b^{3} c^{2} d f h \,x^{3}-24 b^{3} c \,d^{2} e h \,x^{3}-24 b^{3} c \,d^{2} f g \,x^{3}+16 b^{3} d^{3} e g \,x^{3}+45 a^{3} d^{3} f h \,x^{2}-177 a^{2} b c \,d^{2} f h \,x^{2}+36 a^{2} b \,d^{3} e h \,x^{2}+36 a^{2} b \,d^{3} f g \,x^{2}+243 a \,b^{2} c^{2} d f h \,x^{2}-120 a \,b^{2} c \,d^{2} e h \,x^{2}-120 a \,b^{2} c \,d^{2} f g \,x^{2}+72 a \,b^{2} d^{3} e g \,x^{2}-63 b^{3} c^{3} f h \,x^{2}+36 b^{3} c^{2} d e h \,x^{2}+36 b^{3} c^{2} d f g \,x^{2}-24 b^{3} c \,d^{2} e g \,x^{2}-36 a^{3} c \,d^{2} f h x +63 a^{3} d^{3} e h x +63 a^{3} d^{3} f g x +120 a^{2} b \,c^{2} d f h x -243 a^{2} b c \,d^{2} e h x -243 a^{2} b c \,d^{2} f g x +126 a^{2} b \,d^{3} e g x -36 a \,b^{2} c^{3} f h x +177 a \,b^{2} c^{2} d e h x +177 a \,b^{2} c^{2} d f g x -108 a \,b^{2} c \,d^{2} e g x -45 b^{3} c^{3} e h x -45 b^{3} c^{3} f g x +30 b^{3} c^{2} d e g x +24 a^{3} c^{2} d f h -42 a^{3} c \,d^{2} e h -42 a^{3} c \,d^{2} f g +105 a^{3} d^{3} e g -8 a^{2} b \,c^{3} f h +36 a^{2} b \,c^{2} d e h +36 a^{2} b \,c^{2} d f g -189 a^{2} b c \,d^{2} e g -10 a \,b^{2} c^{3} e h -10 a \,b^{2} c^{3} f g +135 a \,b^{2} c^{2} d e g -35 b^{3} c^{3} e g \right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (a d -b c \right )^{4}}\)	\(614\)
gosper	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (10 a^{2} b \,d^{3} f h \,x^{3}-36 a \,b^{2} c \,d^{2} f h \,x^{3}+8 a \,b^{2} d^{3} e h \,x^{3}+8 a \,b^{2} d^{3} f g \,x^{3}+42 b^{3} c^{2} d f h \,x^{3}-24 b^{3} c \,d^{2} e h \,x^{3}-24 b^{3} c \,d^{2} f g \,x^{3}+16 b^{3} d^{3} e g \,x^{3}+45 a^{3} d^{3} f h \,x^{2}-177 a^{2} b c \,d^{2} f h \,x^{2}+36 a^{2} b \,d^{3} e h \,x^{2}+36 a^{2} b \,d^{3} f g \,x^{2}+243 a \,b^{2} c^{2} d f h \,x^{2}-120 a \,b^{2} c \,d^{2} e h \,x^{2}-120 a \,b^{2} c \,d^{2} f g \,x^{2}+72 a \,b^{2} d^{3} e g \,x^{2}-63 b^{3} c^{3} f h \,x^{2}+36 b^{3} c^{2} d e h \,x^{2}+36 b^{3} c^{2} d f g \,x^{2}-24 b^{3} c \,d^{2} e g \,x^{2}-36 a^{3} c \,d^{2} f h x +63 a^{3} d^{3} e h x +63 a^{3} d^{3} f g x +120 a^{2} b \,c^{2} d f h x -243 a^{2} b c \,d^{2} e h x -243 a^{2} b c \,d^{2} f g x +126 a^{2} b \,d^{3} e g x -36 a \,b^{2} c^{3} f h x +177 a \,b^{2} c^{2} d e h x +177 a \,b^{2} c^{2} d f g x -108 a \,b^{2} c \,d^{2} e g x -45 b^{3} c^{3} e h x -45 b^{3} c^{3} f g x +30 b^{3} c^{2} d e g x +24 a^{3} c^{2} d f h -42 a^{3} c \,d^{2} e h -42 a^{3} c \,d^{2} f g +105 a^{3} d^{3} e g -8 a^{2} b \,c^{3} f h +36 a^{2} b \,c^{2} d e h +36 a^{2} b \,c^{2} d f g -189 a^{2} b c \,d^{2} e g -10 a \,b^{2} c^{3} e h -10 a \,b^{2} c^{3} f g +135 a \,b^{2} c^{2} d e g -35 b^{3} c^{3} e g \right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(655\)
orering	\(\frac {2 \left (x d +c \right )^{\frac {3}{2}} \left (10 a^{2} b \,d^{3} f h \,x^{3}-36 a \,b^{2} c \,d^{2} f h \,x^{3}+8 a \,b^{2} d^{3} e h \,x^{3}+8 a \,b^{2} d^{3} f g \,x^{3}+42 b^{3} c^{2} d f h \,x^{3}-24 b^{3} c \,d^{2} e h \,x^{3}-24 b^{3} c \,d^{2} f g \,x^{3}+16 b^{3} d^{3} e g \,x^{3}+45 a^{3} d^{3} f h \,x^{2}-177 a^{2} b c \,d^{2} f h \,x^{2}+36 a^{2} b \,d^{3} e h \,x^{2}+36 a^{2} b \,d^{3} f g \,x^{2}+243 a \,b^{2} c^{2} d f h \,x^{2}-120 a \,b^{2} c \,d^{2} e h \,x^{2}-120 a \,b^{2} c \,d^{2} f g \,x^{2}+72 a \,b^{2} d^{3} e g \,x^{2}-63 b^{3} c^{3} f h \,x^{2}+36 b^{3} c^{2} d e h \,x^{2}+36 b^{3} c^{2} d f g \,x^{2}-24 b^{3} c \,d^{2} e g \,x^{2}-36 a^{3} c \,d^{2} f h x +63 a^{3} d^{3} e h x +63 a^{3} d^{3} f g x +120 a^{2} b \,c^{2} d f h x -243 a^{2} b c \,d^{2} e h x -243 a^{2} b c \,d^{2} f g x +126 a^{2} b \,d^{3} e g x -36 a \,b^{2} c^{3} f h x +177 a \,b^{2} c^{2} d e h x +177 a \,b^{2} c^{2} d f g x -108 a \,b^{2} c \,d^{2} e g x -45 b^{3} c^{3} e h x -45 b^{3} c^{3} f g x +30 b^{3} c^{2} d e g x +24 a^{3} c^{2} d f h -42 a^{3} c \,d^{2} e h -42 a^{3} c \,d^{2} f g +105 a^{3} d^{3} e g -8 a^{2} b \,c^{3} f h +36 a^{2} b \,c^{2} d e h +36 a^{2} b \,c^{2} d f g -189 a^{2} b c \,d^{2} e g -10 a \,b^{2} c^{3} e h -10 a \,b^{2} c^{3} f g +135 a \,b^{2} c^{2} d e g -35 b^{3} c^{3} e g \right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}\)	\(655\)

Input:

int((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

2/315*(d*x+c)^(3/2)/(b*x+a)^(9/2)*(10*a^2*b*d^3*f*h*x^3-36*a*b^2*c*d^2*f*h 
*x^3+8*a*b^2*d^3*e*h*x^3+8*a*b^2*d^3*f*g*x^3+42*b^3*c^2*d*f*h*x^3-24*b^3*c 
*d^2*e*h*x^3-24*b^3*c*d^2*f*g*x^3+16*b^3*d^3*e*g*x^3+45*a^3*d^3*f*h*x^2-17 
7*a^2*b*c*d^2*f*h*x^2+36*a^2*b*d^3*e*h*x^2+36*a^2*b*d^3*f*g*x^2+243*a*b^2* 
c^2*d*f*h*x^2-120*a*b^2*c*d^2*e*h*x^2-120*a*b^2*c*d^2*f*g*x^2+72*a*b^2*d^3 
*e*g*x^2-63*b^3*c^3*f*h*x^2+36*b^3*c^2*d*e*h*x^2+36*b^3*c^2*d*f*g*x^2-24*b 
^3*c*d^2*e*g*x^2-36*a^3*c*d^2*f*h*x+63*a^3*d^3*e*h*x+63*a^3*d^3*f*g*x+120* 
a^2*b*c^2*d*f*h*x-243*a^2*b*c*d^2*e*h*x-243*a^2*b*c*d^2*f*g*x+126*a^2*b*d^ 
3*e*g*x-36*a*b^2*c^3*f*h*x+177*a*b^2*c^2*d*e*h*x+177*a*b^2*c^2*d*f*g*x-108 
*a*b^2*c*d^2*e*g*x-45*b^3*c^3*e*h*x-45*b^3*c^3*f*g*x+30*b^3*c^2*d*e*g*x+24 
*a^3*c^2*d*f*h-42*a^3*c*d^2*e*h-42*a^3*c*d^2*f*g+105*a^3*d^3*e*g-8*a^2*b*c 
^3*f*h+36*a^2*b*c^2*d*e*h+36*a^2*b*c^2*d*f*g-189*a^2*b*c*d^2*e*g-10*a*b^2* 
c^3*e*h-10*a*b^2*c^3*f*g+135*a*b^2*c^2*d*e*g-35*b^3*c^3*e*g)/(a*d-b*c)^4
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(11/2),x, algorithm="frica 
s")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx=\int \frac {\sqrt {c + d x} \left (e + f x\right ) \left (g + h x\right )}{\left (a + b x\right )^{\frac {11}{2}}}\, dx \] Input:

integrate((d*x+c)**(1/2)*(f*x+e)*(h*x+g)/(b*x+a)**(11/2),x)
 

Output:

Integral(sqrt(c + d*x)*(e + f*x)*(g + h*x)/(a + b*x)**(11/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(11/2),x, algorithm="maxim 
a")
 

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*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5705 vs. \(2 (306) = 612\).

Time = 0.69 (sec) , antiderivative size = 5705, normalized size of antiderivative = 17.29 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(11/2),x, algorithm="giac" 
)
 

Output:

8/315*(8*sqrt(b*d)*b^14*c^5*d^4*e*g*abs(b) - 40*sqrt(b*d)*a*b^13*c^4*d^5*e 
*g*abs(b) + 80*sqrt(b*d)*a^2*b^12*c^3*d^6*e*g*abs(b) - 80*sqrt(b*d)*a^3*b^ 
11*c^2*d^7*e*g*abs(b) + 40*sqrt(b*d)*a^4*b^10*c*d^8*e*g*abs(b) - 8*sqrt(b* 
d)*a^5*b^9*d^9*e*g*abs(b) - 12*sqrt(b*d)*b^14*c^6*d^3*f*g*abs(b) + 64*sqrt 
(b*d)*a*b^13*c^5*d^4*f*g*abs(b) - 140*sqrt(b*d)*a^2*b^12*c^4*d^5*f*g*abs(b 
) + 160*sqrt(b*d)*a^3*b^11*c^3*d^6*f*g*abs(b) - 100*sqrt(b*d)*a^4*b^10*c^2 
*d^7*f*g*abs(b) + 32*sqrt(b*d)*a^5*b^9*c*d^8*f*g*abs(b) - 4*sqrt(b*d)*a^6* 
b^8*d^9*f*g*abs(b) - 12*sqrt(b*d)*b^14*c^6*d^3*e*h*abs(b) + 64*sqrt(b*d)*a 
*b^13*c^5*d^4*e*h*abs(b) - 140*sqrt(b*d)*a^2*b^12*c^4*d^5*e*h*abs(b) + 160 
*sqrt(b*d)*a^3*b^11*c^3*d^6*e*h*abs(b) - 100*sqrt(b*d)*a^4*b^10*c^2*d^7*e* 
h*abs(b) + 32*sqrt(b*d)*a^5*b^9*c*d^8*e*h*abs(b) - 4*sqrt(b*d)*a^6*b^8*d^9 
*e*h*abs(b) + 21*sqrt(b*d)*b^14*c^7*d^2*f*h*abs(b) - 123*sqrt(b*d)*a*b^13* 
c^6*d^3*f*h*abs(b) + 305*sqrt(b*d)*a^2*b^12*c^5*d^4*f*h*abs(b) - 415*sqrt( 
b*d)*a^3*b^11*c^4*d^5*f*h*abs(b) + 335*sqrt(b*d)*a^4*b^10*c^3*d^6*f*h*abs( 
b) - 161*sqrt(b*d)*a^5*b^9*c^2*d^7*f*h*abs(b) + 43*sqrt(b*d)*a^6*b^8*c*d^8 
*f*h*abs(b) - 5*sqrt(b*d)*a^7*b^7*d^9*f*h*abs(b) - 72*sqrt(b*d)*(sqrt(b*d) 
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^12*c^4*d^4*e*g*a 
bs(b) + 288*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* 
d - a*b*d))^2*a*b^11*c^3*d^5*e*g*abs(b) - 432*sqrt(b*d)*(sqrt(b*d)*sqrt(b* 
x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^10*c^2*d^6*e*g*ab...
 

Mupad [B] (verification not implemented)

Time = 3.69 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {x^4\,\left (32\,b^3\,d^4\,e\,g+16\,a\,b^2\,d^4\,e\,h+16\,a\,b^2\,d^4\,f\,g+20\,a^2\,b\,d^4\,f\,h-48\,b^3\,c\,d^3\,e\,h-48\,b^3\,c\,d^3\,f\,g+84\,b^3\,c^2\,d^2\,f\,h-72\,a\,b^2\,c\,d^3\,f\,h\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^4}-\frac {x\,\left (90\,b^3\,c^4\,e\,h-210\,a^3\,d^4\,e\,g+90\,b^3\,c^4\,f\,g+72\,a\,b^2\,c^4\,f\,h-42\,a^3\,c\,d^3\,e\,h-42\,a^3\,c\,d^3\,f\,g+10\,b^3\,c^3\,d\,e\,g+24\,a^3\,c^2\,d^2\,f\,h+126\,a^2\,b\,c\,d^3\,e\,g-334\,a\,b^2\,c^3\,d\,e\,h-334\,a\,b^2\,c^3\,d\,f\,g-224\,a^2\,b\,c^3\,d\,f\,h-54\,a\,b^2\,c^2\,d^2\,e\,g+414\,a^2\,b\,c^2\,d^2\,e\,h+414\,a^2\,b\,c^2\,d^2\,f\,g\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^4}-\frac {70\,b^3\,c^4\,e\,g+20\,a\,b^2\,c^4\,e\,h+20\,a\,b^2\,c^4\,f\,g+16\,a^2\,b\,c^4\,f\,h-210\,a^3\,c\,d^3\,e\,g-48\,a^3\,c^3\,d\,f\,h+84\,a^3\,c^2\,d^2\,e\,h+84\,a^3\,c^2\,d^2\,f\,g-270\,a\,b^2\,c^3\,d\,e\,g-72\,a^2\,b\,c^3\,d\,e\,h-72\,a^2\,b\,c^3\,d\,f\,g+378\,a^2\,b\,c^2\,d^2\,e\,g}{315\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {x^2\,\left (126\,a^3\,d^4\,e\,h+126\,a^3\,d^4\,f\,g-126\,b^3\,c^4\,f\,h+252\,a^2\,b\,d^4\,e\,g+18\,a^3\,c\,d^3\,f\,h-18\,b^3\,c^3\,d\,e\,h-18\,b^3\,c^3\,d\,f\,g+12\,b^3\,c^2\,d^2\,e\,g-72\,a\,b^2\,c\,d^3\,e\,g-414\,a^2\,b\,c\,d^3\,e\,h-414\,a^2\,b\,c\,d^3\,f\,g+414\,a\,b^2\,c^3\,d\,f\,h+114\,a\,b^2\,c^2\,d^2\,e\,h+114\,a\,b^2\,c^2\,d^2\,f\,g-114\,a^2\,b\,c^2\,d^2\,f\,h\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,d\,x^3\,\left (9\,a\,d-b\,c\right )\,\left (8\,b^2\,d^2\,e\,g+5\,a^2\,d^2\,f\,h+21\,b^2\,c^2\,f\,h+4\,a\,b\,d^2\,e\,h+4\,a\,b\,d^2\,f\,g-12\,b^2\,c\,d\,e\,h-12\,b^2\,c\,d\,f\,g-18\,a\,b\,c\,d\,f\,h\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \] Input:

int(((e + f*x)*(g + h*x)*(c + d*x)^(1/2))/(a + b*x)^(11/2),x)
 

Output:

((c + d*x)^(1/2)*((x^4*(32*b^3*d^4*e*g + 16*a*b^2*d^4*e*h + 16*a*b^2*d^4*f 
*g + 20*a^2*b*d^4*f*h - 48*b^3*c*d^3*e*h - 48*b^3*c*d^3*f*g + 84*b^3*c^2*d 
^2*f*h - 72*a*b^2*c*d^3*f*h))/(315*b^4*(a*d - b*c)^4) - (x*(90*b^3*c^4*e*h 
 - 210*a^3*d^4*e*g + 90*b^3*c^4*f*g + 72*a*b^2*c^4*f*h - 42*a^3*c*d^3*e*h 
- 42*a^3*c*d^3*f*g + 10*b^3*c^3*d*e*g + 24*a^3*c^2*d^2*f*h + 126*a^2*b*c*d 
^3*e*g - 334*a*b^2*c^3*d*e*h - 334*a*b^2*c^3*d*f*g - 224*a^2*b*c^3*d*f*h - 
 54*a*b^2*c^2*d^2*e*g + 414*a^2*b*c^2*d^2*e*h + 414*a^2*b*c^2*d^2*f*g))/(3 
15*b^4*(a*d - b*c)^4) - (70*b^3*c^4*e*g + 20*a*b^2*c^4*e*h + 20*a*b^2*c^4* 
f*g + 16*a^2*b*c^4*f*h - 210*a^3*c*d^3*e*g - 48*a^3*c^3*d*f*h + 84*a^3*c^2 
*d^2*e*h + 84*a^3*c^2*d^2*f*g - 270*a*b^2*c^3*d*e*g - 72*a^2*b*c^3*d*e*h - 
 72*a^2*b*c^3*d*f*g + 378*a^2*b*c^2*d^2*e*g)/(315*b^4*(a*d - b*c)^4) + (x^ 
2*(126*a^3*d^4*e*h + 126*a^3*d^4*f*g - 126*b^3*c^4*f*h + 252*a^2*b*d^4*e*g 
 + 18*a^3*c*d^3*f*h - 18*b^3*c^3*d*e*h - 18*b^3*c^3*d*f*g + 12*b^3*c^2*d^2 
*e*g - 72*a*b^2*c*d^3*e*g - 414*a^2*b*c*d^3*e*h - 414*a^2*b*c*d^3*f*g + 41 
4*a*b^2*c^3*d*f*h + 114*a*b^2*c^2*d^2*e*h + 114*a*b^2*c^2*d^2*f*g - 114*a^ 
2*b*c^2*d^2*f*h))/(315*b^4*(a*d - b*c)^4) + (2*d*x^3*(9*a*d - b*c)*(8*b^2* 
d^2*e*g + 5*a^2*d^2*f*h + 21*b^2*c^2*f*h + 4*a*b*d^2*e*h + 4*a*b*d^2*f*g - 
 12*b^2*c*d*e*h - 12*b^2*c*d*f*g - 18*a*b*c*d*f*h))/(315*b^4*(a*d - b*c)^4 
)))/(x^4*(a + b*x)^(1/2) + (a^4*(a + b*x)^(1/2))/b^4 + (6*a^2*x^2*(a + b*x 
)^(1/2))/b^2 + (4*a*x^3*(a + b*x)^(1/2))/b + (4*a^3*x*(a + b*x)^(1/2))/...
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 2628, normalized size of antiderivative = 7.96 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{11/2}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(11/2),x)
 

Output:

(2*( - 10*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**6*d**4*f*h + 36*sqrt(d)*sqrt(b) 
*sqrt(a + b*x)*a**5*b*c*d**3*f*h - 8*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*b* 
d**4*e*h - 8*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*b*d**4*f*g - 40*sqrt(d)*sq 
rt(b)*sqrt(a + b*x)*a**5*b*d**4*f*h*x - 42*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a 
**4*b**2*c**2*d**2*f*h + 24*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4*b**2*c*d**3 
*e*h + 24*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4*b**2*c*d**3*f*g + 144*sqrt(d) 
*sqrt(b)*sqrt(a + b*x)*a**4*b**2*c*d**3*f*h*x - 16*sqrt(d)*sqrt(b)*sqrt(a 
+ b*x)*a**4*b**2*d**4*e*g - 32*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4*b**2*d** 
4*e*h*x - 32*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4*b**2*d**4*f*g*x - 60*sqrt( 
d)*sqrt(b)*sqrt(a + b*x)*a**4*b**2*d**4*f*h*x**2 - 168*sqrt(d)*sqrt(b)*sqr 
t(a + b*x)*a**3*b**3*c**2*d**2*f*h*x + 96*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a* 
*3*b**3*c*d**3*e*h*x + 96*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**3*c*d**3*f 
*g*x + 216*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**3*c*d**3*f*h*x**2 - 64*sq 
rt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**3*d**4*e*g*x - 48*sqrt(d)*sqrt(b)*sqrt 
(a + b*x)*a**3*b**3*d**4*e*h*x**2 - 48*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3* 
b**3*d**4*f*g*x**2 - 40*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**3*d**4*f*h*x 
**3 - 252*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b**4*c**2*d**2*f*h*x**2 + 144 
*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b**4*c*d**3*e*h*x**2 + 144*sqrt(d)*sqr 
t(b)*sqrt(a + b*x)*a**2*b**4*c*d**3*f*g*x**2 + 144*sqrt(d)*sqrt(b)*sqrt(a 
+ b*x)*a**2*b**4*c*d**3*f*h*x**3 - 96*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**...