\(\int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx\) [205]

Optimal result

Integrand size = 29, antiderivative size = 432 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx=-\frac {2 (b e-a f) (b g-a h) (c+d x)^{3/2}}{11 b^2 (b c-a d) (a+b x)^{11/2}}-\frac {2 \left (14 a^2 d f h-b^2 (8 d e g-11 c (f g+e h))-a b (22 c f h+3 d (f g+e h))\right ) (c+d x)^{3/2}}{99 b^2 (b c-a d)^2 (a+b x)^{9/2}}-\frac {2 \left (5 a^2 d^2 f h-2 a b d (11 c f h-3 d (f g+e h))+b^2 \left (16 d^2 e g+33 c^2 f h-22 c d (f g+e h)\right )\right ) (c+d x)^{3/2}}{231 b^2 (b c-a d)^3 (a+b x)^{7/2}}+\frac {8 d \left (5 a^2 d^2 f h-2 a b d (11 c f h-3 d (f g+e h))+b^2 \left (16 d^2 e g+33 c^2 f h-22 c d (f g+e h)\right )\right ) (c+d x)^{3/2}}{1155 b^2 (b c-a d)^4 (a+b x)^{5/2}}-\frac {16 d^2 \left (5 a^2 d^2 f h-2 a b d (11 c f h-3 d (f g+e h))+b^2 \left (16 d^2 e g+33 c^2 f h-22 c d (f g+e h)\right )\right ) (c+d x)^{3/2}}{3465 b^2 (b c-a d)^5 (a+b x)^{3/2}} \] Output:

-2/11*(-a*f+b*e)*(-a*h+b*g)*(d*x+c)^(3/2)/b^2/(-a*d+b*c)/(b*x+a)^(11/2)-2/ 
99*(14*a^2*d*f*h-b^2*(8*d*e*g-11*c*(e*h+f*g))-a*b*(22*c*f*h+3*d*(e*h+f*g)) 
)*(d*x+c)^(3/2)/b^2/(-a*d+b*c)^2/(b*x+a)^(9/2)-2/231*(5*a^2*d^2*f*h-2*a*b* 
d*(11*c*f*h-3*d*(e*h+f*g))+b^2*(16*d^2*e*g+33*c^2*f*h-22*c*d*(e*h+f*g)))*( 
d*x+c)^(3/2)/b^2/(-a*d+b*c)^3/(b*x+a)^(7/2)+8/1155*d*(5*a^2*d^2*f*h-2*a*b* 
d*(11*c*f*h-3*d*(e*h+f*g))+b^2*(16*d^2*e*g+33*c^2*f*h-22*c*d*(e*h+f*g)))*( 
d*x+c)^(3/2)/b^2/(-a*d+b*c)^4/(b*x+a)^(5/2)-16/3465*d^2*(5*a^2*d^2*f*h-2*a 
*b*d*(11*c*f*h-3*d*(e*h+f*g))+b^2*(16*d^2*e*g+33*c^2*f*h-22*c*d*(e*h+f*g)) 
)*(d*x+c)^(3/2)/b^2/(-a*d+b*c)^5/(b*x+a)^(3/2)
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx=-\frac {2 (c+d x)^{3/2} \left (22 a^3 b d \left (-8 c^3 f h+3 c^2 d (9 f g+9 e h+26 f h x)+d^3 x \left (84 e g+27 f g x+27 e h x+10 f h x^2\right )-6 c d^2 \left (21 e g+26 f g x+26 e h x+19 f h x^2\right )\right )+2 a b^3 \left (8 d^4 x^3 (44 e g+3 f g x+3 e h x)+5 c^4 (7 f g+7 e h+22 f h x)-8 c d^3 x^2 \left (66 e g+65 f g x+65 e h x+11 f h x^2\right )+3 c^2 d^2 x \left (220 e g+257 f g x+257 e h x+286 f h x^2\right )-2 c^3 d \left (385 e g+480 f g x+480 e h x+627 f h x^2\right )\right )+33 a^4 d^2 \left (8 c^2 f h-2 c d (7 f g+7 e h+6 f h x)+d^2 (7 e (5 g+3 h x)+3 f x (7 g+5 h x))\right )+b^4 \left (128 d^4 e g x^4-16 c d^3 x^3 (12 e g+11 f g x+11 e h x)+5 c^4 (11 f x (7 g+9 h x)+7 e (9 g+11 h x))+24 c^2 d^2 x^2 (11 f x (g+h x)+e (10 g+11 h x))-2 c^3 d x (33 f x (5 g+6 h x)+5 e (28 g+33 h x))\right )+2 a^2 b^2 \left (20 c^4 f h-c^3 d (165 f g+165 e h+514 f h x)+4 d^4 x^2 (33 e (6 g+h x)+f x (33 g+5 h x))-c d^3 x (99 e (12 g+13 h x)+f x (1287 g+514 h x))+3 c^2 d^2 (33 e (15 g+19 h x)+f x (627 g+799 h x))\right )\right )}{3465 (b c-a d)^5 (a+b x)^{11/2}} \] Input:

Integrate[(Sqrt[c + d*x]*(e + f*x)*(g + h*x))/(a + b*x)^(13/2),x]
 

Output:

(-2*(c + d*x)^(3/2)*(22*a^3*b*d*(-8*c^3*f*h + 3*c^2*d*(9*f*g + 9*e*h + 26* 
f*h*x) + d^3*x*(84*e*g + 27*f*g*x + 27*e*h*x + 10*f*h*x^2) - 6*c*d^2*(21*e 
*g + 26*f*g*x + 26*e*h*x + 19*f*h*x^2)) + 2*a*b^3*(8*d^4*x^3*(44*e*g + 3*f 
*g*x + 3*e*h*x) + 5*c^4*(7*f*g + 7*e*h + 22*f*h*x) - 8*c*d^3*x^2*(66*e*g + 
 65*f*g*x + 65*e*h*x + 11*f*h*x^2) + 3*c^2*d^2*x*(220*e*g + 257*f*g*x + 25 
7*e*h*x + 286*f*h*x^2) - 2*c^3*d*(385*e*g + 480*f*g*x + 480*e*h*x + 627*f* 
h*x^2)) + 33*a^4*d^2*(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^4*(128*d^4*e*g*x^4 - 16*c*d^3 
*x^3*(12*e*g + 11*f*g*x + 11*e*h*x) + 5*c^4*(11*f*x*(7*g + 9*h*x) + 7*e*(9 
*g + 11*h*x)) + 24*c^2*d^2*x^2*(11*f*x*(g + h*x) + e*(10*g + 11*h*x)) - 2* 
c^3*d*x*(33*f*x*(5*g + 6*h*x) + 5*e*(28*g + 33*h*x))) + 2*a^2*b^2*(20*c^4* 
f*h - c^3*d*(165*f*g + 165*e*h + 514*f*h*x) + 4*d^4*x^2*(33*e*(6*g + h*x) 
+ f*x*(33*g + 5*h*x)) - c*d^3*x*(99*e*(12*g + 13*h*x) + f*x*(1287*g + 514* 
h*x)) + 3*c^2*d^2*(33*e*(15*g + 19*h*x) + f*x*(627*g + 799*h*x)))))/(3465* 
(b*c - a*d)^5*(a + b*x)^(11/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.79, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {162, 55, 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.

\(\displaystyle \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx\)

\(\Big \downarrow \) 162

\(\displaystyle \frac {\left (5 a^2 d^2 f h-2 a b d (11 c f h-3 d (e h+f g))+b^2 \left (33 c^2 f h-22 c d (e h+f g)+16 d^2 e g\right )\right ) \int \frac {\sqrt {c+d x}}{(a+b x)^{9/2}}dx}{33 b^2 (b c-a d)^2}-\frac {2 (c+d x)^{3/2} \left (5 a^3 d f h+b x \left (14 a^2 d f h-a b (22 c f h+3 d (e h+f g))-b^2 (8 d e g-11 c (e h+f g))\right )-a^2 b (13 c f h-6 d (e h+f g))-a b^2 (17 d e g-2 c (e h+f g))+9 b^3 c e g\right )}{99 b^2 (a+b x)^{11/2} (b c-a d)^2}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {\left (5 a^2 d^2 f h-2 a b d (11 c f h-3 d (e h+f g))+b^2 \left (33 c^2 f h-22 c d (e h+f g)+16 d^2 e g\right )\right ) \left (-\frac {4 d \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}}dx}{7 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{33 b^2 (b c-a d)^2}-\frac {2 (c+d x)^{3/2} \left (5 a^3 d f h+b x \left (14 a^2 d f h-a b (22 c f h+3 d (e h+f g))-b^2 (8 d e g-11 c (e h+f g))\right )-a^2 b (13 c f h-6 d (e h+f g))-a b^2 (17 d e g-2 c (e h+f g))+9 b^3 c e g\right )}{99 b^2 (a+b x)^{11/2} (b c-a d)^2}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {\left (5 a^2 d^2 f h-2 a b d (11 c f h-3 d (e h+f g))+b^2 \left (33 c^2 f h-22 c d (e h+f g)+16 d^2 e g\right )\right ) \left (-\frac {4 d \left (-\frac {2 d \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}}dx}{5 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{33 b^2 (b c-a d)^2}-\frac {2 (c+d x)^{3/2} \left (5 a^3 d f h+b x \left (14 a^2 d f h-a b (22 c f h+3 d (e h+f g))-b^2 (8 d e g-11 c (e h+f g))\right )-a^2 b (13 c f h-6 d (e h+f g))-a b^2 (17 d e g-2 c (e h+f g))+9 b^3 c e g\right )}{99 b^2 (a+b x)^{11/2} (b c-a d)^2}\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {\left (-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}-\frac {4 d \left (\frac {4 d (c+d x)^{3/2}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}\right ) \left (5 a^2 d^2 f h-2 a b d (11 c f h-3 d (e h+f g))+b^2 \left (33 c^2 f h-22 c d (e h+f g)+16 d^2 e g\right )\right )}{33 b^2 (b c-a d)^2}-\frac {2 (c+d x)^{3/2} \left (5 a^3 d f h+b x \left (14 a^2 d f h-a b (22 c f h+3 d (e h+f g))-b^2 (8 d e g-11 c (e h+f g))\right )-a^2 b (13 c f h-6 d (e h+f g))-a b^2 (17 d e g-2 c (e h+f g))+9 b^3 c e g\right )}{99 b^2 (a+b x)^{11/2} (b c-a d)^2}\)

Input:

Int[(Sqrt[c + d*x]*(e + f*x)*(g + h*x))/(a + b*x)^(13/2),x]
 

Output:

(-2*(c + d*x)^(3/2)*(9*b^3*c*e*g + 5*a^3*d*f*h - a*b^2*(17*d*e*g - 2*c*(f* 
g + e*h)) - a^2*b*(13*c*f*h - 6*d*(f*g + e*h)) + b*(14*a^2*d*f*h - b^2*(8* 
d*e*g - 11*c*(f*g + e*h)) - a*b*(22*c*f*h + 3*d*(f*g + e*h)))*x))/(99*b^2* 
(b*c - a*d)^2*(a + b*x)^(11/2)) + ((5*a^2*d^2*f*h - 2*a*b*d*(11*c*f*h - 3* 
d*(f*g + e*h)) + b^2*(16*d^2*e*g + 33*c^2*f*h - 22*c*d*(f*g + e*h)))*((-2* 
(c + d*x)^(3/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) - (4*d*((-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))))/(7*(b*c - a*d))))/(33*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. \(1009\) vs. \(2(402)=804\).

Time = 0.34 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.34

method	result	size
default	\(\text {Expression too large to display}\)	\(1010\)
gosper	\(\text {Expression too large to display}\)	\(1066\)
orering	\(\text {Expression too large to display}\)	\(1066\)

Input:

int((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(13/2),x,method=_RETURNVERBOSE)
 

Output:

2/3465*(d*x+c)^(3/2)/(b*x+a)^(11/2)*(40*a^2*b^2*d^4*f*h*x^4-176*a*b^3*c*d^ 
3*f*h*x^4+48*a*b^3*d^4*e*h*x^4+48*a*b^3*d^4*f*g*x^4+264*b^4*c^2*d^2*f*h*x^ 
4-176*b^4*c*d^3*e*h*x^4-176*b^4*c*d^3*f*g*x^4+128*b^4*d^4*e*g*x^4+220*a^3* 
b*d^4*f*h*x^3-1028*a^2*b^2*c*d^3*f*h*x^3+264*a^2*b^2*d^4*e*h*x^3+264*a^2*b 
^2*d^4*f*g*x^3+1716*a*b^3*c^2*d^2*f*h*x^3-1040*a*b^3*c*d^3*e*h*x^3-1040*a* 
b^3*c*d^3*f*g*x^3+704*a*b^3*d^4*e*g*x^3-396*b^4*c^3*d*f*h*x^3+264*b^4*c^2* 
d^2*e*h*x^3+264*b^4*c^2*d^2*f*g*x^3-192*b^4*c*d^3*e*g*x^3+495*a^4*d^4*f*h* 
x^2-2508*a^3*b*c*d^3*f*h*x^2+594*a^3*b*d^4*e*h*x^2+594*a^3*b*d^4*f*g*x^2+4 
794*a^2*b^2*c^2*d^2*f*h*x^2-2574*a^2*b^2*c*d^3*e*h*x^2-2574*a^2*b^2*c*d^3* 
f*g*x^2+1584*a^2*b^2*d^4*e*g*x^2-2508*a*b^3*c^3*d*f*h*x^2+1542*a*b^3*c^2*d 
^2*e*h*x^2+1542*a*b^3*c^2*d^2*f*g*x^2-1056*a*b^3*c*d^3*e*g*x^2+495*b^4*c^4 
*f*h*x^2-330*b^4*c^3*d*e*h*x^2-330*b^4*c^3*d*f*g*x^2+240*b^4*c^2*d^2*e*g*x 
^2-396*a^4*c*d^3*f*h*x+693*a^4*d^4*e*h*x+693*a^4*d^4*f*g*x+1716*a^3*b*c^2* 
d^2*f*h*x-3432*a^3*b*c*d^3*e*h*x-3432*a^3*b*c*d^3*f*g*x+1848*a^3*b*d^4*e*g 
*x-1028*a^2*b^2*c^3*d*f*h*x+3762*a^2*b^2*c^2*d^2*e*h*x+3762*a^2*b^2*c^2*d^ 
2*f*g*x-2376*a^2*b^2*c*d^3*e*g*x+220*a*b^3*c^4*f*h*x-1920*a*b^3*c^3*d*e*h* 
x-1920*a*b^3*c^3*d*f*g*x+1320*a*b^3*c^2*d^2*e*g*x+385*b^4*c^4*e*h*x+385*b^ 
4*c^4*f*g*x-280*b^4*c^3*d*e*g*x+264*a^4*c^2*d^2*f*h-462*a^4*c*d^3*e*h-462* 
a^4*c*d^3*f*g+1155*a^4*d^4*e*g-176*a^3*b*c^3*d*f*h+594*a^3*b*c^2*d^2*e*h+5 
94*a^3*b*c^2*d^2*f*g-2772*a^3*b*c*d^3*e*g+40*a^2*b^2*c^4*f*h-330*a^2*b^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(13/2),x, algorithm="frica 
s")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx=\int \frac {\sqrt {c + d x} \left (e + f x\right ) \left (g + h x\right )}{\left (a + b x\right )^{\frac {13}{2}}}\, dx \] Input:

integrate((d*x+c)**(1/2)*(f*x+e)*(h*x+g)/(b*x+a)**(13/2),x)
 

Output:

Integral(sqrt(c + d*x)*(e + f*x)*(g + h*x)/(a + b*x)**(13/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(13/2),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. 7493 vs. \(2 (402) = 804\).

Time = 0.91 (sec) , antiderivative size = 7493, normalized size of antiderivative = 17.34 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(13/2),x, algorithm="giac" 
)
 

Output:

-32/3465*(16*sqrt(b*d)*b^16*c^6*d^5*e*g*abs(b) - 96*sqrt(b*d)*a*b^15*c^5*d 
^6*e*g*abs(b) + 240*sqrt(b*d)*a^2*b^14*c^4*d^7*e*g*abs(b) - 320*sqrt(b*d)* 
a^3*b^13*c^3*d^8*e*g*abs(b) + 240*sqrt(b*d)*a^4*b^12*c^2*d^9*e*g*abs(b) - 
96*sqrt(b*d)*a^5*b^11*c*d^10*e*g*abs(b) + 16*sqrt(b*d)*a^6*b^10*d^11*e*g*a 
bs(b) - 22*sqrt(b*d)*b^16*c^7*d^4*f*g*abs(b) + 138*sqrt(b*d)*a*b^15*c^6*d^ 
5*f*g*abs(b) - 366*sqrt(b*d)*a^2*b^14*c^5*d^6*f*g*abs(b) + 530*sqrt(b*d)*a 
^3*b^13*c^4*d^7*f*g*abs(b) - 450*sqrt(b*d)*a^4*b^12*c^3*d^8*f*g*abs(b) + 2 
22*sqrt(b*d)*a^5*b^11*c^2*d^9*f*g*abs(b) - 58*sqrt(b*d)*a^6*b^10*c*d^10*f* 
g*abs(b) + 6*sqrt(b*d)*a^7*b^9*d^11*f*g*abs(b) - 22*sqrt(b*d)*b^16*c^7*d^4 
*e*h*abs(b) + 138*sqrt(b*d)*a*b^15*c^6*d^5*e*h*abs(b) - 366*sqrt(b*d)*a^2* 
b^14*c^5*d^6*e*h*abs(b) + 530*sqrt(b*d)*a^3*b^13*c^4*d^7*e*h*abs(b) - 450* 
sqrt(b*d)*a^4*b^12*c^3*d^8*e*h*abs(b) + 222*sqrt(b*d)*a^5*b^11*c^2*d^9*e*h 
*abs(b) - 58*sqrt(b*d)*a^6*b^10*c*d^10*e*h*abs(b) + 6*sqrt(b*d)*a^7*b^9*d^ 
11*e*h*abs(b) + 33*sqrt(b*d)*b^16*c^8*d^3*f*h*abs(b) - 220*sqrt(b*d)*a*b^1 
5*c^7*d^4*f*h*abs(b) + 632*sqrt(b*d)*a^2*b^14*c^6*d^5*f*h*abs(b) - 1020*sq 
rt(b*d)*a^3*b^13*c^5*d^6*f*h*abs(b) + 1010*sqrt(b*d)*a^4*b^12*c^4*d^7*f*h* 
abs(b) - 628*sqrt(b*d)*a^5*b^11*c^3*d^8*f*h*abs(b) + 240*sqrt(b*d)*a^6*b^1 
0*c^2*d^9*f*h*abs(b) - 52*sqrt(b*d)*a^7*b^9*c*d^10*f*h*abs(b) + 5*sqrt(b*d 
)*a^8*b^8*d^11*f*h*abs(b) - 176*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt( 
b^2*c + (b*x + a)*b*d - a*b*d))^2*b^14*c^5*d^5*e*g*abs(b) + 880*sqrt(b*...
 

Mupad [B] (verification not implemented)

Time = 4.20 (sec) , antiderivative size = 1177, normalized size of antiderivative = 2.72 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx =\text {Too large to display} \] Input:

int(((e + f*x)*(g + h*x)*(c + d*x)^(1/2))/(a + b*x)^(13/2),x)
 

Output:

((c + d*x)^(1/2)*((630*b^4*c^5*e*g + 140*a*b^3*c^5*e*h + 140*a*b^3*c^5*f*g 
 + 2310*a^4*c*d^4*e*g + 80*a^2*b^2*c^5*f*h - 924*a^4*c^2*d^3*e*h - 924*a^4 
*c^2*d^3*f*g + 528*a^4*c^3*d^2*f*h + 5940*a^2*b^2*c^3*d^2*e*g - 3080*a*b^3 
*c^4*d*e*g - 352*a^3*b*c^4*d*f*h - 5544*a^3*b*c^2*d^3*e*g - 660*a^2*b^2*c^ 
4*d*e*h - 660*a^2*b^2*c^4*d*f*g + 1188*a^3*b*c^3*d^2*e*h + 1188*a^3*b*c^3* 
d^2*f*g)/(3465*b^5*(a*d - b*c)^5) + (x^2*(1386*a^4*d^5*e*h + 1386*a^4*d^5* 
f*g + 990*b^4*c^5*f*h + 3696*a^3*b*d^5*e*g + 198*a^4*c*d^4*f*h + 110*b^4*c 
^4*d*e*h + 110*b^4*c^4*d*f*g - 80*b^4*c^3*d^2*e*g + 2376*a^2*b^2*c^2*d^3*e 
*h + 2376*a^2*b^2*c^2*d^3*f*g + 7532*a^2*b^2*c^3*d^2*f*h - 5676*a^3*b*c*d^ 
4*e*h - 5676*a^3*b*c*d^4*f*g - 4576*a*b^3*c^4*d*f*h + 528*a*b^3*c^2*d^3*e* 
g - 1584*a^2*b^2*c*d^4*e*g - 756*a*b^3*c^3*d^2*e*h - 756*a*b^3*c^3*d^2*f*g 
 - 1584*a^3*b*c^2*d^3*f*h))/(3465*b^5*(a*d - b*c)^5) + (x^5*(256*b^4*d^5*e 
*g + 96*a*b^3*d^5*e*h + 96*a*b^3*d^5*f*g - 352*b^4*c*d^4*e*h - 352*b^4*c*d 
^4*f*g + 80*a^2*b^2*d^5*f*h + 528*b^4*c^2*d^3*f*h - 352*a*b^3*c*d^4*f*h))/ 
(3465*b^5*(a*d - b*c)^5) + (x*(2310*a^4*d^5*e*g + 770*b^4*c^5*e*h + 770*b^ 
4*c^5*f*g + 440*a*b^3*c^5*f*h + 462*a^4*c*d^4*e*h + 462*a^4*c*d^4*f*g + 70 
*b^4*c^4*d*e*g - 264*a^4*c^2*d^3*f*h + 1188*a^2*b^2*c^2*d^3*e*g + 6864*a^2 
*b^2*c^3*d^2*e*h + 6864*a^2*b^2*c^3*d^2*f*g - 1848*a^3*b*c*d^4*e*g - 3700* 
a*b^3*c^4*d*e*h - 3700*a*b^3*c^4*d*f*g - 440*a*b^3*c^3*d^2*e*g - 5676*a^3* 
b*c^2*d^3*e*h - 5676*a^3*b*c^2*d^3*f*g - 1976*a^2*b^2*c^4*d*f*h + 3080*...
 

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 3656, normalized size of antiderivative = 8.46 \[ \int \frac {\sqrt {c+d x} (e+f x) (g+h x)}{(a+b x)^{13/2}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(1/2)*(f*x+e)*(h*x+g)/(b*x+a)^(13/2),x)
 

Output:

(2*( - 40*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**7*d**5*f*h + 176*sqrt(d)*sqrt(b 
)*sqrt(a + b*x)*a**6*b*c*d**4*f*h - 48*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**6* 
b*d**5*e*h - 48*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**6*b*d**5*f*g - 200*sqrt(d 
)*sqrt(b)*sqrt(a + b*x)*a**6*b*d**5*f*h*x - 264*sqrt(d)*sqrt(b)*sqrt(a + b 
*x)*a**5*b**2*c**2*d**3*f*h + 176*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*b**2* 
c*d**4*e*h + 176*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*b**2*c*d**4*f*g + 880* 
sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*b**2*c*d**4*f*h*x - 128*sqrt(d)*sqrt(b) 
*sqrt(a + b*x)*a**5*b**2*d**5*e*g - 240*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5 
*b**2*d**5*e*h*x - 240*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*b**2*d**5*f*g*x 
- 400*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*b**2*d**5*f*h*x**2 - 1320*sqrt(d) 
*sqrt(b)*sqrt(a + b*x)*a**4*b**3*c**2*d**3*f*h*x + 880*sqrt(d)*sqrt(b)*sqr 
t(a + b*x)*a**4*b**3*c*d**4*e*h*x + 880*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4 
*b**3*c*d**4*f*g*x + 1760*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4*b**3*c*d**4*f 
*h*x**2 - 640*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**4*b**3*d**5*e*g*x - 480*sqr 
t(d)*sqrt(b)*sqrt(a + b*x)*a**4*b**3*d**5*e*h*x**2 - 480*sqrt(d)*sqrt(b)*s 
qrt(a + b*x)*a**4*b**3*d**5*f*g*x**2 - 400*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a 
**4*b**3*d**5*f*h*x**3 - 2640*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**4*c**2 
*d**3*f*h*x**2 + 1760*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**4*c*d**4*e*h*x 
**2 + 1760*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**4*c*d**4*f*g*x**2 + 1760* 
sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**4*c*d**4*f*h*x**3 - 1280*sqrt(d)*...