\(\int (d+e x)^{3/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\) [205]

Optimal result

Integrand size = 46, antiderivative size = 343 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 (2 c d-b e)^3 (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^5 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e)^2 (3 c e f+5 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^5 e^2 (d+e x)^{7/2}}-\frac {2 (2 c d-b e) (c e f+3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}{3 c^5 e^2 (d+e x)^{9/2}}+\frac {2 (c e f+7 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}{11 c^5 e^2 (d+e x)^{11/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}{13 c^5 e^2 (d+e x)^{13/2}} \] Output:

-2/5*(-b*e+2*c*d)^3*(-b*e*g+c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^ 
(5/2)/c^5/e^2/(e*x+d)^(5/2)+2/7*(-b*e+2*c*d)^2*(-4*b*e*g+5*c*d*g+3*c*e*f)* 
(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^5/e^2/(e*x+d)^(7/2)-2/3*(-b*e+2*c 
*d)*(-2*b*e*g+3*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(9/2)/c^5/e^ 
2/(e*x+d)^(9/2)+2/11*(-4*b*e*g+7*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2* 
x^2)^(11/2)/c^5/e^2/(e*x+d)^(11/2)-2/13*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2) 
^(13/2)/c^5/e^2/(e*x+d)^(13/2)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.77 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (128 b^4 e^4 g-16 b^3 c e^3 (13 e f+71 d g+20 e g x)+8 b^2 c^2 e^2 \left (473 d^2 g+5 e^2 x (13 f+14 g x)+d e (221 f+315 g x)\right )-2 b c^3 e \left (2765 d^3 g+35 e^3 x^2 (13 f+12 g x)+25 d e^2 x (78 f+77 g x)+d^2 e (2743 f+3470 g x)\right )+c^4 \left (2754 d^4 g+105 e^4 x^3 (13 f+11 g x)+35 d e^3 x^2 (169 f+141 g x)+5 d^2 e^2 x (1963 f+1659 g x)+d^3 e (6929 f+6885 g x)\right )\right )}{15015 c^5 e^2 \sqrt {d+e x}} \] Input:

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

Output:

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(128*b 
^4*e^4*g - 16*b^3*c*e^3*(13*e*f + 71*d*g + 20*e*g*x) + 8*b^2*c^2*e^2*(473* 
d^2*g + 5*e^2*x*(13*f + 14*g*x) + d*e*(221*f + 315*g*x)) - 2*b*c^3*e*(2765 
*d^3*g + 35*e^3*x^2*(13*f + 12*g*x) + 25*d*e^2*x*(78*f + 77*g*x) + d^2*e*( 
2743*f + 3470*g*x)) + c^4*(2754*d^4*g + 105*e^4*x^3*(13*f + 11*g*x) + 35*d 
*e^3*x^2*(169*f + 141*g*x) + 5*d^2*e^2*x*(1963*f + 1659*g*x) + d^3*e*(6929 
*f + 6885*g*x))))/(15015*c^5*e^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1221, 1128, 1128, 1128, 1122}

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.

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

\(\Big \downarrow \) 1221

\(\displaystyle \frac {(-8 b e g+3 c d g+13 c e f) \int (d+e x)^{3/2} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{13 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-8 b e g+3 c d g+13 c e f) \left (\frac {6 (2 c d-b e) \int \sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{11 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e}\right )}{13 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}\)

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1128

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

\(\Big \downarrow \) 1122

\(\displaystyle \frac {\left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 c^2 e (d+e x)^{5/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\right )}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )}{11 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e}\right ) (-8 b e g+3 c d g+13 c e f)}{13 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}\)

Input:

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

Output:

(-2*g*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(13*c*e 
^2) + ((13*c*e*f + 3*c*d*g - 8*b*e*g)*((-2*Sqrt[d + e*x]*(d*(c*d - b*e) - 
b*e^2*x - c*e^2*x^2)^(5/2))/(11*c*e) + (6*(2*c*d - b*e)*((-2*(d*(c*d - b*e 
) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*c*e*Sqrt[d + e*x]) + (4*(2*c*d - b*e)*( 
(-4*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(35*c^2*e*( 
d + e*x)^(5/2)) - (2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*c*e*( 
d + e*x)^(3/2))))/(9*c)))/(11*c)))/(13*c*e)
 

Defintions of rubi rules used

rule 1122

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

rule 1128

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

rule 1221

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

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.05

method	result	size
default	\(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{2} \left (1155 g \,e^{4} x^{4} c^{4}-840 b \,c^{3} e^{4} g \,x^{3}+4935 c^{4} d \,e^{3} g \,x^{3}+1365 c^{4} e^{4} f \,x^{3}+560 b^{2} c^{2} e^{4} g \,x^{2}-3850 b \,c^{3} d \,e^{3} g \,x^{2}-910 b \,c^{3} e^{4} f \,x^{2}+8295 c^{4} d^{2} e^{2} g \,x^{2}+5915 c^{4} d \,e^{3} f \,x^{2}-320 b^{3} c \,e^{4} g x +2520 b^{2} c^{2} d \,e^{3} g x +520 b^{2} c^{2} e^{4} f x -6940 b \,c^{3} d^{2} e^{2} g x -3900 b \,c^{3} d \,e^{3} f x +6885 c^{4} d^{3} e g x +9815 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1136 b^{3} c d \,e^{3} g -208 b^{3} c \,e^{4} f +3784 b^{2} c^{2} d^{2} e^{2} g +1768 b^{2} c^{2} d \,e^{3} f -5530 b \,c^{3} d^{3} e g -5486 b \,c^{3} d^{2} e^{2} f +2754 c^{4} d^{4} g +6929 d^{3} f \,c^{4} e \right )}{15015 \sqrt {e x +d}\, c^{5} e^{2}}\)	\(361\)
gosper	\(\frac {2 \left (c e x +b e -c d \right ) \left (1155 g \,e^{4} x^{4} c^{4}-840 b \,c^{3} e^{4} g \,x^{3}+4935 c^{4} d \,e^{3} g \,x^{3}+1365 c^{4} e^{4} f \,x^{3}+560 b^{2} c^{2} e^{4} g \,x^{2}-3850 b \,c^{3} d \,e^{3} g \,x^{2}-910 b \,c^{3} e^{4} f \,x^{2}+8295 c^{4} d^{2} e^{2} g \,x^{2}+5915 c^{4} d \,e^{3} f \,x^{2}-320 b^{3} c \,e^{4} g x +2520 b^{2} c^{2} d \,e^{3} g x +520 b^{2} c^{2} e^{4} f x -6940 b \,c^{3} d^{2} e^{2} g x -3900 b \,c^{3} d \,e^{3} f x +6885 c^{4} d^{3} e g x +9815 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1136 b^{3} c d \,e^{3} g -208 b^{3} c \,e^{4} f +3784 b^{2} c^{2} d^{2} e^{2} g +1768 b^{2} c^{2} d \,e^{3} f -5530 b \,c^{3} d^{3} e g -5486 b \,c^{3} d^{2} e^{2} f +2754 c^{4} d^{4} g +6929 d^{3} f \,c^{4} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{15015 c^{5} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\)	\(367\)
orering	\(\frac {2 \left (c e x +b e -c d \right ) \left (1155 g \,e^{4} x^{4} c^{4}-840 b \,c^{3} e^{4} g \,x^{3}+4935 c^{4} d \,e^{3} g \,x^{3}+1365 c^{4} e^{4} f \,x^{3}+560 b^{2} c^{2} e^{4} g \,x^{2}-3850 b \,c^{3} d \,e^{3} g \,x^{2}-910 b \,c^{3} e^{4} f \,x^{2}+8295 c^{4} d^{2} e^{2} g \,x^{2}+5915 c^{4} d \,e^{3} f \,x^{2}-320 b^{3} c \,e^{4} g x +2520 b^{2} c^{2} d \,e^{3} g x +520 b^{2} c^{2} e^{4} f x -6940 b \,c^{3} d^{2} e^{2} g x -3900 b \,c^{3} d \,e^{3} f x +6885 c^{4} d^{3} e g x +9815 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1136 b^{3} c d \,e^{3} g -208 b^{3} c \,e^{4} f +3784 b^{2} c^{2} d^{2} e^{2} g +1768 b^{2} c^{2} d \,e^{3} f -5530 b \,c^{3} d^{3} e g -5486 b \,c^{3} d^{2} e^{2} f +2754 c^{4} d^{4} g +6929 d^{3} f \,c^{4} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{15015 c^{5} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\)	\(367\)

Input:

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

Output:

-2/15015/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(c*e*x+b*e-c*d)^2* 
(1155*c^4*e^4*g*x^4-840*b*c^3*e^4*g*x^3+4935*c^4*d*e^3*g*x^3+1365*c^4*e^4* 
f*x^3+560*b^2*c^2*e^4*g*x^2-3850*b*c^3*d*e^3*g*x^2-910*b*c^3*e^4*f*x^2+829 
5*c^4*d^2*e^2*g*x^2+5915*c^4*d*e^3*f*x^2-320*b^3*c*e^4*g*x+2520*b^2*c^2*d* 
e^3*g*x+520*b^2*c^2*e^4*f*x-6940*b*c^3*d^2*e^2*g*x-3900*b*c^3*d*e^3*f*x+68 
85*c^4*d^3*e*g*x+9815*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1136*b^3*c*d*e^3*g-208 
*b^3*c*e^4*f+3784*b^2*c^2*d^2*e^2*g+1768*b^2*c^2*d*e^3*f-5530*b*c^3*d^3*e* 
g-5486*b*c^3*d^2*e^2*f+2754*c^4*d^4*g+6929*c^4*d^3*e*f)/c^5/e^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (313) = 626\).

Time = 0.11 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.98 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (1155 \, c^{6} e^{6} g x^{6} + 105 \, {\left (13 \, c^{6} e^{6} f + {\left (25 \, c^{6} d e^{5} + 14 \, b c^{5} e^{6}\right )} g\right )} x^{5} + 35 \, {\left (13 \, {\left (7 \, c^{6} d e^{5} + 4 \, b c^{5} e^{6}\right )} f - {\left (12 \, c^{6} d^{2} e^{4} - 154 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} g\right )} x^{4} - 5 \, {\left (13 \, {\left (10 \, c^{6} d^{2} e^{4} - 108 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} f + {\left (954 \, c^{6} d^{3} e^{3} - 1328 \, b c^{5} d^{2} e^{4} - 63 \, b^{2} c^{4} d e^{5} + 8 \, b^{3} c^{3} e^{6}\right )} g\right )} x^{3} - 3 \, {\left (13 \, {\left (174 \, c^{6} d^{3} e^{3} - 236 \, b c^{5} d^{2} e^{4} - 17 \, b^{2} c^{4} d e^{5} + 2 \, b^{3} c^{3} e^{6}\right )} f + {\left (907 \, c^{6} d^{4} e^{2} - 560 \, b c^{5} d^{3} e^{3} - 473 \, b^{2} c^{4} d^{2} e^{4} + 142 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} g\right )} x^{2} + 13 \, {\left (533 \, c^{6} d^{5} e - 1488 \, b c^{5} d^{4} e^{2} + 1513 \, b^{2} c^{4} d^{3} e^{3} - 710 \, b^{3} c^{3} d^{2} e^{4} + 168 \, b^{4} c^{2} d e^{5} - 16 \, b^{5} c e^{6}\right )} f + 2 \, {\left (1377 \, c^{6} d^{6} - 5519 \, b c^{5} d^{5} e + 8799 \, b^{2} c^{4} d^{4} e^{2} - 7117 \, b^{3} c^{3} d^{3} e^{3} + 3092 \, b^{4} c^{2} d^{2} e^{4} - 696 \, b^{5} c d e^{5} + 64 \, b^{6} e^{6}\right )} g - {\left (13 \, {\left (311 \, c^{6} d^{4} e^{2} - 100 \, b c^{5} d^{3} e^{3} - 279 \, b^{2} c^{4} d^{2} e^{4} + 76 \, b^{3} c^{3} d e^{5} - 8 \, b^{4} c^{2} e^{6}\right )} f - {\left (1377 \, c^{6} d^{5} e - 4142 \, b c^{5} d^{4} e^{2} + 4657 \, b^{2} c^{4} d^{3} e^{3} - 2460 \, b^{3} c^{3} d^{2} e^{4} + 632 \, b^{4} c^{2} d e^{5} - 64 \, b^{5} c e^{6}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15015 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \] Input:

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

Output:

-2/15015*(1155*c^6*e^6*g*x^6 + 105*(13*c^6*e^6*f + (25*c^6*d*e^5 + 14*b*c^ 
5*e^6)*g)*x^5 + 35*(13*(7*c^6*d*e^5 + 4*b*c^5*e^6)*f - (12*c^6*d^2*e^4 - 1 
54*b*c^5*d*e^5 - b^2*c^4*e^6)*g)*x^4 - 5*(13*(10*c^6*d^2*e^4 - 108*b*c^5*d 
*e^5 - b^2*c^4*e^6)*f + (954*c^6*d^3*e^3 - 1328*b*c^5*d^2*e^4 - 63*b^2*c^4 
*d*e^5 + 8*b^3*c^3*e^6)*g)*x^3 - 3*(13*(174*c^6*d^3*e^3 - 236*b*c^5*d^2*e^ 
4 - 17*b^2*c^4*d*e^5 + 2*b^3*c^3*e^6)*f + (907*c^6*d^4*e^2 - 560*b*c^5*d^3 
*e^3 - 473*b^2*c^4*d^2*e^4 + 142*b^3*c^3*d*e^5 - 16*b^4*c^2*e^6)*g)*x^2 + 
13*(533*c^6*d^5*e - 1488*b*c^5*d^4*e^2 + 1513*b^2*c^4*d^3*e^3 - 710*b^3*c^ 
3*d^2*e^4 + 168*b^4*c^2*d*e^5 - 16*b^5*c*e^6)*f + 2*(1377*c^6*d^6 - 5519*b 
*c^5*d^5*e + 8799*b^2*c^4*d^4*e^2 - 7117*b^3*c^3*d^3*e^3 + 3092*b^4*c^2*d^ 
2*e^4 - 696*b^5*c*d*e^5 + 64*b^6*e^6)*g - (13*(311*c^6*d^4*e^2 - 100*b*c^5 
*d^3*e^3 - 279*b^2*c^4*d^2*e^4 + 76*b^3*c^3*d*e^5 - 8*b^4*c^2*e^6)*f - (13 
77*c^6*d^5*e - 4142*b*c^5*d^4*e^2 + 4657*b^2*c^4*d^3*e^3 - 2460*b^3*c^3*d^ 
2*e^4 + 632*b^4*c^2*d*e^5 - 64*b^5*c*e^6)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x 
+ c*d^2 - b*d*e)*sqrt(e*x + d)/(c^5*e^3*x + c^5*d*e^2)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 676 vs. \(2 (313) = 626\).

Time = 0.11 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.97 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (105 \, c^{5} e^{5} x^{5} + 533 \, c^{5} d^{5} - 1488 \, b c^{4} d^{4} e + 1513 \, b^{2} c^{3} d^{3} e^{2} - 710 \, b^{3} c^{2} d^{2} e^{3} + 168 \, b^{4} c d e^{4} - 16 \, b^{5} e^{5} + 35 \, {\left (7 \, c^{5} d e^{4} + 4 \, b c^{4} e^{5}\right )} x^{4} - 5 \, {\left (10 \, c^{5} d^{2} e^{3} - 108 \, b c^{4} d e^{4} - b^{2} c^{3} e^{5}\right )} x^{3} - 3 \, {\left (174 \, c^{5} d^{3} e^{2} - 236 \, b c^{4} d^{2} e^{3} - 17 \, b^{2} c^{3} d e^{4} + 2 \, b^{3} c^{2} e^{5}\right )} x^{2} - {\left (311 \, c^{5} d^{4} e - 100 \, b c^{4} d^{3} e^{2} - 279 \, b^{2} c^{3} d^{2} e^{3} + 76 \, b^{3} c^{2} d e^{4} - 8 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{1155 \, {\left (c^{4} e^{2} x + c^{4} d e\right )}} - \frac {2 \, {\left (1155 \, c^{6} e^{6} x^{6} + 2754 \, c^{6} d^{6} - 11038 \, b c^{5} d^{5} e + 17598 \, b^{2} c^{4} d^{4} e^{2} - 14234 \, b^{3} c^{3} d^{3} e^{3} + 6184 \, b^{4} c^{2} d^{2} e^{4} - 1392 \, b^{5} c d e^{5} + 128 \, b^{6} e^{6} + 105 \, {\left (25 \, c^{6} d e^{5} + 14 \, b c^{5} e^{6}\right )} x^{5} - 35 \, {\left (12 \, c^{6} d^{2} e^{4} - 154 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} x^{4} - 5 \, {\left (954 \, c^{6} d^{3} e^{3} - 1328 \, b c^{5} d^{2} e^{4} - 63 \, b^{2} c^{4} d e^{5} + 8 \, b^{3} c^{3} e^{6}\right )} x^{3} - 3 \, {\left (907 \, c^{6} d^{4} e^{2} - 560 \, b c^{5} d^{3} e^{3} - 473 \, b^{2} c^{4} d^{2} e^{4} + 142 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} x^{2} + {\left (1377 \, c^{6} d^{5} e - 4142 \, b c^{5} d^{4} e^{2} + 4657 \, b^{2} c^{4} d^{3} e^{3} - 2460 \, b^{3} c^{3} d^{2} e^{4} + 632 \, b^{4} c^{2} d e^{5} - 64 \, b^{5} c e^{6}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{15015 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \] Input:

integrate((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, 
algorithm="maxima")
 

Output:

-2/1155*(105*c^5*e^5*x^5 + 533*c^5*d^5 - 1488*b*c^4*d^4*e + 1513*b^2*c^3*d 
^3*e^2 - 710*b^3*c^2*d^2*e^3 + 168*b^4*c*d*e^4 - 16*b^5*e^5 + 35*(7*c^5*d* 
e^4 + 4*b*c^4*e^5)*x^4 - 5*(10*c^5*d^2*e^3 - 108*b*c^4*d*e^4 - b^2*c^3*e^5 
)*x^3 - 3*(174*c^5*d^3*e^2 - 236*b*c^4*d^2*e^3 - 17*b^2*c^3*d*e^4 + 2*b^3* 
c^2*e^5)*x^2 - (311*c^5*d^4*e - 100*b*c^4*d^3*e^2 - 279*b^2*c^3*d^2*e^3 + 
76*b^3*c^2*d*e^4 - 8*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c 
^4*e^2*x + c^4*d*e) - 2/15015*(1155*c^6*e^6*x^6 + 2754*c^6*d^6 - 11038*b*c 
^5*d^5*e + 17598*b^2*c^4*d^4*e^2 - 14234*b^3*c^3*d^3*e^3 + 6184*b^4*c^2*d^ 
2*e^4 - 1392*b^5*c*d*e^5 + 128*b^6*e^6 + 105*(25*c^6*d*e^5 + 14*b*c^5*e^6) 
*x^5 - 35*(12*c^6*d^2*e^4 - 154*b*c^5*d*e^5 - b^2*c^4*e^6)*x^4 - 5*(954*c^ 
6*d^3*e^3 - 1328*b*c^5*d^2*e^4 - 63*b^2*c^4*d*e^5 + 8*b^3*c^3*e^6)*x^3 - 3 
*(907*c^6*d^4*e^2 - 560*b*c^5*d^3*e^3 - 473*b^2*c^4*d^2*e^4 + 142*b^3*c^3* 
d*e^5 - 16*b^4*c^2*e^6)*x^2 + (1377*c^6*d^5*e - 4142*b*c^5*d^4*e^2 + 4657* 
b^2*c^4*d^3*e^3 - 2460*b^3*c^3*d^2*e^4 + 632*b^4*c^2*d*e^5 - 64*b^5*c*e^6) 
*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^5*e^3*x + c^5*d*e^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7604 vs. \(2 (313) = 626\).

Time = 0.41 (sec) , antiderivative size = 7604, normalized size of antiderivative = 22.17 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/45045*(45045*sqrt(-c*e*x + c*d - b*e)*c^2*d^5*e*f - 90090*sqrt(-c*e*x + 
 c*d - b*e)*b*c*d^4*e^2*f + 45045*sqrt(-c*e*x + c*d - b*e)*b^2*d^3*e^3*f + 
 15015*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - 
(-c*e*x + c*d - b*e)^(3/2))*c*d^4*e*f - 60060*(3*sqrt(-c*e*x + c*d - b*e)* 
c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*b*d^3*e 
^2*f + 45045*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)* 
b*e - (-c*e*x + c*d - b*e)^(3/2))*b^2*d^2*e^3*f/c + 15015*(3*sqrt(-c*e*x + 
 c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3 
/2))*c*d^5*g - 30030*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d 
 - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*b*d^4*e*g + 15015*(3*sqrt(-c*e*x 
 + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^ 
(3/2))*b^2*d^3*e^2*g/c - 6006*(15*sqrt(-c*e*x + c*d - b*e)*c^2*d^2 - 30*sq 
rt(-c*e*x + c*d - b*e)*b*c*d*e + 15*sqrt(-c*e*x + c*d - b*e)*b^2*e^2 - 10* 
(-c*e*x + c*d - b*e)^(3/2)*c*d + 10*(-c*e*x + c*d - b*e)^(3/2)*b*e + 3*(c* 
e*x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b*e))*d^3*e*f + 9009*(15*sqrt(-c*e* 
x + c*d - b*e)*c^2*d^2 - 30*sqrt(-c*e*x + c*d - b*e)*b*c*d*e + 15*sqrt(-c* 
e*x + c*d - b*e)*b^2*e^2 - 10*(-c*e*x + c*d - b*e)^(3/2)*c*d + 10*(-c*e*x 
+ c*d - b*e)^(3/2)*b*e + 3*(c*e*x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b*e)) 
*b^2*d*e^3*f/c^2 + 3003*(15*sqrt(-c*e*x + c*d - b*e)*c^2*d^2 - 30*sqrt(-c* 
e*x + c*d - b*e)*b*c*d*e + 15*sqrt(-c*e*x + c*d - b*e)*b^2*e^2 - 10*(-c...
 

Mupad [B] (verification not implemented)

Time = 12.64 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.86 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^2\,x^5\,\sqrt {d+e\,x}\,\left (14\,b\,e\,g+25\,c\,d\,g+13\,c\,e\,f\right )}{143}+\frac {2\,c\,e^3\,g\,x^6\,\sqrt {d+e\,x}}{13}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,g\,b^4\,c^2\,e^6-852\,g\,b^3\,c^3\,d\,e^5-156\,f\,b^3\,c^3\,e^6+2838\,g\,b^2\,c^4\,d^2\,e^4+1326\,f\,b^2\,c^4\,d\,e^5+3360\,g\,b\,c^5\,d^3\,e^3+18408\,f\,b\,c^5\,d^2\,e^4-5442\,g\,c^6\,d^4\,e^2-13572\,f\,c^6\,d^3\,e^3\right )}{15015\,c^5\,e^3}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1136\,g\,b^3\,c\,d\,e^3-208\,f\,b^3\,c\,e^4+3784\,g\,b^2\,c^2\,d^2\,e^2+1768\,f\,b^2\,c^2\,d\,e^3-5530\,g\,b\,c^3\,d^3\,e-5486\,f\,b\,c^3\,d^2\,e^2+2754\,g\,c^4\,d^4+6929\,f\,c^4\,d^3\,e\right )}{15015\,c^5\,e^3}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (g\,b^2\,e^2+154\,g\,b\,c\,d\,e+52\,f\,b\,c\,e^2-12\,g\,c^2\,d^2+91\,f\,c^2\,d\,e\right )}{429\,c}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-80\,g\,b^3\,c^3\,e^6+630\,g\,b^2\,c^4\,d\,e^5+130\,f\,b^2\,c^4\,e^6+13280\,g\,b\,c^5\,d^2\,e^4+14040\,f\,b\,c^5\,d\,e^5-9540\,g\,c^6\,d^3\,e^3-1300\,f\,c^6\,d^2\,e^4\right )}{15015\,c^5\,e^3}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (-64\,g\,b^4\,e^4+568\,g\,b^3\,c\,d\,e^3+104\,f\,b^3\,c\,e^4-1892\,g\,b^2\,c^2\,d^2\,e^2-884\,f\,b^2\,c^2\,d\,e^3+2765\,g\,b\,c^3\,d^3\,e+2743\,f\,b\,c^3\,d^2\,e^2-1377\,g\,c^4\,d^4+4043\,f\,c^4\,d^3\,e\right )}{15015\,c^4\,e^2}\right )}{x+\frac {d}{e}} \] Input:

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

Output:

-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e^2*x^5*(d + e*x)^(1/2)* 
(14*b*e*g + 25*c*d*g + 13*c*e*f))/143 + (2*c*e^3*g*x^6*(d + e*x)^(1/2))/13 
 + (x^2*(d + e*x)^(1/2)*(96*b^4*c^2*e^6*g - 156*b^3*c^3*e^6*f - 13572*c^6* 
d^3*e^3*f - 5442*c^6*d^4*e^2*g + 18408*b*c^5*d^2*e^4*f + 1326*b^2*c^4*d*e^ 
5*f + 3360*b*c^5*d^3*e^3*g - 852*b^3*c^3*d*e^5*g + 2838*b^2*c^4*d^2*e^4*g) 
)/(15015*c^5*e^3) + (2*(b*e - c*d)^2*(d + e*x)^(1/2)*(128*b^4*e^4*g + 2754 
*c^4*d^4*g - 208*b^3*c*e^4*f + 6929*c^4*d^3*e*f - 5530*b*c^3*d^3*e*g - 113 
6*b^3*c*d*e^3*g - 5486*b*c^3*d^2*e^2*f + 1768*b^2*c^2*d*e^3*f + 3784*b^2*c 
^2*d^2*e^2*g))/(15015*c^5*e^3) + (2*e*x^4*(d + e*x)^(1/2)*(b^2*e^2*g - 12* 
c^2*d^2*g + 52*b*c*e^2*f + 91*c^2*d*e*f + 154*b*c*d*e*g))/(429*c) + (x^3*( 
d + e*x)^(1/2)*(130*b^2*c^4*e^6*f - 80*b^3*c^3*e^6*g - 1300*c^6*d^2*e^4*f 
- 9540*c^6*d^3*e^3*g + 14040*b*c^5*d*e^5*f + 13280*b*c^5*d^2*e^4*g + 630*b 
^2*c^4*d*e^5*g))/(15015*c^5*e^3) + (2*x*(b*e - c*d)*(d + e*x)^(1/2)*(104*b 
^3*c*e^4*f - 1377*c^4*d^4*g - 64*b^4*e^4*g + 4043*c^4*d^3*e*f + 2765*b*c^3 
*d^3*e*g + 568*b^3*c*d*e^3*g + 2743*b*c^3*d^2*e^2*f - 884*b^2*c^2*d*e^3*f 
- 1892*b^2*c^2*d^2*e^2*g))/(15015*c^4*e^2)))/(x + d/e)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.05 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {-c e x -b e +c d}\, \left (-1155 c^{6} e^{6} g \,x^{6}-1470 b \,c^{5} e^{6} g \,x^{5}-2625 c^{6} d \,e^{5} g \,x^{5}-1365 c^{6} e^{6} f \,x^{5}-35 b^{2} c^{4} e^{6} g \,x^{4}-5390 b \,c^{5} d \,e^{5} g \,x^{4}-1820 b \,c^{5} e^{6} f \,x^{4}+420 c^{6} d^{2} e^{4} g \,x^{4}-3185 c^{6} d \,e^{5} f \,x^{4}+40 b^{3} c^{3} e^{6} g \,x^{3}-315 b^{2} c^{4} d \,e^{5} g \,x^{3}-65 b^{2} c^{4} e^{6} f \,x^{3}-6640 b \,c^{5} d^{2} e^{4} g \,x^{3}-7020 b \,c^{5} d \,e^{5} f \,x^{3}+4770 c^{6} d^{3} e^{3} g \,x^{3}+650 c^{6} d^{2} e^{4} f \,x^{3}-48 b^{4} c^{2} e^{6} g \,x^{2}+426 b^{3} c^{3} d \,e^{5} g \,x^{2}+78 b^{3} c^{3} e^{6} f \,x^{2}-1419 b^{2} c^{4} d^{2} e^{4} g \,x^{2}-663 b^{2} c^{4} d \,e^{5} f \,x^{2}-1680 b \,c^{5} d^{3} e^{3} g \,x^{2}-9204 b \,c^{5} d^{2} e^{4} f \,x^{2}+2721 c^{6} d^{4} e^{2} g \,x^{2}+6786 c^{6} d^{3} e^{3} f \,x^{2}+64 b^{5} c \,e^{6} g x -632 b^{4} c^{2} d \,e^{5} g x -104 b^{4} c^{2} e^{6} f x +2460 b^{3} c^{3} d^{2} e^{4} g x +988 b^{3} c^{3} d \,e^{5} f x -4657 b^{2} c^{4} d^{3} e^{3} g x -3627 b^{2} c^{4} d^{2} e^{4} f x +4142 b \,c^{5} d^{4} e^{2} g x -1300 b \,c^{5} d^{3} e^{3} f x -1377 c^{6} d^{5} e g x +4043 c^{6} d^{4} e^{2} f x -128 b^{6} e^{6} g +1392 b^{5} c d \,e^{5} g +208 b^{5} c \,e^{6} f -6184 b^{4} c^{2} d^{2} e^{4} g -2184 b^{4} c^{2} d \,e^{5} f +14234 b^{3} c^{3} d^{3} e^{3} g +9230 b^{3} c^{3} d^{2} e^{4} f -17598 b^{2} c^{4} d^{4} e^{2} g -19669 b^{2} c^{4} d^{3} e^{3} f +11038 b \,c^{5} d^{5} e g +19344 b \,c^{5} d^{4} e^{2} f -2754 c^{6} d^{6} g -6929 c^{6} d^{5} e f \right )}{15015 c^{5} e^{2}} \] Input:

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

Output:

(2*sqrt( - b*e + c*d - c*e*x)*( - 128*b**6*e**6*g + 1392*b**5*c*d*e**5*g + 
 208*b**5*c*e**6*f + 64*b**5*c*e**6*g*x - 6184*b**4*c**2*d**2*e**4*g - 218 
4*b**4*c**2*d*e**5*f - 632*b**4*c**2*d*e**5*g*x - 104*b**4*c**2*e**6*f*x - 
 48*b**4*c**2*e**6*g*x**2 + 14234*b**3*c**3*d**3*e**3*g + 9230*b**3*c**3*d 
**2*e**4*f + 2460*b**3*c**3*d**2*e**4*g*x + 988*b**3*c**3*d*e**5*f*x + 426 
*b**3*c**3*d*e**5*g*x**2 + 78*b**3*c**3*e**6*f*x**2 + 40*b**3*c**3*e**6*g* 
x**3 - 17598*b**2*c**4*d**4*e**2*g - 19669*b**2*c**4*d**3*e**3*f - 4657*b* 
*2*c**4*d**3*e**3*g*x - 3627*b**2*c**4*d**2*e**4*f*x - 1419*b**2*c**4*d**2 
*e**4*g*x**2 - 663*b**2*c**4*d*e**5*f*x**2 - 315*b**2*c**4*d*e**5*g*x**3 - 
 65*b**2*c**4*e**6*f*x**3 - 35*b**2*c**4*e**6*g*x**4 + 11038*b*c**5*d**5*e 
*g + 19344*b*c**5*d**4*e**2*f + 4142*b*c**5*d**4*e**2*g*x - 1300*b*c**5*d* 
*3*e**3*f*x - 1680*b*c**5*d**3*e**3*g*x**2 - 9204*b*c**5*d**2*e**4*f*x**2 
- 6640*b*c**5*d**2*e**4*g*x**3 - 7020*b*c**5*d*e**5*f*x**3 - 5390*b*c**5*d 
*e**5*g*x**4 - 1820*b*c**5*e**6*f*x**4 - 1470*b*c**5*e**6*g*x**5 - 2754*c* 
*6*d**6*g - 6929*c**6*d**5*e*f - 1377*c**6*d**5*e*g*x + 4043*c**6*d**4*e** 
2*f*x + 2721*c**6*d**4*e**2*g*x**2 + 6786*c**6*d**3*e**3*f*x**2 + 4770*c** 
6*d**3*e**3*g*x**3 + 650*c**6*d**2*e**4*f*x**3 + 420*c**6*d**2*e**4*g*x**4 
 - 3185*c**6*d*e**5*f*x**4 - 2625*c**6*d*e**5*g*x**5 - 1365*c**6*e**6*f*x* 
*5 - 1155*c**6*e**6*g*x**6))/(15015*c**5*e**2)