\(\int (d+e x)^{5/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\) [204]

Optimal result

Integrand size = 46, antiderivative size = 421 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 (2 c d-b e)^4 (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^6 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e)^3 (4 c e f+6 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^6 e^2 (d+e x)^{7/2}}-\frac {4 (2 c d-b e)^2 (3 c e f+7 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}{9 c^6 e^2 (d+e x)^{9/2}}+\frac {4 (2 c d-b e) (2 c e f+8 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}{11 c^6 e^2 (d+e x)^{11/2}}-\frac {2 (c e f+9 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}{13 c^6 e^2 (d+e x)^{13/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{15/2}}{15 c^6 e^2 (d+e x)^{15/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.86 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-256 b^5 e^5 g+128 b^4 c e^4 (3 e f+22 d g+5 e g x)-32 b^3 c^2 e^3 \left (389 d^2 g+5 e^2 x (6 f+7 g x)+2 d e (63 f+100 g x)\right )+16 b^2 c^3 e^2 \left (1724 d^3 g+105 e^3 x^2 (f+g x)+30 d e^2 x (19 f+21 g x)+3 d^2 e (347 f+515 g x)\right )-2 b c^4 e \left (15191 d^4 g+105 e^4 x^3 (12 f+11 g x)+420 d e^3 x^2 (17 f+16 g x)+30 d^2 e^2 x (542 f+553 g x)+4 d^3 e (4131 f+5530 g x)\right )+c^5 \left (12686 d^5 g+231 e^5 x^4 (15 f+13 g x)+210 d e^4 x^3 (90 f+77 g x)+210 d^2 e^3 x^2 (203 f+173 g x)+20 d^3 e^2 x (2505 f+2212 g x)+d^4 e (29049 f+31715 g x)\right )\right )}{45045 c^6 e^2 \sqrt {d+e x}} \] Input:

Integrate[(d + e*x)^(5/2)*(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))]*(-256* 
b^5*e^5*g + 128*b^4*c*e^4*(3*e*f + 22*d*g + 5*e*g*x) - 32*b^3*c^2*e^3*(389 
*d^2*g + 5*e^2*x*(6*f + 7*g*x) + 2*d*e*(63*f + 100*g*x)) + 16*b^2*c^3*e^2* 
(1724*d^3*g + 105*e^3*x^2*(f + g*x) + 30*d*e^2*x*(19*f + 21*g*x) + 3*d^2*e 
*(347*f + 515*g*x)) - 2*b*c^4*e*(15191*d^4*g + 105*e^4*x^3*(12*f + 11*g*x) 
 + 420*d*e^3*x^2*(17*f + 16*g*x) + 30*d^2*e^2*x*(542*f + 553*g*x) + 4*d^3* 
e*(4131*f + 5530*g*x)) + c^5*(12686*d^5*g + 231*e^5*x^4*(15*f + 13*g*x) + 
210*d*e^4*x^3*(90*f + 77*g*x) + 210*d^2*e^3*x^2*(203*f + 173*g*x) + 20*d^3 
*e^2*x*(2505*f + 2212*g*x) + d^4*e*(29049*f + 31715*g*x))))/(45045*c^6*e^2 
*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1221, 1128, 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.

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

\(\Big \downarrow \) 1221

\(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \int (d+e x)^{5/2} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{3 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {8 (2 c d-b e) \int (d+e x)^{3/2} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{13 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e}\right )}{3 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \int \sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{11 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e}\right )}{13 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e}\right )}{3 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{\sqrt {d+e x}}dx}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )}{11 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e}\right )}{13 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e}\right )}{3 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{3/2}}dx}{7 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\right )}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )}{11 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e}\right )}{13 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e}\right )}{3 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}\)

\(\Big \downarrow \) 1122

\(\displaystyle \frac {\left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 c^2 e (d+e x)^{5/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\right )}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )}{11 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e}\right )}{13 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e}\right ) (-2 b e g+c d g+3 c e f)}{3 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 c e^2}\)

Input:

Int[(d + e*x)^(5/2)*(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)^(5/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15*c*e 
^2) + ((3*c*e*f + c*d*g - 2*b*e*g)*((-2*(d + e*x)^(3/2)*(d*(c*d - b*e) - b 
*e^2*x - c*e^2*x^2)^(5/2))/(13*c*e) + (8*(2*c*d - b*e)*((-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)))/(3*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 = 2.11 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.26

method	result	size
default	\(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{2} \left (-3003 g \,e^{5} x^{5} c^{5}+2310 b \,c^{4} e^{5} g \,x^{4}-16170 c^{5} d \,e^{4} g \,x^{4}-3465 c^{5} e^{5} f \,x^{4}-1680 b^{2} c^{3} e^{5} g \,x^{3}+13440 b \,c^{4} d \,e^{4} g \,x^{3}+2520 b \,c^{4} e^{5} f \,x^{3}-36330 c^{5} d^{2} e^{3} g \,x^{3}-18900 c^{5} d \,e^{4} f \,x^{3}+1120 b^{3} c^{2} e^{5} g \,x^{2}-10080 b^{2} c^{3} d \,e^{4} g \,x^{2}-1680 b^{2} c^{3} e^{5} f \,x^{2}+33180 b \,c^{4} d^{2} e^{3} g \,x^{2}+14280 b \,c^{4} d \,e^{4} f \,x^{2}-44240 c^{5} d^{3} e^{2} g \,x^{2}-42630 c^{5} d^{2} e^{3} f \,x^{2}-640 b^{4} c \,e^{5} g x +6400 b^{3} c^{2} d \,e^{4} g x +960 b^{3} c^{2} e^{5} f x -24720 b^{2} c^{3} d^{2} e^{3} g x -9120 b^{2} c^{3} d \,e^{4} f x +44240 b \,c^{4} d^{3} e^{2} g x +32520 b \,c^{4} d^{2} e^{3} f x -31715 c^{5} d^{4} e g x -50100 c^{5} d^{3} e^{2} f x +256 b^{5} e^{5} g -2816 b^{4} c d \,e^{4} g -384 b^{4} c \,e^{5} f +12448 b^{3} c^{2} d^{2} e^{3} g +4032 b^{3} c^{2} d \,e^{4} f -27584 b^{2} c^{3} d^{3} e^{2} g -16656 b^{2} c^{3} d^{2} e^{3} f +30382 b \,c^{4} d^{4} e g +33048 b \,c^{4} d^{3} e^{2} f -12686 c^{5} d^{5} g -29049 f \,d^{4} c^{5} e \right )}{45045 \sqrt {e x +d}\, c^{6} e^{2}}\)	\(529\)
gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-3003 g \,e^{5} x^{5} c^{5}+2310 b \,c^{4} e^{5} g \,x^{4}-16170 c^{5} d \,e^{4} g \,x^{4}-3465 c^{5} e^{5} f \,x^{4}-1680 b^{2} c^{3} e^{5} g \,x^{3}+13440 b \,c^{4} d \,e^{4} g \,x^{3}+2520 b \,c^{4} e^{5} f \,x^{3}-36330 c^{5} d^{2} e^{3} g \,x^{3}-18900 c^{5} d \,e^{4} f \,x^{3}+1120 b^{3} c^{2} e^{5} g \,x^{2}-10080 b^{2} c^{3} d \,e^{4} g \,x^{2}-1680 b^{2} c^{3} e^{5} f \,x^{2}+33180 b \,c^{4} d^{2} e^{3} g \,x^{2}+14280 b \,c^{4} d \,e^{4} f \,x^{2}-44240 c^{5} d^{3} e^{2} g \,x^{2}-42630 c^{5} d^{2} e^{3} f \,x^{2}-640 b^{4} c \,e^{5} g x +6400 b^{3} c^{2} d \,e^{4} g x +960 b^{3} c^{2} e^{5} f x -24720 b^{2} c^{3} d^{2} e^{3} g x -9120 b^{2} c^{3} d \,e^{4} f x +44240 b \,c^{4} d^{3} e^{2} g x +32520 b \,c^{4} d^{2} e^{3} f x -31715 c^{5} d^{4} e g x -50100 c^{5} d^{3} e^{2} f x +256 b^{5} e^{5} g -2816 b^{4} c d \,e^{4} g -384 b^{4} c \,e^{5} f +12448 b^{3} c^{2} d^{2} e^{3} g +4032 b^{3} c^{2} d \,e^{4} f -27584 b^{2} c^{3} d^{3} e^{2} g -16656 b^{2} c^{3} d^{2} e^{3} f +30382 b \,c^{4} d^{4} e g +33048 b \,c^{4} d^{3} e^{2} f -12686 c^{5} d^{5} g -29049 f \,d^{4} c^{5} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{45045 c^{6} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\)	\(535\)
orering	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-3003 g \,e^{5} x^{5} c^{5}+2310 b \,c^{4} e^{5} g \,x^{4}-16170 c^{5} d \,e^{4} g \,x^{4}-3465 c^{5} e^{5} f \,x^{4}-1680 b^{2} c^{3} e^{5} g \,x^{3}+13440 b \,c^{4} d \,e^{4} g \,x^{3}+2520 b \,c^{4} e^{5} f \,x^{3}-36330 c^{5} d^{2} e^{3} g \,x^{3}-18900 c^{5} d \,e^{4} f \,x^{3}+1120 b^{3} c^{2} e^{5} g \,x^{2}-10080 b^{2} c^{3} d \,e^{4} g \,x^{2}-1680 b^{2} c^{3} e^{5} f \,x^{2}+33180 b \,c^{4} d^{2} e^{3} g \,x^{2}+14280 b \,c^{4} d \,e^{4} f \,x^{2}-44240 c^{5} d^{3} e^{2} g \,x^{2}-42630 c^{5} d^{2} e^{3} f \,x^{2}-640 b^{4} c \,e^{5} g x +6400 b^{3} c^{2} d \,e^{4} g x +960 b^{3} c^{2} e^{5} f x -24720 b^{2} c^{3} d^{2} e^{3} g x -9120 b^{2} c^{3} d \,e^{4} f x +44240 b \,c^{4} d^{3} e^{2} g x +32520 b \,c^{4} d^{2} e^{3} f x -31715 c^{5} d^{4} e g x -50100 c^{5} d^{3} e^{2} f x +256 b^{5} e^{5} g -2816 b^{4} c d \,e^{4} g -384 b^{4} c \,e^{5} f +12448 b^{3} c^{2} d^{2} e^{3} g +4032 b^{3} c^{2} d \,e^{4} f -27584 b^{2} c^{3} d^{3} e^{2} g -16656 b^{2} c^{3} d^{2} e^{3} f +30382 b \,c^{4} d^{4} e g +33048 b \,c^{4} d^{3} e^{2} f -12686 c^{5} d^{5} g -29049 f \,d^{4} c^{5} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{45045 c^{6} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\)	\(535\)

Input:

int((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method= 
_RETURNVERBOSE)
 

Output:

2/45045/(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*( 
-3003*c^5*e^5*g*x^5+2310*b*c^4*e^5*g*x^4-16170*c^5*d*e^4*g*x^4-3465*c^5*e^ 
5*f*x^4-1680*b^2*c^3*e^5*g*x^3+13440*b*c^4*d*e^4*g*x^3+2520*b*c^4*e^5*f*x^ 
3-36330*c^5*d^2*e^3*g*x^3-18900*c^5*d*e^4*f*x^3+1120*b^3*c^2*e^5*g*x^2-100 
80*b^2*c^3*d*e^4*g*x^2-1680*b^2*c^3*e^5*f*x^2+33180*b*c^4*d^2*e^3*g*x^2+14 
280*b*c^4*d*e^4*f*x^2-44240*c^5*d^3*e^2*g*x^2-42630*c^5*d^2*e^3*f*x^2-640* 
b^4*c*e^5*g*x+6400*b^3*c^2*d*e^4*g*x+960*b^3*c^2*e^5*f*x-24720*b^2*c^3*d^2 
*e^3*g*x-9120*b^2*c^3*d*e^4*f*x+44240*b*c^4*d^3*e^2*g*x+32520*b*c^4*d^2*e^ 
3*f*x-31715*c^5*d^4*e*g*x-50100*c^5*d^3*e^2*f*x+256*b^5*e^5*g-2816*b^4*c*d 
*e^4*g-384*b^4*c*e^5*f+12448*b^3*c^2*d^2*e^3*g+4032*b^3*c^2*d*e^4*f-27584* 
b^2*c^3*d^3*e^2*g-16656*b^2*c^3*d^2*e^3*f+30382*b*c^4*d^4*e*g+33048*b*c^4* 
d^3*e^2*f-12686*c^5*d^5*g-29049*c^5*d^4*e*f)/c^6/e^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (385) = 770\).

Time = 0.13 (sec) , antiderivative size = 880, normalized size of antiderivative = 2.09 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, 
algorithm="fricas")
 

Output:

-2/45045*(3003*c^7*e^7*g*x^7 + 231*(15*c^7*e^7*f + 4*(11*c^7*d*e^6 + 4*b*c 
^6*e^7)*g)*x^6 + 63*(10*(19*c^7*d*e^6 + 7*b*c^6*e^7)*f + (111*c^7*d^2*e^5 
+ 278*b*c^6*d*e^6 + b^2*c^5*e^7)*g)*x^5 + 35*(3*(79*c^7*d^2*e^5 + 206*b*c^ 
6*d*e^6 + b^2*c^5*e^7)*f - 2*(175*c^7*d^3*e^4 - 453*b*c^6*d^2*e^5 - 9*b^2* 
c^5*d*e^6 + b^3*c^4*e^7)*g)*x^4 - 5*(12*(271*c^7*d^3*e^4 - 683*b*c^6*d^2*e 
^5 - 19*b^2*c^5*d*e^6 + 2*b^3*c^4*e^7)*f + (4087*c^7*d^4*e^3 - 4900*b*c^6* 
d^3*e^4 - 618*b^2*c^5*d^2*e^5 + 160*b^3*c^4*d*e^6 - 16*b^4*c^3*e^7)*g)*x^3 
 - 3*(3*(3169*c^7*d^4*e^3 - 3628*b*c^6*d^3*e^4 - 694*b^2*c^5*d^2*e^5 + 168 
*b^3*c^4*d*e^6 - 16*b^4*c^3*e^7)*f + 4*(542*c^7*d^5*e^2 + 11*b*c^6*d^4*e^3 
 - 862*b^2*c^5*d^3*e^4 + 389*b^3*c^4*d^2*e^5 - 88*b^4*c^3*d*e^6 + 8*b^5*c^ 
2*e^7)*g)*x^2 + 3*(9683*c^7*d^6*e - 30382*b*c^6*d^5*e^2 + 37267*b^2*c^5*d^ 
4*e^3 - 23464*b^3*c^4*d^3*e^4 + 8368*b^4*c^3*d^2*e^5 - 1600*b^5*c^2*d*e^6 
+ 128*b^6*c*e^7)*f + 2*(6343*c^7*d^7 - 27877*b*c^6*d^6*e + 50517*b^2*c^5*d 
^5*e^2 - 48999*b^3*c^4*d^4*e^3 + 27648*b^4*c^3*d^3*e^4 - 9168*b^5*c^2*d^2* 
e^5 + 1664*b^6*c*d*e^6 - 128*b^7*e^7)*g - (6*(1333*c^7*d^5*e^2 + 1421*b*c^ 
6*d^4*e^3 - 4142*b^2*c^5*d^3*e^4 + 1724*b^3*c^4*d^2*e^5 - 368*b^4*c^3*d*e^ 
6 + 32*b^5*c^2*e^7)*f - (6343*c^7*d^6*e - 21534*b*c^6*d^5*e^2 + 28983*b^2* 
c^5*d^4*e^3 - 20016*b^3*c^4*d^3*e^4 + 7632*b^4*c^3*d^2*e^5 - 1536*b^5*c^2* 
d*e^6 + 128*b^6*c*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sq 
rt(e*x + d)/(c^6*e^3*x + c^6*d*e^2)
 

Sympy [F]

\[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )\, dx \] Input:

integrate((e*x+d)**(5/2)*(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)**(5/2)*(f + g*x 
), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (385) = 770\).

Time = 0.11 (sec) , antiderivative size = 875, normalized size of antiderivative = 2.08 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, 
algorithm="maxima")
 

Output:

-2/15015*(1155*c^6*e^6*x^6 + 9683*c^6*d^6 - 30382*b*c^5*d^5*e + 37267*b^2* 
c^4*d^4*e^2 - 23464*b^3*c^3*d^3*e^3 + 8368*b^4*c^2*d^2*e^4 - 1600*b^5*c*d* 
e^5 + 128*b^6*e^6 + 210*(19*c^6*d*e^5 + 7*b*c^5*e^6)*x^5 + 35*(79*c^6*d^2* 
e^4 + 206*b*c^5*d*e^5 + b^2*c^4*e^6)*x^4 - 20*(271*c^6*d^3*e^3 - 683*b*c^5 
*d^2*e^4 - 19*b^2*c^4*d*e^5 + 2*b^3*c^3*e^6)*x^3 - 3*(3169*c^6*d^4*e^2 - 3 
628*b*c^5*d^3*e^3 - 694*b^2*c^4*d^2*e^4 + 168*b^3*c^3*d*e^5 - 16*b^4*c^2*e 
^6)*x^2 - 2*(1333*c^6*d^5*e + 1421*b*c^5*d^4*e^2 - 4142*b^2*c^4*d^3*e^3 + 
1724*b^3*c^3*d^2*e^4 - 368*b^4*c^2*d*e^5 + 32*b^5*c*e^6)*x)*sqrt(-c*e*x + 
c*d - b*e)*(e*x + d)*f/(c^5*e^2*x + c^5*d*e) - 2/45045*(3003*c^7*e^7*x^7 + 
 12686*c^7*d^7 - 55754*b*c^6*d^6*e + 101034*b^2*c^5*d^5*e^2 - 97998*b^3*c^ 
4*d^4*e^3 + 55296*b^4*c^3*d^3*e^4 - 18336*b^5*c^2*d^2*e^5 + 3328*b^6*c*d*e 
^6 - 256*b^7*e^7 + 924*(11*c^7*d*e^6 + 4*b*c^6*e^7)*x^6 + 63*(111*c^7*d^2* 
e^5 + 278*b*c^6*d*e^6 + b^2*c^5*e^7)*x^5 - 70*(175*c^7*d^3*e^4 - 453*b*c^6 
*d^2*e^5 - 9*b^2*c^5*d*e^6 + b^3*c^4*e^7)*x^4 - 5*(4087*c^7*d^4*e^3 - 4900 
*b*c^6*d^3*e^4 - 618*b^2*c^5*d^2*e^5 + 160*b^3*c^4*d*e^6 - 16*b^4*c^3*e^7) 
*x^3 - 12*(542*c^7*d^5*e^2 + 11*b*c^6*d^4*e^3 - 862*b^2*c^5*d^3*e^4 + 389* 
b^3*c^4*d^2*e^5 - 88*b^4*c^3*d*e^6 + 8*b^5*c^2*e^7)*x^2 + (6343*c^7*d^6*e 
- 21534*b*c^6*d^5*e^2 + 28983*b^2*c^5*d^4*e^3 - 20016*b^3*c^4*d^3*e^4 + 76 
32*b^4*c^3*d^2*e^5 - 1536*b^5*c^2*d*e^6 + 128*b^6*c*e^7)*x)*sqrt(-c*e*x + 
c*d - b*e)*(e*x + d)*g/(c^6*e^3*x + c^6*d*e^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 12772 vs. \(2 (385) = 770\).

Time = 0.48 (sec) , antiderivative size = 12772, normalized size of antiderivative = 30.34 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)*(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^6*e*f - 90090*sqrt(-c*e*x + 
 c*d - b*e)*b*c*d^5*e^2*f + 45045*sqrt(-c*e*x + c*d - b*e)*b^2*d^4*e^3*f + 
 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))*c*d^5*e*f - 90090*(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 
^2*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^2*d^3*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^6*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^5*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^4*e^2*g/c - 3003*(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^4*e*f - 12012*(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*d^3*e^2*f/c + 18018*(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 = 12.98 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.05 \[ \int (d+e x)^{5/2} (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^3\,x^6\,\sqrt {d+e\,x}\,\left (16\,b\,e\,g+44\,c\,d\,g+15\,c\,e\,f\right )}{195}+\frac {2\,e^2\,x^5\,\sqrt {d+e\,x}\,\left (g\,b^2\,e^2+278\,g\,b\,c\,d\,e+70\,f\,b\,c\,e^2+111\,g\,c^2\,d^2+190\,f\,c^2\,d\,e\right )}{715\,c}+\frac {2\,c\,e^4\,g\,x^7\,\sqrt {d+e\,x}}{15}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (-256\,g\,b^5\,e^5+2816\,g\,b^4\,c\,d\,e^4+384\,f\,b^4\,c\,e^5-12448\,g\,b^3\,c^2\,d^2\,e^3-4032\,f\,b^3\,c^2\,d\,e^4+27584\,g\,b^2\,c^3\,d^3\,e^2+16656\,f\,b^2\,c^3\,d^2\,e^3-30382\,g\,b\,c^4\,d^4\,e-33048\,f\,b\,c^4\,d^3\,e^2+12686\,g\,c^5\,d^5+29049\,f\,c^5\,d^4\,e\right )}{45045\,c^6\,e^3}+\frac {x^3\,\sqrt {d+e\,x}\,\left (160\,g\,b^4\,c^3\,e^7-1600\,g\,b^3\,c^4\,d\,e^6-240\,f\,b^3\,c^4\,e^7+6180\,g\,b^2\,c^5\,d^2\,e^5+2280\,f\,b^2\,c^5\,d\,e^6+49000\,g\,b\,c^6\,d^3\,e^4+81960\,f\,b\,c^6\,d^2\,e^5-40870\,g\,c^7\,d^4\,e^3-32520\,f\,c^7\,d^3\,e^4\right )}{45045\,c^6\,e^3}+\frac {x^4\,\sqrt {d+e\,x}\,\left (-140\,g\,b^3\,c^4\,e^7+1260\,g\,b^2\,c^5\,d\,e^6+210\,f\,b^2\,c^5\,e^7+63420\,g\,b\,c^6\,d^2\,e^5+43260\,f\,b\,c^6\,d\,e^6-24500\,g\,c^7\,d^3\,e^4+16590\,f\,c^7\,d^2\,e^5\right )}{45045\,c^6\,e^3}-\frac {x^2\,\sqrt {d+e\,x}\,\left (192\,g\,b^5\,c^2\,e^7-2112\,g\,b^4\,c^3\,d\,e^6-288\,f\,b^4\,c^3\,e^7+9336\,g\,b^3\,c^4\,d^2\,e^5+3024\,f\,b^3\,c^4\,d\,e^6-20688\,g\,b^2\,c^5\,d^3\,e^4-12492\,f\,b^2\,c^5\,d^2\,e^5+264\,g\,b\,c^6\,d^4\,e^3-65304\,f\,b\,c^6\,d^3\,e^4+13008\,g\,c^7\,d^5\,e^2+57042\,f\,c^7\,d^4\,e^3\right )}{45045\,c^6\,e^3}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (128\,g\,b^5\,e^5-1408\,g\,b^4\,c\,d\,e^4-192\,f\,b^4\,c\,e^5+6224\,g\,b^3\,c^2\,d^2\,e^3+2016\,f\,b^3\,c^2\,d\,e^4-13792\,g\,b^2\,c^3\,d^3\,e^2-8328\,f\,b^2\,c^3\,d^2\,e^3+15191\,g\,b\,c^4\,d^4\,e+16524\,f\,b\,c^4\,d^3\,e^2-6343\,g\,c^5\,d^5+7998\,f\,c^5\,d^4\,e\right )}{45045\,c^5\,e^2}\right )}{x+\frac {d}{e}} \] Input:

int((f + g*x)*(d + e*x)^(5/2)*(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^3*x^6*(d + e*x)^(1/2)* 
(16*b*e*g + 44*c*d*g + 15*c*e*f))/195 + (2*e^2*x^5*(d + e*x)^(1/2)*(b^2*e^ 
2*g + 111*c^2*d^2*g + 70*b*c*e^2*f + 190*c^2*d*e*f + 278*b*c*d*e*g))/(715* 
c) + (2*c*e^4*g*x^7*(d + e*x)^(1/2))/15 + (2*(b*e - c*d)^2*(d + e*x)^(1/2) 
*(12686*c^5*d^5*g - 256*b^5*e^5*g + 384*b^4*c*e^5*f + 29049*c^5*d^4*e*f - 
30382*b*c^4*d^4*e*g + 2816*b^4*c*d*e^4*g - 33048*b*c^4*d^3*e^2*f - 4032*b^ 
3*c^2*d*e^4*f + 16656*b^2*c^3*d^2*e^3*f + 27584*b^2*c^3*d^3*e^2*g - 12448* 
b^3*c^2*d^2*e^3*g))/(45045*c^6*e^3) + (x^3*(d + e*x)^(1/2)*(160*b^4*c^3*e^ 
7*g - 240*b^3*c^4*e^7*f - 32520*c^7*d^3*e^4*f - 40870*c^7*d^4*e^3*g + 8196 
0*b*c^6*d^2*e^5*f + 2280*b^2*c^5*d*e^6*f + 49000*b*c^6*d^3*e^4*g - 1600*b^ 
3*c^4*d*e^6*g + 6180*b^2*c^5*d^2*e^5*g))/(45045*c^6*e^3) + (x^4*(d + e*x)^ 
(1/2)*(210*b^2*c^5*e^7*f - 140*b^3*c^4*e^7*g + 16590*c^7*d^2*e^5*f - 24500 
*c^7*d^3*e^4*g + 43260*b*c^6*d*e^6*f + 63420*b*c^6*d^2*e^5*g + 1260*b^2*c^ 
5*d*e^6*g))/(45045*c^6*e^3) - (x^2*(d + e*x)^(1/2)*(192*b^5*c^2*e^7*g - 28 
8*b^4*c^3*e^7*f + 57042*c^7*d^4*e^3*f + 13008*c^7*d^5*e^2*g - 65304*b*c^6* 
d^3*e^4*f + 3024*b^3*c^4*d*e^6*f + 264*b*c^6*d^4*e^3*g - 2112*b^4*c^3*d*e^ 
6*g - 12492*b^2*c^5*d^2*e^5*f - 20688*b^2*c^5*d^3*e^4*g + 9336*b^3*c^4*d^2 
*e^5*g))/(45045*c^6*e^3) + (2*x*(b*e - c*d)*(d + e*x)^(1/2)*(128*b^5*e^5*g 
 - 6343*c^5*d^5*g - 192*b^4*c*e^5*f + 7998*c^5*d^4*e*f + 15191*b*c^4*d^4*e 
*g - 1408*b^4*c*d*e^4*g + 16524*b*c^4*d^3*e^2*f + 2016*b^3*c^2*d*e^4*f ...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 944, normalized size of antiderivative = 2.24 \[ \int (d+e x)^{5/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^(5/2)*(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)*(256*b**7*e**7*g - 3328*b**6*c*d*e**6*g - 38 
4*b**6*c*e**7*f - 128*b**6*c*e**7*g*x + 18336*b**5*c**2*d**2*e**5*g + 4800 
*b**5*c**2*d*e**6*f + 1536*b**5*c**2*d*e**6*g*x + 192*b**5*c**2*e**7*f*x + 
 96*b**5*c**2*e**7*g*x**2 - 55296*b**4*c**3*d**3*e**4*g - 25104*b**4*c**3* 
d**2*e**5*f - 7632*b**4*c**3*d**2*e**5*g*x - 2208*b**4*c**3*d*e**6*f*x - 1 
056*b**4*c**3*d*e**6*g*x**2 - 144*b**4*c**3*e**7*f*x**2 - 80*b**4*c**3*e** 
7*g*x**3 + 97998*b**3*c**4*d**4*e**3*g + 70392*b**3*c**4*d**3*e**4*f + 200 
16*b**3*c**4*d**3*e**4*g*x + 10344*b**3*c**4*d**2*e**5*f*x + 4668*b**3*c** 
4*d**2*e**5*g*x**2 + 1512*b**3*c**4*d*e**6*f*x**2 + 800*b**3*c**4*d*e**6*g 
*x**3 + 120*b**3*c**4*e**7*f*x**3 + 70*b**3*c**4*e**7*g*x**4 - 101034*b**2 
*c**5*d**5*e**2*g - 111801*b**2*c**5*d**4*e**3*f - 28983*b**2*c**5*d**4*e* 
*3*g*x - 24852*b**2*c**5*d**3*e**4*f*x - 10344*b**2*c**5*d**3*e**4*g*x**2 
- 6246*b**2*c**5*d**2*e**5*f*x**2 - 3090*b**2*c**5*d**2*e**5*g*x**3 - 1140 
*b**2*c**5*d*e**6*f*x**3 - 630*b**2*c**5*d*e**6*g*x**4 - 105*b**2*c**5*e** 
7*f*x**4 - 63*b**2*c**5*e**7*g*x**5 + 55754*b*c**6*d**6*e*g + 91146*b*c**6 
*d**5*e**2*f + 21534*b*c**6*d**5*e**2*g*x + 8526*b*c**6*d**4*e**3*f*x + 13 
2*b*c**6*d**4*e**3*g*x**2 - 32652*b*c**6*d**3*e**4*f*x**2 - 24500*b*c**6*d 
**3*e**4*g*x**3 - 40980*b*c**6*d**2*e**5*f*x**3 - 31710*b*c**6*d**2*e**5*g 
*x**4 - 21630*b*c**6*d*e**6*f*x**4 - 17514*b*c**6*d*e**6*g*x**5 - 4410*b*c 
**6*e**7*f*x**5 - 3696*b*c**6*e**7*g*x**6 - 12686*c**7*d**7*g - 29049*c...