\(\int (d+e x)^{3/2} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\) [215]

Optimal result

Integrand size = 46, antiderivative size = 421 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=-\frac {2 (2 c d-b e)^4 (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^6 e^2 (d+e x)^{7/2}}+\frac {2 (2 c d-b e)^3 (4 c e f+6 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}{9 c^6 e^2 (d+e x)^{9/2}}-\frac {4 (2 c d-b e)^2 (3 c e f+7 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}{11 c^6 e^2 (d+e x)^{11/2}}+\frac {4 (2 c d-b e) (2 c e f+8 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}{13 c^6 e^2 (d+e x)^{13/2}}-\frac {2 (c e f+9 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{15/2}}{15 c^6 e^2 (d+e x)^{15/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{17/2}}{17 c^6 e^2 (d+e x)^{17/2}} \] Output:

-2/7*(-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)^ 
(7/2)/c^6/e^2/(e*x+d)^(7/2)+2/9*(-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)^(9/2)/c^6/e^2/(e*x+d)^(9/2)-4/11*(-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)^(11/2)/ 
c^6/e^2/(e*x+d)^(11/2)+4/13*(-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)^(13/2)/c^6/e^2/(e*x+d)^(13/2)-2/15*(-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)^(15/2)/c^6/e^2/(e*x+d)^(15/2 
)+2/17*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(17/2)/c^6/e^2/(e*x+d)^(17/2)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-1280 b^5 e^5 g+128 b^4 c e^4 (17 e f+118 d g+35 e g x)-32 b^3 c^2 e^3 \left (2253 d^2 g+7 e^2 x (34 f+45 g x)+2 d e (391 f+756 g x)\right )+16 b^2 c^3 e^2 \left (10864 d^3 g+294 d e^2 x (17 f+21 g x)+21 e^3 x^2 (51 f+55 g x)+3 d^2 e (2397 f+4249 g x)\right )-2 b c^4 e \left (104843 d^4 g+231 e^4 x^3 (68 f+65 g x)+84 d e^3 x^2 (969 f+968 g x)+42 d^2 e^2 x (3842 f+4287 g x)+4 d^3 e (32623 f+50554 g x)\right )+c^5 \left (94134 d^5 g+3003 e^5 x^4 (17 f+15 g x)+462 d e^4 x^3 (578 f+507 g x)+126 d^2 e^3 x^2 (4471 f+3949 g x)+28 d^3 e^2 x (21097 f+19638 g x)+d^4 e (278171 f+329469 g x)\right )\right )}{765765 c^6 e^2 \sqrt {d+e x}} \] 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)^ 
(5/2),x]
 

Output:

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-1280* 
b^5*e^5*g + 128*b^4*c*e^4*(17*e*f + 118*d*g + 35*e*g*x) - 32*b^3*c^2*e^3*( 
2253*d^2*g + 7*e^2*x*(34*f + 45*g*x) + 2*d*e*(391*f + 756*g*x)) + 16*b^2*c 
^3*e^2*(10864*d^3*g + 294*d*e^2*x*(17*f + 21*g*x) + 21*e^3*x^2*(51*f + 55* 
g*x) + 3*d^2*e*(2397*f + 4249*g*x)) - 2*b*c^4*e*(104843*d^4*g + 231*e^4*x^ 
3*(68*f + 65*g*x) + 84*d*e^3*x^2*(969*f + 968*g*x) + 42*d^2*e^2*x*(3842*f 
+ 4287*g*x) + 4*d^3*e*(32623*f + 50554*g*x)) + c^5*(94134*d^5*g + 3003*e^5 
*x^4*(17*f + 15*g*x) + 462*d*e^4*x^3*(578*f + 507*g*x) + 126*d^2*e^3*x^2*( 
4471*f + 3949*g*x) + 28*d^3*e^2*x*(21097*f + 19638*g*x) + d^4*e*(278171*f 
+ 329469*g*x))))/(765765*c^6*e^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 389, 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.

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

\(\Big \downarrow \) 1221

\(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \int (d+e x)^{3/2} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \left (\frac {8 (2 c d-b e) \int \sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{\sqrt {d+e x}}dx}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{3/2}}dx}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {(-10 b e g+3 c d g+17 c e f) \left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right )}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\)

\(\Big \downarrow \) 1122

\(\displaystyle \frac {\left (\frac {8 (2 c d-b e) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 c^2 e (d+e x)^{7/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e (d+e x)^{3/2}}\right )}{13 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e \sqrt {d+e x}}\right )}{15 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e}\right ) (-10 b e g+3 c d g+17 c e f)}{17 c e}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{17 c e^2}\)

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)^(5/2), 
x]
 

Output:

(-2*g*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(17*c*e 
^2) + ((17*c*e*f + 3*c*d*g - 10*b*e*g)*((-2*Sqrt[d + e*x]*(d*(c*d - b*e) - 
 b*e^2*x - c*e^2*x^2)^(7/2))/(15*c*e) + (8*(2*c*d - b*e)*((-2*(d*(c*d - b* 
e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*c*e*Sqrt[d + e*x]) + (6*(2*c*d - b*e) 
*((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*c*e*(d + e*x)^(3/2) 
) + (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)^(7/2))/(63*c^2*e*(d + e*x)^(7/2)) - (2*(d*(c*d - b*e) - b*e^2*x - c*e^ 
2*x^2)^(7/2))/(9*c*e*(d + e*x)^(5/2))))/(11*c)))/(13*c)))/(15*c)))/(17*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 = 1.96 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.26

method	result	size
default	\(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{3} \left (-45045 g \,e^{5} x^{5} c^{5}+30030 b \,c^{4} e^{5} g \,x^{4}-234234 c^{5} d \,e^{4} g \,x^{4}-51051 c^{5} e^{5} f \,x^{4}-18480 b^{2} c^{3} e^{5} g \,x^{3}+162624 b \,c^{4} d \,e^{4} g \,x^{3}+31416 b \,c^{4} e^{5} f \,x^{3}-497574 c^{5} d^{2} e^{3} g \,x^{3}-267036 c^{5} d \,e^{4} f \,x^{3}+10080 b^{3} c^{2} e^{5} g \,x^{2}-98784 b^{2} c^{3} d \,e^{4} g \,x^{2}-17136 b^{2} c^{3} e^{5} f \,x^{2}+360108 b \,c^{4} d^{2} e^{3} g \,x^{2}+162792 b \,c^{4} d \,e^{4} f \,x^{2}-549864 c^{5} d^{3} e^{2} g \,x^{2}-563346 c^{5} d^{2} e^{3} f \,x^{2}-4480 b^{4} c \,e^{5} g x +48384 b^{3} c^{2} d \,e^{4} g x +7616 b^{3} c^{2} e^{5} f x -203952 b^{2} c^{3} d^{2} e^{3} g x -79968 b^{2} c^{3} d \,e^{4} f x +404432 b \,c^{4} d^{3} e^{2} g x +322728 b \,c^{4} d^{2} e^{3} f x -329469 c^{5} d^{4} e g x -590716 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -15104 b^{4} c d \,e^{4} g -2176 b^{4} c \,e^{5} f +72096 b^{3} c^{2} d^{2} e^{3} g +25024 b^{3} c^{2} d \,e^{4} f -173824 b^{2} c^{3} d^{3} e^{2} g -115056 b^{2} c^{3} d^{2} e^{3} f +209686 b \,c^{4} d^{4} e g +260984 b \,c^{4} d^{3} e^{2} f -94134 c^{5} d^{5} g -278171 f \,d^{4} c^{5} e \right )}{765765 \sqrt {e x +d}\, c^{6} e^{2}}\)	\(529\)
gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-45045 g \,e^{5} x^{5} c^{5}+30030 b \,c^{4} e^{5} g \,x^{4}-234234 c^{5} d \,e^{4} g \,x^{4}-51051 c^{5} e^{5} f \,x^{4}-18480 b^{2} c^{3} e^{5} g \,x^{3}+162624 b \,c^{4} d \,e^{4} g \,x^{3}+31416 b \,c^{4} e^{5} f \,x^{3}-497574 c^{5} d^{2} e^{3} g \,x^{3}-267036 c^{5} d \,e^{4} f \,x^{3}+10080 b^{3} c^{2} e^{5} g \,x^{2}-98784 b^{2} c^{3} d \,e^{4} g \,x^{2}-17136 b^{2} c^{3} e^{5} f \,x^{2}+360108 b \,c^{4} d^{2} e^{3} g \,x^{2}+162792 b \,c^{4} d \,e^{4} f \,x^{2}-549864 c^{5} d^{3} e^{2} g \,x^{2}-563346 c^{5} d^{2} e^{3} f \,x^{2}-4480 b^{4} c \,e^{5} g x +48384 b^{3} c^{2} d \,e^{4} g x +7616 b^{3} c^{2} e^{5} f x -203952 b^{2} c^{3} d^{2} e^{3} g x -79968 b^{2} c^{3} d \,e^{4} f x +404432 b \,c^{4} d^{3} e^{2} g x +322728 b \,c^{4} d^{2} e^{3} f x -329469 c^{5} d^{4} e g x -590716 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -15104 b^{4} c d \,e^{4} g -2176 b^{4} c \,e^{5} f +72096 b^{3} c^{2} d^{2} e^{3} g +25024 b^{3} c^{2} d \,e^{4} f -173824 b^{2} c^{3} d^{3} e^{2} g -115056 b^{2} c^{3} d^{2} e^{3} f +209686 b \,c^{4} d^{4} e g +260984 b \,c^{4} d^{3} e^{2} f -94134 c^{5} d^{5} g -278171 f \,d^{4} c^{5} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{765765 c^{6} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\)	\(535\)
orering	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-45045 g \,e^{5} x^{5} c^{5}+30030 b \,c^{4} e^{5} g \,x^{4}-234234 c^{5} d \,e^{4} g \,x^{4}-51051 c^{5} e^{5} f \,x^{4}-18480 b^{2} c^{3} e^{5} g \,x^{3}+162624 b \,c^{4} d \,e^{4} g \,x^{3}+31416 b \,c^{4} e^{5} f \,x^{3}-497574 c^{5} d^{2} e^{3} g \,x^{3}-267036 c^{5} d \,e^{4} f \,x^{3}+10080 b^{3} c^{2} e^{5} g \,x^{2}-98784 b^{2} c^{3} d \,e^{4} g \,x^{2}-17136 b^{2} c^{3} e^{5} f \,x^{2}+360108 b \,c^{4} d^{2} e^{3} g \,x^{2}+162792 b \,c^{4} d \,e^{4} f \,x^{2}-549864 c^{5} d^{3} e^{2} g \,x^{2}-563346 c^{5} d^{2} e^{3} f \,x^{2}-4480 b^{4} c \,e^{5} g x +48384 b^{3} c^{2} d \,e^{4} g x +7616 b^{3} c^{2} e^{5} f x -203952 b^{2} c^{3} d^{2} e^{3} g x -79968 b^{2} c^{3} d \,e^{4} f x +404432 b \,c^{4} d^{3} e^{2} g x +322728 b \,c^{4} d^{2} e^{3} f x -329469 c^{5} d^{4} e g x -590716 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -15104 b^{4} c d \,e^{4} g -2176 b^{4} c \,e^{5} f +72096 b^{3} c^{2} d^{2} e^{3} g +25024 b^{3} c^{2} d \,e^{4} f -173824 b^{2} c^{3} d^{3} e^{2} g -115056 b^{2} c^{3} d^{2} e^{3} f +209686 b \,c^{4} d^{4} e g +260984 b \,c^{4} d^{3} e^{2} f -94134 c^{5} d^{5} g -278171 f \,d^{4} c^{5} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{765765 c^{6} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\)	\(535\)

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)^(5/2),x,method= 
_RETURNVERBOSE)
 

Output:

-2/765765/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(c*e*x+b*e-c*d)^3 
*(-45045*c^5*e^5*g*x^5+30030*b*c^4*e^5*g*x^4-234234*c^5*d*e^4*g*x^4-51051* 
c^5*e^5*f*x^4-18480*b^2*c^3*e^5*g*x^3+162624*b*c^4*d*e^4*g*x^3+31416*b*c^4 
*e^5*f*x^3-497574*c^5*d^2*e^3*g*x^3-267036*c^5*d*e^4*f*x^3+10080*b^3*c^2*e 
^5*g*x^2-98784*b^2*c^3*d*e^4*g*x^2-17136*b^2*c^3*e^5*f*x^2+360108*b*c^4*d^ 
2*e^3*g*x^2+162792*b*c^4*d*e^4*f*x^2-549864*c^5*d^3*e^2*g*x^2-563346*c^5*d 
^2*e^3*f*x^2-4480*b^4*c*e^5*g*x+48384*b^3*c^2*d*e^4*g*x+7616*b^3*c^2*e^5*f 
*x-203952*b^2*c^3*d^2*e^3*g*x-79968*b^2*c^3*d*e^4*f*x+404432*b*c^4*d^3*e^2 
*g*x+322728*b*c^4*d^2*e^3*f*x-329469*c^5*d^4*e*g*x-590716*c^5*d^3*e^2*f*x+ 
1280*b^5*e^5*g-15104*b^4*c*d*e^4*g-2176*b^4*c*e^5*f+72096*b^3*c^2*d^2*e^3* 
g+25024*b^3*c^2*d*e^4*f-173824*b^2*c^3*d^3*e^2*g-115056*b^2*c^3*d^2*e^3*f+ 
209686*b*c^4*d^4*e*g+260984*b*c^4*d^3*e^2*f-94134*c^5*d^5*g-278171*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. 1112 vs. \(2 (385) = 770\).

Time = 0.16 (sec) , antiderivative size = 1112, normalized size of antiderivative = 2.64 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] 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)^(5/2),x, 
algorithm="fricas")
 

Output:

2/765765*(45045*c^8*e^8*g*x^8 + 3003*(17*c^8*e^8*f + (33*c^8*d*e^7 + 35*b* 
c^7*e^8)*g)*x^7 + 231*(17*(29*c^8*d*e^7 + 31*b*c^7*e^8)*f - (303*c^8*d^2*e 
^6 - 1558*b*c^7*d*e^7 - 275*b^2*c^6*e^8)*g)*x^6 - 63*(17*(79*c^8*d^2*e^6 - 
 398*b*c^7*d*e^7 - 71*b^2*c^6*e^8)*f + (4527*c^8*d^3*e^5 - 4129*b*c^7*d^2* 
e^6 - 4813*b^2*c^6*d*e^7 - 5*b^3*c^5*e^8)*g)*x^5 - 35*(17*(587*c^8*d^3*e^5 
 - 525*b*c^7*d^2*e^6 - 633*b^2*c^6*d*e^7 - b^3*c^5*e^8)*f + (1761*c^8*d^4* 
e^4 + 11860*b*c^7*d^3*e^5 - 15954*b^2*c^6*d^2*e^6 - 108*b^3*c^5*d*e^7 + 10 
*b^4*c^4*e^8)*g)*x^4 - 5*(17*(835*c^8*d^4*e^4 + 6548*b*c^7*d^3*e^5 - 8586* 
b^2*c^6*d^2*e^6 - 92*b^3*c^5*d*e^7 + 8*b^4*c^4*e^8)*f - (51549*c^8*d^5*e^3 
 - 146429*b*c^7*d^4*e^4 + 91238*b^2*c^6*d^3*e^5 + 4506*b^3*c^5*d^2*e^6 - 9 
44*b^4*c^4*d*e^7 + 80*b^5*c^3*e^8)*g)*x^3 + 3*(17*(7339*c^8*d^5*e^3 - 2043 
5*b*c^7*d^4*e^4 + 12250*b^2*c^6*d^3*e^5 + 1030*b^3*c^5*d^2*e^6 - 200*b^4*c 
^4*d*e^7 + 16*b^5*c^3*e^8)*f + (52047*c^8*d^6*e^2 - 89650*b*c^7*d^5*e^3 + 
15875*b^2*c^6*d^4*e^4 + 30740*b^3*c^5*d^3*e^5 - 10900*b^4*c^4*d^2*e^6 + 20 
48*b^5*c^3*d*e^7 - 160*b^6*c^2*e^8)*g)*x^2 - 17*(16363*c^8*d^7*e - 64441*b 
*c^7*d^6*e^2 + 101913*b^2*c^6*d^5*e^3 - 84195*b^3*c^5*d^4*e^4 + 40200*b^4* 
c^4*d^3*e^5 - 11568*b^5*c^3*d^2*e^6 + 1856*b^6*c^2*d*e^7 - 128*b^7*c*e^8)* 
f - 2*(47067*c^8*d^8 - 246044*b*c^7*d^7*e + 542642*b^2*c^6*d^6*e^2 - 65838 
0*b^3*c^5*d^5*e^3 + 481275*b^4*c^4*d^4*e^4 - 218352*b^5*c^3*d^3*e^5 + 6062 
4*b^6*c^2*d^2*e^6 - 9472*b^7*c*d*e^7 + 640*b^8*e^8)*g + (17*(14341*c^8*...
 

Sympy [F(-1)]

Timed out. \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Timed out} \] 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)**(5/ 
2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (385) = 770\).

Time = 0.11 (sec) , antiderivative size = 1108, normalized size of antiderivative = 2.63 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] 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)^(5/2),x, 
algorithm="maxima")
 

Output:

2/45045*(3003*c^7*e^7*x^7 - 16363*c^7*d^7 + 64441*b*c^6*d^6*e - 101913*b^2 
*c^5*d^5*e^2 + 84195*b^3*c^4*d^4*e^3 - 40200*b^4*c^3*d^3*e^4 + 11568*b^5*c 
^2*d^2*e^5 - 1856*b^6*c*d*e^6 + 128*b^7*e^7 + 231*(29*c^7*d*e^6 + 31*b*c^6 
*e^7)*x^6 - 63*(79*c^7*d^2*e^5 - 398*b*c^6*d*e^6 - 71*b^2*c^5*e^7)*x^5 - 3 
5*(587*c^7*d^3*e^4 - 525*b*c^6*d^2*e^5 - 633*b^2*c^5*d*e^6 - b^3*c^4*e^7)* 
x^4 - 5*(835*c^7*d^4*e^3 + 6548*b*c^6*d^3*e^4 - 8586*b^2*c^5*d^2*e^5 - 92* 
b^3*c^4*d*e^6 + 8*b^4*c^3*e^7)*x^3 + 3*(7339*c^7*d^5*e^2 - 20435*b*c^6*d^4 
*e^3 + 12250*b^2*c^5*d^3*e^4 + 1030*b^3*c^4*d^2*e^5 - 200*b^4*c^3*d*e^6 + 
16*b^5*c^2*e^7)*x^2 + (14341*c^7*d^6*e - 21006*b*c^6*d^5*e^2 - 4395*b^2*c^ 
5*d^4*e^3 + 15180*b^3*c^4*d^3*e^4 - 4920*b^4*c^3*d^2*e^5 + 864*b^5*c^2*d*e 
^6 - 64*b^6*c*e^7)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^5*e^2*x + c^ 
5*d*e) + 2/765765*(45045*c^8*e^8*x^8 - 94134*c^8*d^8 + 492088*b*c^7*d^7*e 
- 1085284*b^2*c^6*d^6*e^2 + 1316760*b^3*c^5*d^5*e^3 - 962550*b^4*c^4*d^4*e 
^4 + 436704*b^5*c^3*d^3*e^5 - 121248*b^6*c^2*d^2*e^6 + 18944*b^7*c*d*e^7 - 
 1280*b^8*e^8 + 3003*(33*c^8*d*e^7 + 35*b*c^7*e^8)*x^7 - 231*(303*c^8*d^2* 
e^6 - 1558*b*c^7*d*e^7 - 275*b^2*c^6*e^8)*x^6 - 63*(4527*c^8*d^3*e^5 - 412 
9*b*c^7*d^2*e^6 - 4813*b^2*c^6*d*e^7 - 5*b^3*c^5*e^8)*x^5 - 35*(1761*c^8*d 
^4*e^4 + 11860*b*c^7*d^3*e^5 - 15954*b^2*c^6*d^2*e^6 - 108*b^3*c^5*d*e^7 + 
 10*b^4*c^4*e^8)*x^4 + 5*(51549*c^8*d^5*e^3 - 146429*b*c^7*d^4*e^4 + 91238 
*b^2*c^6*d^3*e^5 + 4506*b^3*c^5*d^2*e^6 - 944*b^4*c^4*d*e^7 + 80*b^5*c^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 21712 vs. \(2 (385) = 770\).

Time = 0.52 (sec) , antiderivative size = 21712, normalized size of antiderivative = 51.57 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] 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)^(5/2),x, 
algorithm="giac")
 

Output:

-2/765765*(765765*sqrt(-c*e*x + c*d - b*e)*c^3*d^7*e*f - 2297295*sqrt(-c*e 
*x + c*d - b*e)*b*c^2*d^6*e^2*f + 2297295*sqrt(-c*e*x + c*d - b*e)*b^2*c*d 
^5*e^3*f - 765765*sqrt(-c*e*x + c*d - b*e)*b^3*d^4*e^4*f + 255255*(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^2*d^6*e*f - 1531530*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqr 
t(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*b*c*d^5*e^2*f + 22 
97295*(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^3*f - 1021020*(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^ 
3*d^3*e^4*f/c + 255255*(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^2*d^7*g - 765765*(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*c*d^6*e*g + 765765*(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^5*e^2*g - 255255 
*(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^3*d^4*e^3*g/c - 153153*(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))*c*d^5*e*f + 
153153*(15*sqrt(-c*e*x + c*d - b*e)*c^2*d^2 - 30*sqrt(-c*e*x + c*d - b*...
 

Mupad [B] (verification not implemented)

Time = 13.14 (sec) , antiderivative size = 1023, normalized size of antiderivative = 2.43 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] 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)^(5/2), 
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)*( 
275*b^2*e^2*g - 303*c^2*d^2*g + 527*b*c*e^2*f + 493*c^2*d*e*f + 1558*b*c*d 
*e*g))/3315 + (2*(b*e - c*d)^3*(d + e*x)^(1/2)*(94134*c^5*d^5*g - 1280*b^5 
*e^5*g + 2176*b^4*c*e^5*f + 278171*c^5*d^4*e*f - 209686*b*c^4*d^4*e*g + 15 
104*b^4*c*d*e^4*g - 260984*b*c^4*d^3*e^2*f - 25024*b^3*c^2*d*e^4*f + 11505 
6*b^2*c^3*d^2*e^3*f + 173824*b^2*c^3*d^3*e^2*g - 72096*b^3*c^2*d^2*e^3*g)) 
/(765765*c^6*e^3) + (x^4*(d + e*x)^(1/2)*(1190*b^3*c^5*e^8*f - 700*b^4*c^4 
*e^8*g - 698530*c^8*d^3*e^5*f - 123270*c^8*d^4*e^4*g + 624750*b*c^7*d^2*e^ 
6*f + 753270*b^2*c^6*d*e^7*f - 830200*b*c^7*d^3*e^5*g + 7560*b^3*c^5*d*e^7 
*g + 1116780*b^2*c^6*d^2*e^6*g))/(765765*c^6*e^3) + (2*c^2*e^5*g*x^8*(d + 
e*x)^(1/2))/17 + (x^5*(d + e*x)^(1/2)*(152082*b^2*c^6*e^8*f + 630*b^3*c^5* 
e^8*g - 169218*c^8*d^2*e^6*f - 570402*c^8*d^3*e^5*g + 852516*b*c^7*d*e^7*f 
 + 520254*b*c^7*d^2*e^6*g + 606438*b^2*c^6*d*e^7*g))/(765765*c^6*e^3) + (2 
*c*e^4*x^7*(d + e*x)^(1/2)*(35*b*e*g + 33*c*d*g + 17*c*e*f))/255 + (x^3*(d 
 + e*x)^(1/2)*(800*b^5*c^3*e^8*g - 1360*b^4*c^4*e^8*f - 141950*c^8*d^4*e^4 
*f + 515490*c^8*d^5*e^3*g - 1113160*b*c^7*d^3*e^5*f + 15640*b^3*c^5*d*e^7* 
f - 1464290*b*c^7*d^4*e^4*g - 9440*b^4*c^4*d*e^7*g + 1459620*b^2*c^6*d^2*e 
^6*f + 912380*b^2*c^6*d^3*e^5*g + 45060*b^3*c^5*d^2*e^6*g))/(765765*c^6*e^ 
3) + (2*x^2*(b*e - c*d)*(d + e*x)^(1/2)*(272*b^4*c*e^5*f - 52047*c^5*d^5*g 
 - 160*b^5*e^5*g - 124763*c^5*d^4*e*f + 37603*b*c^4*d^4*e*g + 1888*b^4*...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1220, normalized size of antiderivative = 2.90 \[ \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] 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)^(5/2),x)
 

Output:

(2*sqrt( - b*e + c*d - c*e*x)*( - 1280*b**8*e**8*g + 18944*b**7*c*d*e**7*g 
 + 2176*b**7*c*e**8*f + 640*b**7*c*e**8*g*x - 121248*b**6*c**2*d**2*e**6*g 
 - 31552*b**6*c**2*d*e**7*f - 8832*b**6*c**2*d*e**7*g*x - 1088*b**6*c**2*e 
**8*f*x - 480*b**6*c**2*e**8*g*x**2 + 436704*b**5*c**3*d**3*e**5*g + 19665 
6*b**5*c**3*d**2*e**6*f + 51792*b**5*c**3*d**2*e**6*g*x + 14688*b**5*c**3* 
d*e**7*f*x + 6144*b**5*c**3*d*e**7*g*x**2 + 816*b**5*c**3*e**8*f*x**2 + 40 
0*b**5*c**3*e**8*g*x**3 - 962550*b**4*c**4*d**4*e**4*g - 683400*b**4*c**4* 
d**3*e**5*f - 166560*b**4*c**4*d**3*e**5*g*x - 83640*b**4*c**4*d**2*e**6*f 
*x - 32700*b**4*c**4*d**2*e**6*g*x**2 - 10200*b**4*c**4*d*e**7*f*x**2 - 47 
20*b**4*c**4*d*e**7*g*x**3 - 680*b**4*c**4*e**8*f*x**3 - 350*b**4*c**4*e** 
8*g*x**4 + 1316760*b**3*c**5*d**5*e**3*g + 1431315*b**3*c**5*d**4*e**4*f + 
 314715*b**3*c**5*d**4*e**4*g*x + 258060*b**3*c**5*d**3*e**5*f*x + 92220*b 
**3*c**5*d**3*e**5*g*x**2 + 52530*b**3*c**5*d**2*e**6*f*x**2 + 22530*b**3* 
c**5*d**2*e**6*g*x**3 + 7820*b**3*c**5*d*e**7*f*x**3 + 3780*b**3*c**5*d*e* 
*7*g*x**4 + 595*b**3*c**5*e**8*f*x**4 + 315*b**3*c**5*e**8*g*x**5 - 108528 
4*b**2*c**6*d**6*e**2*g - 1732521*b**2*c**6*d**5*e**3*f - 343665*b**2*c**6 
*d**5*e**3*g*x - 74715*b**2*c**6*d**4*e**4*f*x + 47625*b**2*c**6*d**4*e**4 
*g*x**2 + 624750*b**2*c**6*d**3*e**5*f*x**2 + 456190*b**2*c**6*d**3*e**5*g 
*x**3 + 729810*b**2*c**6*d**2*e**6*f*x**3 + 558390*b**2*c**6*d**2*e**6*g*x 
**4 + 376635*b**2*c**6*d*e**7*f*x**4 + 303219*b**2*c**6*d*e**7*g*x**5 +...