\(\int \frac {d+e x}{(a+b x+c x^2)^{9/2}} \, dx\) [171]

Optimal result

Integrand size = 20, antiderivative size = 181 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1024 c^2 (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^4 \sqrt {a+b x+c x^2}} \] Output:

1/7*(-2*b*d+4*a*e-2*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(7/2)+24/35 
*(-b*e+2*c*d)*(2*c*x+b)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(5/2)-128/35*c*(-b*e+ 
2*c*d)*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^(3/2)+1024/35*c^2*(-b*e+2*c* 
d)*(2*c*x+b)/(-4*a*c+b^2)^4/(c*x^2+b*x+a)^(1/2)
 

Mathematica [A] (verified)

Time = 8.13 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.93 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \left (b^7 (5 d+7 e x)-128 c^3 \left (-5 a^4 e+35 a^3 c d x+70 a^2 c^2 d x^3+56 a c^3 d x^5+16 c^4 d x^7\right )-64 b c^3 \left (35 a^3 (d-e x)-16 c^3 x^6 (-7 d+e x)-56 a c^2 x^4 (-5 d+e x)-70 a^2 c x^2 (-3 d+e x)\right )+32 b^2 c^2 \left (15 a^3 e-105 a^2 c x (d-2 e x)+56 c^3 x^5 (-5 d+2 e x)+140 a c^2 x^3 (-3 d+2 e x)\right )+2 b^6 (a e-7 c x (d+2 e x))+560 b^3 c^2 \left (-4 a c x^2 (d-3 e x)+8 c^2 x^4 (-d+e x)+a^2 (d+3 e x)\right )-40 b^4 c \left (a^2 e+14 c^2 x^3 (d-4 e x)-7 a c x (d+4 e x)\right )-28 b^5 c \left (-2 c x^2 (d+5 e x)+a (3 d+5 e x)\right )\right )}{35 \left (b^2-4 a c\right )^4 (a+x (b+c x))^{7/2}} \] Input:

Integrate[(d + e*x)/(a + b*x + c*x^2)^(9/2),x]
 

Output:

(-2*(b^7*(5*d + 7*e*x) - 128*c^3*(-5*a^4*e + 35*a^3*c*d*x + 70*a^2*c^2*d*x 
^3 + 56*a*c^3*d*x^5 + 16*c^4*d*x^7) - 64*b*c^3*(35*a^3*(d - e*x) - 16*c^3* 
x^6*(-7*d + e*x) - 56*a*c^2*x^4*(-5*d + e*x) - 70*a^2*c*x^2*(-3*d + e*x)) 
+ 32*b^2*c^2*(15*a^3*e - 105*a^2*c*x*(d - 2*e*x) + 56*c^3*x^5*(-5*d + 2*e* 
x) + 140*a*c^2*x^3*(-3*d + 2*e*x)) + 2*b^6*(a*e - 7*c*x*(d + 2*e*x)) + 560 
*b^3*c^2*(-4*a*c*x^2*(d - 3*e*x) + 8*c^2*x^4*(-d + e*x) + a^2*(d + 3*e*x)) 
 - 40*b^4*c*(a^2*e + 14*c^2*x^3*(d - 4*e*x) - 7*a*c*x*(d + 4*e*x)) - 28*b^ 
5*c*(-2*c*x^2*(d + 5*e*x) + a*(3*d + 5*e*x))))/(35*(b^2 - 4*a*c)^4*(a + x* 
(b + c*x))^(7/2))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1159, 1089, 1089, 1088}

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.

\(\displaystyle \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 1159

\(\displaystyle -\frac {12 (2 c d-b e) \int \frac {1}{\left (c x^2+b x+a\right )^{7/2}}dx}{7 \left (b^2-4 a c\right )}-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 1089

\(\displaystyle -\frac {12 (2 c d-b e) \left (-\frac {16 c \int \frac {1}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\right )}{7 \left (b^2-4 a c\right )}-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 1089

\(\displaystyle -\frac {12 (2 c d-b e) \left (-\frac {16 c \left (-\frac {8 c \int \frac {1}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\right )}{7 \left (b^2-4 a c\right )}-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 1088

\(\displaystyle -\frac {2 (-2 a e+x (2 c d-b e)+b d)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}-\frac {12 \left (-\frac {2 (b+2 c x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {16 c \left (\frac {16 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}\right ) (2 c d-b e)}{7 \left (b^2-4 a c\right )}\)

Input:

Int[(d + e*x)/(a + b*x + c*x^2)^(9/2),x]
 

Output:

(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(7*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(7 
/2)) - (12*(2*c*d - b*e)*((-2*(b + 2*c*x))/(5*(b^2 - 4*a*c)*(a + b*x + c*x 
^2)^(5/2)) - (16*c*((-2*(b + 2*c*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3 
/2)) + (16*c*(b + 2*c*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])))/(5*( 
b^2 - 4*a*c))))/(7*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 1088

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 
2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0]
 

rule 1089

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1159

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* 
x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* 
c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & 
& LtQ[p, -1] && NeQ[p, -3/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(353\) vs. \(2(167)=334\).

Time = 1.21 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.96

method	result	size
default	\(d \left (\frac {\frac {4 c x}{7}+\frac {2 b}{7}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}+\frac {24 c \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{7 \left (4 a c -b^{2}\right )}\right )+e \left (-\frac {1}{7 c \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}-\frac {b \left (\frac {\frac {4 c x}{7}+\frac {2 b}{7}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}+\frac {24 c \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{7 \left (4 a c -b^{2}\right )}\right )}{2 c}\right )\)	\(354\)
trager	\(-\frac {2 \left (1024 b \,c^{6} e \,x^{7}-2048 c^{7} d \,x^{7}+3584 b^{2} c^{5} e \,x^{6}-7168 b \,c^{6} d \,x^{6}+3584 a b \,c^{5} e \,x^{5}-7168 a \,c^{6} d \,x^{5}+4480 b^{3} c^{4} e \,x^{5}-8960 b^{2} c^{5} d \,x^{5}+8960 a \,b^{2} c^{4} e \,x^{4}-17920 a b \,c^{5} d \,x^{4}+2240 b^{4} c^{3} e \,x^{4}-4480 b^{3} c^{4} d \,x^{4}+4480 a^{2} b \,c^{4} e \,x^{3}-8960 a^{2} c^{5} d \,x^{3}+6720 a \,b^{3} c^{3} e \,x^{3}-13440 a \,b^{2} c^{4} d \,x^{3}+280 b^{5} c^{2} e \,x^{3}-560 b^{4} c^{3} d \,x^{3}+6720 a^{2} b^{2} c^{3} e \,x^{2}-13440 a^{2} b \,c^{4} d \,x^{2}+1120 a \,b^{4} c^{2} e \,x^{2}-2240 a \,b^{3} c^{3} d \,x^{2}-28 b^{6} c e \,x^{2}+56 b^{5} c^{2} d \,x^{2}+2240 a^{3} b \,c^{3} e x -4480 a^{3} c^{4} d x +1680 a^{2} b^{3} c^{2} e x -3360 a^{2} b^{2} c^{3} d x -140 a \,b^{5} c e x +280 a \,b^{4} c^{2} d x +7 b^{7} e x -14 b^{6} c d x +640 a^{4} c^{3} e +480 a^{3} b^{2} c^{2} e -2240 a^{3} b \,c^{3} d -40 a^{2} b^{4} c e +560 a^{2} b^{3} c^{2} d +2 a \,b^{6} e -84 a \,b^{5} c d +5 b^{7} d \right )}{35 \left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}\)	\(469\)
gosper	\(-\frac {2 \left (1024 b \,c^{6} e \,x^{7}-2048 c^{7} d \,x^{7}+3584 b^{2} c^{5} e \,x^{6}-7168 b \,c^{6} d \,x^{6}+3584 a b \,c^{5} e \,x^{5}-7168 a \,c^{6} d \,x^{5}+4480 b^{3} c^{4} e \,x^{5}-8960 b^{2} c^{5} d \,x^{5}+8960 a \,b^{2} c^{4} e \,x^{4}-17920 a b \,c^{5} d \,x^{4}+2240 b^{4} c^{3} e \,x^{4}-4480 b^{3} c^{4} d \,x^{4}+4480 a^{2} b \,c^{4} e \,x^{3}-8960 a^{2} c^{5} d \,x^{3}+6720 a \,b^{3} c^{3} e \,x^{3}-13440 a \,b^{2} c^{4} d \,x^{3}+280 b^{5} c^{2} e \,x^{3}-560 b^{4} c^{3} d \,x^{3}+6720 a^{2} b^{2} c^{3} e \,x^{2}-13440 a^{2} b \,c^{4} d \,x^{2}+1120 a \,b^{4} c^{2} e \,x^{2}-2240 a \,b^{3} c^{3} d \,x^{2}-28 b^{6} c e \,x^{2}+56 b^{5} c^{2} d \,x^{2}+2240 a^{3} b \,c^{3} e x -4480 a^{3} c^{4} d x +1680 a^{2} b^{3} c^{2} e x -3360 a^{2} b^{2} c^{3} d x -140 a \,b^{5} c e x +280 a \,b^{4} c^{2} d x +7 b^{7} e x -14 b^{6} c d x +640 a^{4} c^{3} e +480 a^{3} b^{2} c^{2} e -2240 a^{3} b \,c^{3} d -40 a^{2} b^{4} c e +560 a^{2} b^{3} c^{2} d +2 a \,b^{6} e -84 a \,b^{5} c d +5 b^{7} d \right )}{35 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}\)	\(500\)
orering	\(-\frac {2 \left (1024 b \,c^{6} e \,x^{7}-2048 c^{7} d \,x^{7}+3584 b^{2} c^{5} e \,x^{6}-7168 b \,c^{6} d \,x^{6}+3584 a b \,c^{5} e \,x^{5}-7168 a \,c^{6} d \,x^{5}+4480 b^{3} c^{4} e \,x^{5}-8960 b^{2} c^{5} d \,x^{5}+8960 a \,b^{2} c^{4} e \,x^{4}-17920 a b \,c^{5} d \,x^{4}+2240 b^{4} c^{3} e \,x^{4}-4480 b^{3} c^{4} d \,x^{4}+4480 a^{2} b \,c^{4} e \,x^{3}-8960 a^{2} c^{5} d \,x^{3}+6720 a \,b^{3} c^{3} e \,x^{3}-13440 a \,b^{2} c^{4} d \,x^{3}+280 b^{5} c^{2} e \,x^{3}-560 b^{4} c^{3} d \,x^{3}+6720 a^{2} b^{2} c^{3} e \,x^{2}-13440 a^{2} b \,c^{4} d \,x^{2}+1120 a \,b^{4} c^{2} e \,x^{2}-2240 a \,b^{3} c^{3} d \,x^{2}-28 b^{6} c e \,x^{2}+56 b^{5} c^{2} d \,x^{2}+2240 a^{3} b \,c^{3} e x -4480 a^{3} c^{4} d x +1680 a^{2} b^{3} c^{2} e x -3360 a^{2} b^{2} c^{3} d x -140 a \,b^{5} c e x +280 a \,b^{4} c^{2} d x +7 b^{7} e x -14 b^{6} c d x +640 a^{4} c^{3} e +480 a^{3} b^{2} c^{2} e -2240 a^{3} b \,c^{3} d -40 a^{2} b^{4} c e +560 a^{2} b^{3} c^{2} d +2 a \,b^{6} e -84 a \,b^{5} c d +5 b^{7} d \right )}{35 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}\)	\(500\)

Input:

int((e*x+d)/(c*x^2+b*x+a)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

d*(2/7*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(7/2)+24/7*c/(4*a*c-b^2)*(2/5*( 
2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b) 
/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x 
+a)^(1/2))))+e*(-1/7/c/(c*x^2+b*x+a)^(7/2)-1/2*b/c*(2/7*(2*c*x+b)/(4*a*c-b 
^2)/(c*x^2+b*x+a)^(7/2)+24/7*c/(4*a*c-b^2)*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x 
^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a 
)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 941 vs. \(2 (165) = 330\).

Time = 13.60 (sec) , antiderivative size = 941, normalized size of antiderivative = 5.20 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)/(c*x^2+b*x+a)^(9/2),x, algorithm="fricas")
 

Output:

2/35*(1024*(2*c^7*d - b*c^6*e)*x^7 + 3584*(2*b*c^6*d - b^2*c^5*e)*x^6 + 89 
6*(2*(5*b^2*c^5 + 4*a*c^6)*d - (5*b^3*c^4 + 4*a*b*c^5)*e)*x^5 + 2240*(2*(b 
^3*c^4 + 4*a*b*c^5)*d - (b^4*c^3 + 4*a*b^2*c^4)*e)*x^4 + 280*(2*(b^4*c^3 + 
 24*a*b^2*c^4 + 16*a^2*c^5)*d - (b^5*c^2 + 24*a*b^3*c^3 + 16*a^2*b*c^4)*e) 
*x^3 - 28*(2*(b^5*c^2 - 40*a*b^3*c^3 - 240*a^2*b*c^4)*d - (b^6*c - 40*a*b^ 
4*c^2 - 240*a^2*b^2*c^3)*e)*x^2 - (5*b^7 - 84*a*b^5*c + 560*a^2*b^3*c^2 - 
2240*a^3*b*c^3)*d - 2*(a*b^6 - 20*a^2*b^4*c + 240*a^3*b^2*c^2 + 320*a^4*c^ 
3)*e + 7*(2*(b^6*c - 20*a*b^4*c^2 + 240*a^2*b^2*c^3 + 320*a^3*c^4)*d - (b^ 
7 - 20*a*b^5*c + 240*a^2*b^3*c^2 + 320*a^3*b*c^3)*e)*x)*sqrt(c*x^2 + b*x + 
 a)/(a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3 + 256*a^8*c 
^4 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4* 
c^8)*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 
256*a^4*b*c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3 + 256*a^2*b^6*c^4 - 576* 
a^3*b^4*c^5 + 256*a^4*b^2*c^6 + 512*a^5*c^7)*x^6 + 4*(b^11*c - 13*a*b^9*c^ 
2 + 48*a^2*b^7*c^3 + 32*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 768*a^5*b*c^6)*x^5 
 + (b^12 - 4*a*b^10*c - 90*a^2*b^8*c^2 + 800*a^3*b^6*c^3 - 2240*a^4*b^4*c^ 
4 + 1536*a^5*b^2*c^5 + 1536*a^6*c^6)*x^4 + 4*(a*b^11 - 13*a^2*b^9*c + 48*a 
^3*b^7*c^2 + 32*a^4*b^5*c^3 - 512*a^5*b^3*c^4 + 768*a^6*b*c^5)*x^3 + 2*(3* 
a^2*b^10 - 46*a^3*b^8*c + 256*a^4*b^6*c^2 - 576*a^5*b^4*c^3 + 256*a^6*b^2* 
c^4 + 512*a^7*c^5)*x^2 + 4*(a^3*b^9 - 16*a^4*b^7*c + 96*a^5*b^5*c^2 - 2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)/(c*x**2+b*x+a)**(9/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)/(c*x^2+b*x+a)^(9/2),x, algorithm="maxima")
 

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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (165) = 330\).

Time = 0.25 (sec) , antiderivative size = 768, normalized size of antiderivative = 4.24 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)/(c*x^2+b*x+a)^(9/2),x, algorithm="giac")
 

Output:

2/35*((4*(2*(8*(2*(4*(2*(2*c^7*d - b*c^6*e)*x/(b^8 - 16*a*b^6*c + 96*a^2*b 
^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4) + 7*(2*b*c^6*d - b^2*c^5*e)/(b^8 - 
 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*x + 7*(10*b 
^2*c^5*d + 8*a*c^6*d - 5*b^3*c^4*e - 4*a*b*c^5*e)/(b^8 - 16*a*b^6*c + 96*a 
^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*x + 35*(2*b^3*c^4*d + 8*a*b*c 
^5*d - b^4*c^3*e - 4*a*b^2*c^4*e)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256 
*a^3*b^2*c^3 + 256*a^4*c^4))*x + 35*(2*b^4*c^3*d + 48*a*b^2*c^4*d + 32*a^2 
*c^5*d - b^5*c^2*e - 24*a*b^3*c^3*e - 16*a^2*b*c^4*e)/(b^8 - 16*a*b^6*c + 
96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*x - 7*(2*b^5*c^2*d - 80*a 
*b^3*c^3*d - 480*a^2*b*c^4*d - b^6*c*e + 40*a*b^4*c^2*e + 240*a^2*b^2*c^3* 
e)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*x 
+ 7*(2*b^6*c*d - 40*a*b^4*c^2*d + 480*a^2*b^2*c^3*d + 640*a^3*c^4*d - b^7* 
e + 20*a*b^5*c*e - 240*a^2*b^3*c^2*e - 320*a^3*b*c^3*e)/(b^8 - 16*a*b^6*c 
+ 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*x - (5*b^7*d - 84*a*b^5 
*c*d + 560*a^2*b^3*c^2*d - 2240*a^3*b*c^3*d + 2*a*b^6*e - 40*a^2*b^4*c*e + 
 480*a^3*b^2*c^2*e + 640*a^4*c^3*e)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 2 
56*a^3*b^2*c^3 + 256*a^4*c^4))/(c*x^2 + b*x + a)^(7/2)
 

Mupad [B] (verification not implemented)

Time = 11.79 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.31 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\frac {x\,\left (\frac {2\,c^2\,\left (768\,c^2\,d-368\,b\,c\,e\right )}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}-\frac {32\,b\,c^3\,e}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}\right )+\frac {b\,c\,\left (768\,c^2\,d-368\,b\,c\,e\right )}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}-\frac {64\,a\,c^3\,e}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}+\frac {x\,\left (\frac {4\,c^2\,d}{7\,\left (4\,a\,c^2-b^2\,c\right )}-\frac {2\,b\,c\,e}{7\,\left (4\,a\,c^2-b^2\,c\right )}\right )-\frac {4\,a\,c\,e}{7\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,b\,c\,d}{7\,\left (4\,a\,c^2-b^2\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{7/2}}-\frac {x\,\left (\frac {2\,c^2\,\left (28\,b\,e-48\,c\,d\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,b\,c^2\,e}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )+\frac {b\,c\,\left (28\,b\,e-48\,c\,d\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {16\,a\,c^2\,e}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {\frac {2\,c^2\,x\,\left (2048\,c^3\,d-1024\,b\,c^2\,e\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^3}+\frac {b\,c\,\left (2048\,c^3\,d-1024\,b\,c^2\,e\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^3}}{\sqrt {c\,x^2+b\,x+a}}-\frac {4\,e}{\left (140\,a\,c-35\,b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {16\,c\,e}{105\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \] Input:

int((d + e*x)/(a + b*x + c*x^2)^(9/2),x)
 

Output:

(x*((2*c^2*(768*c^2*d - 368*b*c*e))/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2 
) - (32*b*c^3*e)/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2)) + (b*c*(768*c^2* 
d - 368*b*c*e))/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) - (64*a*c^3*e)/(10 
5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2))/(a + b*x + c*x^2)^(3/2) + (x*((4*c^2 
*d)/(7*(4*a*c^2 - b^2*c)) - (2*b*c*e)/(7*(4*a*c^2 - b^2*c))) - (4*a*c*e)/( 
7*(4*a*c^2 - b^2*c)) + (2*b*c*d)/(7*(4*a*c^2 - b^2*c)))/(a + b*x + c*x^2)^ 
(7/2) - (x*((2*c^2*(28*b*e - 48*c*d))/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) 
 - (8*b*c^2*e)/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (b*c*(28*b*e - 48*c 
*d))/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (16*a*c^2*e)/(35*(4*a*c^2 - b^ 
2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(5/2) + ((2*c^2*x*(2048*c^3*d - 102 
4*b*c^2*e))/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^3) + (b*c*(2048*c^3*d - 10 
24*b*c^2*e))/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^3))/(a + b*x + c*x^2)^(1/ 
2) - (4*e)/((140*a*c - 35*b^2)*(a + b*x + c*x^2)^(5/2)) + (16*c*e)/(105*(4 
*a*c - b^2)^2*(a + b*x + c*x^2)^(3/2))
 

Reduce [B] (verification not implemented)

Time = 41.34 (sec) , antiderivative size = 1920, normalized size of antiderivative = 10.61 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)/(c*x^2+b*x+a)^(9/2),x)
 

Output:

(2*( - 640*sqrt(a + b*x + c*x**2)*a**4*c**3*e - 480*sqrt(a + b*x + c*x**2) 
*a**3*b**2*c**2*e + 2240*sqrt(a + b*x + c*x**2)*a**3*b*c**3*d - 2240*sqrt( 
a + b*x + c*x**2)*a**3*b*c**3*e*x + 4480*sqrt(a + b*x + c*x**2)*a**3*c**4* 
d*x + 40*sqrt(a + b*x + c*x**2)*a**2*b**4*c*e - 560*sqrt(a + b*x + c*x**2) 
*a**2*b**3*c**2*d - 1680*sqrt(a + b*x + c*x**2)*a**2*b**3*c**2*e*x + 3360* 
sqrt(a + b*x + c*x**2)*a**2*b**2*c**3*d*x - 6720*sqrt(a + b*x + c*x**2)*a* 
*2*b**2*c**3*e*x**2 + 13440*sqrt(a + b*x + c*x**2)*a**2*b*c**4*d*x**2 - 44 
80*sqrt(a + b*x + c*x**2)*a**2*b*c**4*e*x**3 + 8960*sqrt(a + b*x + c*x**2) 
*a**2*c**5*d*x**3 - 2*sqrt(a + b*x + c*x**2)*a*b**6*e + 84*sqrt(a + b*x + 
c*x**2)*a*b**5*c*d + 140*sqrt(a + b*x + c*x**2)*a*b**5*c*e*x - 280*sqrt(a 
+ b*x + c*x**2)*a*b**4*c**2*d*x - 1120*sqrt(a + b*x + c*x**2)*a*b**4*c**2* 
e*x**2 + 2240*sqrt(a + b*x + c*x**2)*a*b**3*c**3*d*x**2 - 6720*sqrt(a + b* 
x + c*x**2)*a*b**3*c**3*e*x**3 + 13440*sqrt(a + b*x + c*x**2)*a*b**2*c**4* 
d*x**3 - 8960*sqrt(a + b*x + c*x**2)*a*b**2*c**4*e*x**4 + 17920*sqrt(a + b 
*x + c*x**2)*a*b*c**5*d*x**4 - 3584*sqrt(a + b*x + c*x**2)*a*b*c**5*e*x**5 
 + 7168*sqrt(a + b*x + c*x**2)*a*c**6*d*x**5 - 5*sqrt(a + b*x + c*x**2)*b* 
*7*d - 7*sqrt(a + b*x + c*x**2)*b**7*e*x + 14*sqrt(a + b*x + c*x**2)*b**6* 
c*d*x + 28*sqrt(a + b*x + c*x**2)*b**6*c*e*x**2 - 56*sqrt(a + b*x + c*x**2 
)*b**5*c**2*d*x**2 - 280*sqrt(a + b*x + c*x**2)*b**5*c**2*e*x**3 + 560*sqr 
t(a + b*x + c*x**2)*b**4*c**3*d*x**3 - 2240*sqrt(a + b*x + c*x**2)*b**4...