\(\int \frac {(A+B x) (d+e x)^3}{(a+b x+c x^2)^{7/2}} \, dx\) [997]

Optimal result

Integrand size = 27, antiderivative size = 264 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 (A b-2 a B-(b B-2 A c) x) (d+e x)^3}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {4 (d+e x)^2 \left (4 a A c e+b^2 (4 B d+3 A e)-8 b (A c d+a B e)-\left (b^2 B e-8 b c (B d+A e)+4 c (4 A c d+3 a B e)\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 \left (b^2 e (5 B d+3 A e)+4 c \left (4 A c d^2+3 a B d e+a A e^2\right )-8 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}} \] Output:

-2/5*(A*b-2*B*a-(-2*A*c+B*b)*x)*(e*x+d)^3/(-4*a*c+b^2)/(c*x^2+b*x+a)^(5/2) 
-4/15*(e*x+d)^2*(4*a*A*c*e+b^2*(3*A*e+4*B*d)-8*b*(A*c*d+B*a*e)-(b^2*B*e-8* 
b*c*(A*e+B*d)+4*c*(4*A*c*d+3*B*a*e))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(3/2) 
-16/15*(b^2*e*(3*A*e+5*B*d)+4*c*(A*a*e^2+4*A*c*d^2+3*B*a*d*e)-8*b*(2*A*c*d 
*e+B*a*e^2+B*c*d^2))*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)^3/(c*x^2+b*x+ 
a)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(963\) vs. \(2(264)=528\).

Time = 11.87 (sec) , antiderivative size = 963, normalized size of antiderivative = 3.65 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {2 \left (A \left (-3 b^5 \left (d^3+5 d^2 e x+15 d e^2 x^2-5 e^3 x^3\right )-32 c \left (-2 a^4 e^3+8 c^4 d^3 x^5+15 a^2 c^2 d x \left (d^2+e^2 x^2\right )+2 a c^3 d x^3 \left (10 d^2+3 e^2 x^2\right )-a^3 c e \left (9 d^2+5 e^2 x^2\right )\right )+16 b c \left (-15 a^2 c (d-e x)^3+8 c^3 d^2 x^4 (-5 d+3 e x)+2 a^3 e^2 (-9 d+5 e x)+6 a c^2 x^2 \left (-10 d^3+10 d^2 e x-5 d e^2 x^2+e^3 x^3\right )\right )+48 b^2 \left (a^3 e^3+c^3 d x^3 \left (-10 d^2+20 d e x-3 e^2 x^2\right )+a^2 c e \left (3 d^2-15 d e x+5 e^2 x^2\right )+5 a c^2 x \left (-d^3+6 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+b^4 \left (-6 a e \left (d^2+10 d e x-15 e^2 x^2\right )+10 c x \left (d^3+12 d^2 e x-27 d e^2 x^2+2 e^3 x^3\right )\right )-8 b^3 \left (3 a^2 e^2 (d-5 e x)+c^2 x^2 \left (10 d^3-90 d^2 e x+45 d e^2 x^2-e^3 x^3\right )-5 a c \left (d^3+9 d^2 e x-15 d e^2 x^2+5 e^3 x^3\right )\right )\right )+B \left (64 a^4 e^2 (3 c d-2 b e)+16 a^3 \left (b^2 e^2 (9 d-20 e x)-2 b c e \left (9 d^2-15 d e x+10 e^2 x^2\right )+6 c^2 \left (d^3+5 d e^2 x^2\right )\right )-24 a^2 \left (10 b c^2 x (-d+e x)^3+4 c^3 e x^3 \left (5 d^2+e^2 x^2\right )+b^3 e \left (d^2-15 d e x+10 e^2 x^2\right )-2 b^2 c \left (d^3-15 d^2 e x+15 d e^2 x^2-10 e^3 x^3\right )\right )+b x \left (128 c^4 d^3 x^4+16 b c^3 d^2 x^3 (20 d-9 e x)+24 b^2 c^2 d x^2 \left (10 d^2-15 d e x+e^2 x^2\right )-5 b^4 \left (d^3+9 d^2 e x-9 d e^2 x^2-e^3 x^3\right )+2 b^3 c x \left (20 d^3-135 d^2 e x+30 d e^2 x^2+e^3 x^3\right )\right )-2 a \left (96 c^4 d^2 e x^5-16 b c^3 d x^3 \left (10 d^2-15 d e x+9 e^2 x^2\right )+24 b^2 c^2 x^2 \left (-10 d^3+15 d^2 e x-15 d e^2 x^2+e^3 x^3\right )+60 b^3 c x \left (-d^3+5 d^2 e x-5 d e^2 x^2+e^3 x^3\right )+b^4 \left (d^3+30 d^2 e x-135 d e^2 x^2+20 e^3 x^3\right )\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \] Input:

Integrate[((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(7/2),x]
 

Output:

(2*(A*(-3*b^5*(d^3 + 5*d^2*e*x + 15*d*e^2*x^2 - 5*e^3*x^3) - 32*c*(-2*a^4* 
e^3 + 8*c^4*d^3*x^5 + 15*a^2*c^2*d*x*(d^2 + e^2*x^2) + 2*a*c^3*d*x^3*(10*d 
^2 + 3*e^2*x^2) - a^3*c*e*(9*d^2 + 5*e^2*x^2)) + 16*b*c*(-15*a^2*c*(d - e* 
x)^3 + 8*c^3*d^2*x^4*(-5*d + 3*e*x) + 2*a^3*e^2*(-9*d + 5*e*x) + 6*a*c^2*x 
^2*(-10*d^3 + 10*d^2*e*x - 5*d*e^2*x^2 + e^3*x^3)) + 48*b^2*(a^3*e^3 + c^3 
*d*x^3*(-10*d^2 + 20*d*e*x - 3*e^2*x^2) + a^2*c*e*(3*d^2 - 15*d*e*x + 5*e^ 
2*x^2) + 5*a*c^2*x*(-d^3 + 6*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + b^4*(-6*a 
*e*(d^2 + 10*d*e*x - 15*e^2*x^2) + 10*c*x*(d^3 + 12*d^2*e*x - 27*d*e^2*x^2 
 + 2*e^3*x^3)) - 8*b^3*(3*a^2*e^2*(d - 5*e*x) + c^2*x^2*(10*d^3 - 90*d^2*e 
*x + 45*d*e^2*x^2 - e^3*x^3) - 5*a*c*(d^3 + 9*d^2*e*x - 15*d*e^2*x^2 + 5*e 
^3*x^3))) + B*(64*a^4*e^2*(3*c*d - 2*b*e) + 16*a^3*(b^2*e^2*(9*d - 20*e*x) 
 - 2*b*c*e*(9*d^2 - 15*d*e*x + 10*e^2*x^2) + 6*c^2*(d^3 + 5*d*e^2*x^2)) - 
24*a^2*(10*b*c^2*x*(-d + e*x)^3 + 4*c^3*e*x^3*(5*d^2 + e^2*x^2) + b^3*e*(d 
^2 - 15*d*e*x + 10*e^2*x^2) - 2*b^2*c*(d^3 - 15*d^2*e*x + 15*d*e^2*x^2 - 1 
0*e^3*x^3)) + b*x*(128*c^4*d^3*x^4 + 16*b*c^3*d^2*x^3*(20*d - 9*e*x) + 24* 
b^2*c^2*d*x^2*(10*d^2 - 15*d*e*x + e^2*x^2) - 5*b^4*(d^3 + 9*d^2*e*x - 9*d 
*e^2*x^2 - e^3*x^3) + 2*b^3*c*x*(20*d^3 - 135*d^2*e*x + 30*d*e^2*x^2 + e^3 
*x^3)) - 2*a*(96*c^4*d^2*e*x^5 - 16*b*c^3*d*x^3*(10*d^2 - 15*d*e*x + 9*e^2 
*x^2) + 24*b^2*c^2*x^2*(-10*d^3 + 15*d^2*e*x - 15*d*e^2*x^2 + e^3*x^3) + 6 
0*b^3*c*x*(-d^3 + 5*d^2*e*x - 5*d*e^2*x^2 + e^3*x^3) + b^4*(d^3 + 30*d^...
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1234, 25, 1227, 1158}

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.

\(\displaystyle \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1234

\(\displaystyle -\frac {2 \int -\frac {(d+e x)^2 (4 b B d-8 A c d+3 A b e-6 a B e+(b B-2 A c) e x)}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^3 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {(d+e x)^2 (4 b B d-8 A c d+3 A b e-6 a B e+(b B-2 A c) e x)}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^3 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\)

\(\Big \downarrow \) 1227

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

\(\Big \downarrow \) 1158

\(\displaystyle \frac {2 \left (-\frac {8 (-2 a e+x (2 c d-b e)+b d) \left (-8 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (a A e^2+3 a B d e+4 A c d^2\right )+b^2 e (3 A e+5 B d)\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2 \left (-x \left (4 c (3 a B e+4 A c d)-8 b c (A e+B d)+b^2 B e\right )-8 b (a B e+A c d)+4 a A c e+b^2 (3 A e+4 B d)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^3 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\)

Input:

Int[((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(7/2),x]
 

Output:

(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^3)/(5*(b^2 - 4*a*c)*(a + b*x 
 + c*x^2)^(5/2)) + (2*((-2*(d + e*x)^2*(4*a*A*c*e + b^2*(4*B*d + 3*A*e) - 
8*b*(A*c*d + a*B*e) - (b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + 3*a*B* 
e))*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (8*(b^2*e*(5*B*d + 3*A 
*e) + 4*c*(4*A*c*d^2 + 3*a*B*d*e + a*A*e^2) - 8*b*(B*c*d^2 + 2*A*c*d*e + a 
*B*e^2))*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x 
+ c*x^2])))/(5*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1158

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x 
+ c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1227

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[m*((b*( 
e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c)))   Int[(d + e*x)^(m 
 - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] 
&& EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
 

rule 1234

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 
 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g 
*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* 
(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 
] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1479\) vs. \(2(252)=504\).

Time = 2.40 (sec) , antiderivative size = 1480, normalized size of antiderivative = 5.61

method	result	size
trager	\(\text {Expression too large to display}\)	\(1480\)
gosper	\(\text {Expression too large to display}\)	\(1502\)
orering	\(\text {Expression too large to display}\)	\(1502\)
default	\(\text {Expression too large to display}\)	\(1806\)

Input:

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-2/15*(96*A*a*b*c^3*e^3*x^5-192*A*a*c^4*d*e^2*x^5+8*A*b^3*c^2*e^3*x^5-144* 
A*b^2*c^3*d*e^2*x^5+384*A*b*c^4*d^2*e*x^5-256*A*c^5*d^3*x^5-96*B*a^2*c^3*e 
^3*x^5-48*B*a*b^2*c^2*e^3*x^5+288*B*a*b*c^3*d*e^2*x^5-192*B*a*c^4*d^2*e*x^ 
5+2*B*b^4*c*e^3*x^5+24*B*b^3*c^2*d*e^2*x^5-144*B*b^2*c^3*d^2*e*x^5+128*B*b 
*c^4*d^3*x^5+240*A*a*b^2*c^2*e^3*x^4-480*A*a*b*c^3*d*e^2*x^4+20*A*b^4*c*e^ 
3*x^4-360*A*b^3*c^2*d*e^2*x^4+960*A*b^2*c^3*d^2*e*x^4-640*A*b*c^4*d^3*x^4- 
240*B*a^2*b*c^2*e^3*x^4-120*B*a*b^3*c*e^3*x^4+720*B*a*b^2*c^2*d*e^2*x^4-48 
0*B*a*b*c^3*d^2*e*x^4+5*B*b^5*e^3*x^4+60*B*b^4*c*d*e^2*x^4-360*B*b^3*c^2*d 
^2*e*x^4+320*B*b^2*c^3*d^3*x^4+240*A*a^2*b*c^2*e^3*x^3-480*A*a^2*c^3*d*e^2 
*x^3+200*A*a*b^3*c*e^3*x^3-720*A*a*b^2*c^2*d*e^2*x^3+960*A*a*b*c^3*d^2*e*x 
^3-640*A*a*c^4*d^3*x^3+15*A*b^5*e^3*x^3-270*A*b^4*c*d*e^2*x^3+720*A*b^3*c^ 
2*d^2*e*x^3-480*A*b^2*c^3*d^3*x^3-480*B*a^2*b^2*c*e^3*x^3+720*B*a^2*b*c^2* 
d*e^2*x^3-480*B*a^2*c^3*d^2*e*x^3-40*B*a*b^4*e^3*x^3+600*B*a*b^3*c*d*e^2*x 
^3-720*B*a*b^2*c^2*d^2*e*x^3+320*B*a*b*c^3*d^3*x^3+45*B*b^5*d*e^2*x^3-270* 
B*b^4*c*d^2*e*x^3+240*B*b^3*c^2*d^3*x^3+160*A*a^3*c^2*e^3*x^2+240*A*a^2*b^ 
2*c*e^3*x^2-720*A*a^2*b*c^2*d*e^2*x^2+90*A*a*b^4*e^3*x^2-600*A*a*b^3*c*d*e 
^2*x^2+1440*A*a*b^2*c^2*d^2*e*x^2-960*A*a*b*c^3*d^3*x^2-45*A*b^5*d*e^2*x^2 
+120*A*b^4*c*d^2*e*x^2-80*A*b^3*c^2*d^3*x^2-320*B*a^3*b*c*e^3*x^2+480*B*a^ 
3*c^2*d*e^2*x^2-240*B*a^2*b^3*e^3*x^2+720*B*a^2*b^2*c*d*e^2*x^2-720*B*a^2* 
b*c^2*d^2*e*x^2+270*B*a*b^4*d*e^2*x^2-600*B*a*b^3*c*d^2*e*x^2+480*B*a*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1370 vs. \(2 (252) = 504\).

Time = 76.42 (sec) , antiderivative size = 1370, normalized size of antiderivative = 5.19 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

2/15*(2*(64*(B*b*c^4 - 2*A*c^5)*d^3 - 24*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^ 
4)*d^2*e + 12*(B*b^3*c^2 - 8*A*a*c^4 + 6*(2*B*a*b - A*b^2)*c^3)*d*e^2 + (B 
*b^4*c - 48*(B*a^2 - A*a*b)*c^3 - 4*(6*B*a*b^2 - A*b^3)*c^2)*e^3)*x^5 + 5* 
(64*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 24*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^ 
3)*d^2*e + 12*(B*b^4*c - 8*A*a*b*c^3 + 6*(2*B*a*b^2 - A*b^3)*c^2)*d*e^2 + 
(B*b^5 - 48*(B*a^2*b - A*a*b^2)*c^2 - 4*(6*B*a*b^3 - A*b^4)*c)*e^3)*x^4 - 
(2*B*a*b^4 + 3*A*b^5 - 48*(2*B*a^3 - 5*A*a^2*b)*c^2 - 8*(6*B*a^2*b^2 + 5*A 
*a*b^3)*c)*d^3 - 6*(4*B*a^2*b^3 + A*a*b^4 - 48*A*a^3*c^2 + 24*(2*B*a^3*b - 
 A*a^2*b^2)*c)*d^2*e + 24*(6*B*a^3*b^2 - A*a^2*b^3 + 4*(2*B*a^4 - 3*A*a^3* 
b)*c)*d*e^2 - 16*(8*B*a^4*b - 3*A*a^3*b^2 - 4*A*a^4*c)*e^3 + 5*(16*(3*B*b^ 
3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b - 3*A*b^2)*c^3)*d^3 - 6*(9*B*b^4*c + 16*(B* 
a^2 - 2*A*a*b)*c^3 + 24*(B*a*b^2 - A*b^3)*c^2)*d^2*e + 3*(3*B*b^5 - 32*A*a 
^2*c^3 + 48*(B*a^2*b - A*a*b^2)*c^2 + 2*(20*B*a*b^3 - 9*A*b^4)*c)*d*e^2 - 
(8*B*a*b^4 - 3*A*b^5 - 48*A*a^2*b*c^2 + 8*(12*B*a^2*b^2 - 5*A*a*b^3)*c)*e^ 
3)*x^3 + 5*(8*(B*b^4*c - 24*A*a*b*c^3 + 2*(6*B*a*b^2 - A*b^3)*c^2)*d^3 - 3 
*(3*B*b^5 + 48*(B*a^2*b - 2*A*a*b^2)*c^2 + 8*(5*B*a*b^3 - A*b^4)*c)*d^2*e 
+ 3*(18*B*a*b^4 - 3*A*b^5 + 16*(2*B*a^3 - 3*A*a^2*b)*c^2 + 8*(6*B*a^2*b^2 
- 5*A*a*b^3)*c)*d*e^2 - 2*(24*B*a^2*b^3 - 9*A*a*b^4 - 16*A*a^3*c^2 + 8*(4* 
B*a^3*b - 3*A*a^2*b^2)*c)*e^3)*x^2 - 5*((B*b^5 + 96*A*a^2*c^3 - 48*(B*a^2* 
b - A*a*b^2)*c^2 - 2*(12*B*a*b^3 + A*b^4)*c)*d^3 + 3*(4*B*a*b^4 + A*b^5...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x+a)**(7/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),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. 1465 vs. \(2 (252) = 504\).

Time = 0.42 (sec) , antiderivative size = 1465, normalized size of antiderivative = 5.55 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")
 

Output:

2/15*(((((2*(64*B*b*c^4*d^3 - 128*A*c^5*d^3 - 72*B*b^2*c^3*d^2*e - 96*B*a* 
c^4*d^2*e + 192*A*b*c^4*d^2*e + 12*B*b^3*c^2*d*e^2 + 144*B*a*b*c^3*d*e^2 - 
 72*A*b^2*c^3*d*e^2 - 96*A*a*c^4*d*e^2 + B*b^4*c*e^3 - 24*B*a*b^2*c^2*e^3 
+ 4*A*b^3*c^2*e^3 - 48*B*a^2*c^3*e^3 + 48*A*a*b*c^3*e^3)*x/(b^6 - 12*a*b^4 
*c + 48*a^2*b^2*c^2 - 64*a^3*c^3) + 5*(64*B*b^2*c^3*d^3 - 128*A*b*c^4*d^3 
- 72*B*b^3*c^2*d^2*e - 96*B*a*b*c^3*d^2*e + 192*A*b^2*c^3*d^2*e + 12*B*b^4 
*c*d*e^2 + 144*B*a*b^2*c^2*d*e^2 - 72*A*b^3*c^2*d*e^2 - 96*A*a*b*c^3*d*e^2 
 + B*b^5*e^3 - 24*B*a*b^3*c*e^3 + 4*A*b^4*c*e^3 - 48*B*a^2*b*c^2*e^3 + 48* 
A*a*b^2*c^2*e^3)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*( 
48*B*b^3*c^2*d^3 + 64*B*a*b*c^3*d^3 - 96*A*b^2*c^3*d^3 - 128*A*a*c^4*d^3 - 
 54*B*b^4*c*d^2*e - 144*B*a*b^2*c^2*d^2*e + 144*A*b^3*c^2*d^2*e - 96*B*a^2 
*c^3*d^2*e + 192*A*a*b*c^3*d^2*e + 9*B*b^5*d*e^2 + 120*B*a*b^3*c*d*e^2 - 5 
4*A*b^4*c*d*e^2 + 144*B*a^2*b*c^2*d*e^2 - 144*A*a*b^2*c^2*d*e^2 - 96*A*a^2 
*c^3*d*e^2 - 8*B*a*b^4*e^3 + 3*A*b^5*e^3 - 96*B*a^2*b^2*c*e^3 + 40*A*a*b^3 
*c*e^3 + 48*A*a^2*b*c^2*e^3)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c 
^3))*x + 5*(8*B*b^4*c*d^3 + 96*B*a*b^2*c^2*d^3 - 16*A*b^3*c^2*d^3 - 192*A* 
a*b*c^3*d^3 - 9*B*b^5*d^2*e - 120*B*a*b^3*c*d^2*e + 24*A*b^4*c*d^2*e - 144 
*B*a^2*b*c^2*d^2*e + 288*A*a*b^2*c^2*d^2*e + 54*B*a*b^4*d*e^2 - 9*A*b^5*d* 
e^2 + 144*B*a^2*b^2*c*d*e^2 - 120*A*a*b^3*c*d*e^2 + 96*B*a^3*c^2*d*e^2 - 1 
44*A*a^2*b*c^2*d*e^2 - 48*B*a^2*b^3*e^3 + 18*A*a*b^4*e^3 - 64*B*a^3*b*c...
 

Mupad [B] (verification not implemented)

Time = 14.29 (sec) , antiderivative size = 4090, normalized size of antiderivative = 15.49 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(7/2),x)
 

Output:

(x*((2*A*b*c^2*e^3 + 4*B*a*c^2*e^3 - 2*B*b^2*c*e^3 - 12*A*c^3*d*e^2 - 12*B 
*c^3*d^2*e + 6*B*b*c^2*d*e^2)/(15*c^3*(4*a*c - b^2)) + (b*((2*e^2*(2*A*c*e 
 - B*b*e + 6*B*c*d))/(15*c*(4*a*c - b^2)) - (4*B*b*e^3)/(15*c*(4*a*c - b^2 
))))/c + (4*B*a*e^3)/(15*c*(4*a*c - b^2))) + (2*B*b^3*e^3 - 4*B*c^3*d^3 + 
4*A*a*c^2*e^3 - 2*A*b^2*c*e^3 - 12*A*c^3*d^2*e - 6*B*a*b*c*e^3 + 6*A*b*c^2 
*d*e^2 + 12*B*a*c^2*d*e^2 + 6*B*b*c^2*d^2*e - 6*B*b^2*c*d*e^2)/(15*c^3*(4* 
a*c - b^2)) + (a*((2*e^2*(2*A*c*e - B*b*e + 6*B*c*d))/(15*c*(4*a*c - b^2)) 
 - (4*B*b*e^3)/(15*c*(4*a*c - b^2))))/c)/(a + b*x + c*x^2)^(3/2) - (x*((b* 
((16*c*e^2*(A*c*e - B*b*e + 3*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) 
- (8*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(2*B*b^2*e^3 
+ 16*B*a*c*e^3 - 24*A*c^2*d*e^2 - 24*B*c^2*d^2*e))/(15*(4*a*c^2 - b^2*c)*( 
4*a*c - b^2)) - (8*b*e^2*(A*c*e - B*b*e + 3*B*c*d))/(5*(4*a*c^2 - b^2*c)*( 
4*a*c - b^2)) + (16*B*a*c*e^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (a*( 
(16*c*e^2*(A*c*e - B*b*e + 3*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - 
 (8*B*b*c*e^3)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (b*(2*B*b^2*e^3 + 
 16*B*a*c*e^3 - 24*A*c^2*d*e^2 - 24*B*c^2*d^2*e))/(15*c*(4*a*c^2 - b^2*c)* 
(4*a*c - b^2)))/(a + b*x + c*x^2)^(1/2) + (x*((b*((b*((b*((16*c^3*e^2*(A*e 
 + 3*B*d))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c^2*e^3)/(15*(4*a 
*c^2 - b^2*c)*(4*a*c - b^2))))/c - (2*(24*A*c^4*d*e^2 - 48*B*a*c^3*e^3 + 2 
4*B*c^4*d^2*e + 12*B*b^2*c^2*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2...
 

Reduce [F]

\[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\int \frac {\left (B x +A \right ) \left (e x +d \right )^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}d x \] Input:

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x)
 

Output:

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(7/2),x)