Integrand size = 27, antiderivative size = 210 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 (A b-2 a B-(b B-2 A c) x) (d+e x)^4}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {16 (b B d-2 A c d+A b e-2 a B e) (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {128 (b B d-2 A c d+A b e-2 a B e) \left (c d^2-b d e+a e^2\right ) (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}} \]
-2/5*(A*b-2*B*a-(-2*A*c+B*b)*x)*(e*x+d)^4/(-4*a*c+b^2)/(c*x^2+b*x+a)^(5/2) -16/15*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)*(e*x+d)^2*(b*d-2*a*e+(-b*e+2*c*d)*x)/ (-4*a*c+b^2)^2/(c*x^2+b*x+a)^(3/2)+128/15*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)*(a *e^2-b*d*e+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)
Leaf count is larger than twice the leaf count of optimal. \(1196\) vs. \(2(210)=420\).
Time = 14.98 (sec) , antiderivative size = 1196, normalized size of antiderivative = 5.70 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {-2 A \left (b^5 \left (3 d^4+20 d^3 e x+90 d^2 e^2 x^2-60 d e^3 x^3-5 e^4 x^4\right )+16 b \left (8 a^4 e^4+8 c^4 d^3 x^4 (5 d-4 e x)+15 a^2 c^2 (d-e x)^4+4 a^3 c e^2 \left (9 d^2-10 d e x+5 e^2 x^2\right )+4 a c^3 d x^2 \left (15 d^3-20 d^2 e x+15 d e^2 x^2-6 e^3 x^3\right )\right )+8 b^3 \left (-5 a c (d-e x)^2 \left (d^2+14 d e x-3 e^2 x^2\right )+6 a^2 e^2 \left (d^2-10 d e x+5 e^2 x^2\right )+2 c^2 d x^2 \left (5 d^3-60 d^2 e x+45 d e^2 x^2-2 e^3 x^3\right )\right )+32 c \left (-8 a^4 d e^3+8 c^4 d^4 x^5+4 a c^3 d^2 x^3 \left (5 d^2+3 e^2 x^2\right )-4 a^3 c d e \left (3 d^2+5 e^2 x^2\right )+3 a^2 c^2 x \left (5 d^4+10 d^2 e^2 x^2+e^4 x^4\right )\right )+16 b^2 \left (4 a^3 e^3 (-3 d+5 e x)+2 c^3 d^2 x^3 \left (15 d^2-40 d e x+9 e^2 x^2\right )+6 a^2 c e \left (-2 d^3+15 d^2 e x-10 d e^2 x^2+5 e^3 x^3\right )+3 a c^2 x \left (5 d^4-40 d^3 e x+30 d^2 e^2 x^2-20 d e^3 x^3+e^4 x^4\right )\right )+2 b^4 \left (4 a e \left (d^3+15 d^2 e x-45 d e^2 x^2+5 e^3 x^3\right )-c x \left (5 d^4+80 d^3 e x-270 d^2 e^2 x^2+40 d e^3 x^3+e^4 x^4\right )\right )\right )+2 B \left (256 a^5 e^4+128 a^4 e^2 \left (b e (-4 d+5 e x)+c \left (3 d^2+5 e^2 x^2\right )\right )+b x \left (128 c^4 d^4 x^4+64 b c^3 d^3 x^3 (5 d-3 e x)+48 b^2 c^2 d^2 x^2 \left (5 d^2-10 d e x+e^2 x^2\right )+8 b^3 c d x \left (5 d^3-45 d^2 e x+15 d e^2 x^2+e^3 x^3\right )+b^4 \left (-5 d^4-60 d^3 e x+90 d^2 e^2 x^2+20 d e^3 x^3+3 e^4 x^4\right )\right )+32 a^3 \left (b^2 e^2 \left (9 d^2-40 d e x+15 e^2 x^2\right )+2 b c e \left (-6 d^3+15 d^2 e x-20 d e^2 x^2+15 e^3 x^3\right )+3 c^2 \left (d^4+10 d^2 e^2 x^2+5 e^4 x^4\right )\right )-16 a^2 \left (-15 b c^2 x (d-e x)^4+8 c^3 d e x^3 \left (5 d^2+3 e^2 x^2\right )+b^3 e \left (2 d^3-45 d^2 e x+60 d e^2 x^2-5 e^3 x^3\right )-3 b^2 c \left (d^4-20 d^3 e x+30 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right )\right )-2 a \left (128 c^4 d^3 e x^5+20 b^3 c x (d-e x)^2 \left (-3 d^2+14 d e x+e^2 x^2\right )-32 b c^3 d^2 x^3 \left (5 d^2-10 d e x+9 e^2 x^2\right )+48 b^2 c^2 d x^2 \left (-5 d^3+10 d^2 e x-15 d e^2 x^2+2 e^3 x^3\right )+b^4 \left (d^4+40 d^3 e x-270 d^2 e^2 x^2+80 d e^3 x^3+5 e^4 x^4\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]
(-2*A*(b^5*(3*d^4 + 20*d^3*e*x + 90*d^2*e^2*x^2 - 60*d*e^3*x^3 - 5*e^4*x^4 ) + 16*b*(8*a^4*e^4 + 8*c^4*d^3*x^4*(5*d - 4*e*x) + 15*a^2*c^2*(d - e*x)^4 + 4*a^3*c*e^2*(9*d^2 - 10*d*e*x + 5*e^2*x^2) + 4*a*c^3*d*x^2*(15*d^3 - 20 *d^2*e*x + 15*d*e^2*x^2 - 6*e^3*x^3)) + 8*b^3*(-5*a*c*(d - e*x)^2*(d^2 + 1 4*d*e*x - 3*e^2*x^2) + 6*a^2*e^2*(d^2 - 10*d*e*x + 5*e^2*x^2) + 2*c^2*d*x^ 2*(5*d^3 - 60*d^2*e*x + 45*d*e^2*x^2 - 2*e^3*x^3)) + 32*c*(-8*a^4*d*e^3 + 8*c^4*d^4*x^5 + 4*a*c^3*d^2*x^3*(5*d^2 + 3*e^2*x^2) - 4*a^3*c*d*e*(3*d^2 + 5*e^2*x^2) + 3*a^2*c^2*x*(5*d^4 + 10*d^2*e^2*x^2 + e^4*x^4)) + 16*b^2*(4* a^3*e^3*(-3*d + 5*e*x) + 2*c^3*d^2*x^3*(15*d^2 - 40*d*e*x + 9*e^2*x^2) + 6 *a^2*c*e*(-2*d^3 + 15*d^2*e*x - 10*d*e^2*x^2 + 5*e^3*x^3) + 3*a*c^2*x*(5*d ^4 - 40*d^3*e*x + 30*d^2*e^2*x^2 - 20*d*e^3*x^3 + e^4*x^4)) + 2*b^4*(4*a*e *(d^3 + 15*d^2*e*x - 45*d*e^2*x^2 + 5*e^3*x^3) - c*x*(5*d^4 + 80*d^3*e*x - 270*d^2*e^2*x^2 + 40*d*e^3*x^3 + e^4*x^4))) + 2*B*(256*a^5*e^4 + 128*a^4* e^2*(b*e*(-4*d + 5*e*x) + c*(3*d^2 + 5*e^2*x^2)) + b*x*(128*c^4*d^4*x^4 + 64*b*c^3*d^3*x^3*(5*d - 3*e*x) + 48*b^2*c^2*d^2*x^2*(5*d^2 - 10*d*e*x + e^ 2*x^2) + 8*b^3*c*d*x*(5*d^3 - 45*d^2*e*x + 15*d*e^2*x^2 + e^3*x^3) + b^4*( -5*d^4 - 60*d^3*e*x + 90*d^2*e^2*x^2 + 20*d*e^3*x^3 + 3*e^4*x^4)) + 32*a^3 *(b^2*e^2*(9*d^2 - 40*d*e*x + 15*e^2*x^2) + 2*b*c*e*(-6*d^3 + 15*d^2*e*x - 20*d*e^2*x^2 + 15*e^3*x^3) + 3*c^2*(d^4 + 10*d^2*e^2*x^2 + 5*e^4*x^4)) - 16*a^2*(-15*b*c^2*x*(d - e*x)^4 + 8*c^3*d*e*x^3*(5*d^2 + 3*e^2*x^2) + b...
Time = 0.33 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1227, 1153, 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.
\(\displaystyle \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 1227 |
\(\displaystyle \frac {8 (-2 a B e+A b e-2 A c d+b B d) \int \frac {(d+e x)^3}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^4 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 1153 |
\(\displaystyle \frac {8 (-2 a B e+A b e-2 A c d+b B d) \left (-\frac {8 \left (a e^2-b d e+c d^2\right ) \int \frac {d+e x}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^4 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle \frac {8 \left (\frac {16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^4 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^4)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (8*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*((-2*(d + e*x)^2 *(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2) ) + (16*(c*d^2 - b*d*e + a*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))
3.25.89.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*(2*p + 3)*((c*d^2 - b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && LtQ[p, -1]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1891\) vs. \(2(198)=396\).
Time = 0.95 (sec) , antiderivative size = 1892, normalized size of antiderivative = 9.01
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1892\) |
gosper | \(\text {Expression too large to display}\) | \(1914\) |
default | \(\text {Expression too large to display}\) | \(3092\) |
2/15*(96*A*a^2*c^3*e^4*x^5+48*A*a*b^2*c^2*e^4*x^5-384*A*a*b*c^3*d*e^3*x^5+ 384*A*a*c^4*d^2*e^2*x^5-2*A*b^4*c*e^4*x^5-32*A*b^3*c^2*d*e^3*x^5+288*A*b^2 *c^3*d^2*e^2*x^5-512*A*b*c^4*d^3*e*x^5+256*A*c^5*d^4*x^5-240*B*a^2*b*c^2*e ^4*x^5+384*B*a^2*c^3*d*e^3*x^5+40*B*a*b^3*c*e^4*x^5+192*B*a*b^2*c^2*d*e^3* x^5-576*B*a*b*c^3*d^2*e^2*x^5+256*B*a*c^4*d^3*e*x^5-3*B*b^5*e^4*x^5-8*B*b^ 4*c*d*e^3*x^5-48*B*b^3*c^2*d^2*e^2*x^5+192*B*b^2*c^3*d^3*e*x^5-128*B*b*c^4 *d^4*x^5+240*A*a^2*b*c^2*e^4*x^4+120*A*a*b^3*c*e^4*x^4-960*A*a*b^2*c^2*d*e ^3*x^4+960*A*a*b*c^3*d^2*e^2*x^4-5*A*b^5*e^4*x^4-80*A*b^4*c*d*e^3*x^4+720* A*b^3*c^2*d^2*e^2*x^4-1280*A*b^2*c^3*d^3*e*x^4+640*A*b*c^4*d^4*x^4-480*B*a ^3*c^2*e^4*x^4-240*B*a^2*b^2*c*e^4*x^4+960*B*a^2*b*c^2*d*e^3*x^4+10*B*a*b^ 4*e^4*x^4+480*B*a*b^3*c*d*e^3*x^4-1440*B*a*b^2*c^2*d^2*e^2*x^4+640*B*a*b*c ^3*d^3*e*x^4-20*B*b^5*d*e^3*x^4-120*B*b^4*c*d^2*e^2*x^4+480*B*b^3*c^2*d^3* e*x^4-320*B*b^2*c^3*d^4*x^4+480*A*a^2*b^2*c*e^4*x^3-960*A*a^2*b*c^2*d*e^3* x^3+960*A*a^2*c^3*d^2*e^2*x^3+40*A*a*b^4*e^4*x^3-800*A*a*b^3*c*d*e^3*x^3+1 440*A*a*b^2*c^2*d^2*e^2*x^3-1280*A*a*b*c^3*d^3*e*x^3+640*A*a*c^4*d^4*x^3-6 0*A*b^5*d*e^3*x^3+540*A*b^4*c*d^2*e^2*x^3-960*A*b^3*c^2*d^3*e*x^3+480*A*b^ 2*c^3*d^4*x^3-960*B*a^3*b*c*e^4*x^3-80*B*a^2*b^3*e^4*x^3+1920*B*a^2*b^2*c* d*e^3*x^3-1440*B*a^2*b*c^2*d^2*e^2*x^3+640*B*a^2*c^3*d^3*e*x^3+160*B*a*b^4 *d*e^3*x^3-1200*B*a*b^3*c*d^2*e^2*x^3+960*B*a*b^2*c^2*d^3*e*x^3-320*B*a*b* c^3*d^4*x^3-90*B*b^5*d^2*e^2*x^3+360*B*b^4*c*d^3*e*x^3-240*B*b^3*c^2*d^...
Leaf count of result is larger than twice the leaf count of optimal. 1628 vs. \(2 (198) = 396\).
Time = 38.95 (sec) , antiderivative size = 1628, normalized size of antiderivative = 7.75 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
2/15*((128*(B*b*c^4 - 2*A*c^5)*d^4 - 64*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4 )*d^3*e + 48*(B*b^3*c^2 - 8*A*a*c^4 + 6*(2*B*a*b - A*b^2)*c^3)*d^2*e^2 + 8 *(B*b^4*c - 48*(B*a^2 - A*a*b)*c^3 - 4*(6*B*a*b^2 - A*b^3)*c^2)*d*e^3 + (3 *B*b^5 - 96*A*a^2*c^3 + 48*(5*B*a^2*b - A*a*b^2)*c^2 - 2*(20*B*a*b^3 - A*b ^4)*c)*e^4)*x^5 - (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^4 - 8*(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^3*e + 48*(6*B*a^3*b^2 - A*a^2*b^3 + 4*( 2*B*a^4 - 3*A*a^3*b)*c)*d^2*e^2 - 64*(8*B*a^4*b - 3*A*a^3*b^2 - 4*A*a^4*c) *d*e^3 + 128*(2*B*a^5 - A*a^4*b)*e^4 + 5*(64*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 32*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^3)*d^3*e + 24*(B*b^4*c - 8*A*a*b* c^3 + 6*(2*B*a*b^2 - A*b^3)*c^2)*d^2*e^2 + 4*(B*b^5 - 48*(B*a^2*b - A*a*b^ 2)*c^2 - 4*(6*B*a*b^3 - A*b^4)*c)*d*e^3 - (2*B*a*b^4 - A*b^5 - 48*(2*B*a^3 - A*a^2*b)*c^2 - 24*(2*B*a^2*b^2 - A*a*b^3)*c)*e^4)*x^4 + 10*(8*(3*B*b^3* c^2 - 8*A*a*c^4 + 2*(2*B*a*b - 3*A*b^2)*c^3)*d^4 - 4*(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^3*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^2*e^2 - 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)* d*e^3 + 4*(2*B*a^2*b^3 - A*a*b^4 + 12*(2*B*a^3*b - A*a^2*b^2)*c)*e^4)*x^3 + 10*(4*(B*b^4*c - 24*A*a*b*c^3 + 2*(6*B*a*b^2 - A*b^3)*c^2)*d^4 - 2*(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^3*e + 3...
Timed out. \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 1815 vs. \(2 (198) = 396\).
Time = 0.30 (sec) , antiderivative size = 1815, normalized size of antiderivative = 8.64 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
2/15*((((((128*B*b*c^4*d^4 - 256*A*c^5*d^4 - 192*B*b^2*c^3*d^3*e - 256*B*a *c^4*d^3*e + 512*A*b*c^4*d^3*e + 48*B*b^3*c^2*d^2*e^2 + 576*B*a*b*c^3*d^2* e^2 - 288*A*b^2*c^3*d^2*e^2 - 384*A*a*c^4*d^2*e^2 + 8*B*b^4*c*d*e^3 - 192* B*a*b^2*c^2*d*e^3 + 32*A*b^3*c^2*d*e^3 - 384*B*a^2*c^3*d*e^3 + 384*A*a*b*c ^3*d*e^3 + 3*B*b^5*e^4 - 40*B*a*b^3*c*e^4 + 2*A*b^4*c*e^4 + 240*B*a^2*b*c^ 2*e^4 - 48*A*a*b^2*c^2*e^4 - 96*A*a^2*c^3*e^4)*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^4 - 128*A*b*c^4*d^4 - 96*B*b^3 *c^2*d^3*e - 128*B*a*b*c^3*d^3*e + 256*A*b^2*c^3*d^3*e + 24*B*b^4*c*d^2*e^ 2 + 288*B*a*b^2*c^2*d^2*e^2 - 144*A*b^3*c^2*d^2*e^2 - 192*A*a*b*c^3*d^2*e^ 2 + 4*B*b^5*d*e^3 - 96*B*a*b^3*c*d*e^3 + 16*A*b^4*c*d*e^3 - 192*B*a^2*b*c^ 2*d*e^3 + 192*A*a*b^2*c^2*d*e^3 - 2*B*a*b^4*e^4 + A*b^5*e^4 + 48*B*a^2*b^2 *c*e^4 - 24*A*a*b^3*c*e^4 + 96*B*a^3*c^2*e^4 - 48*A*a^2*b*c^2*e^4)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 10*(24*B*b^3*c^2*d^4 + 32*B *a*b*c^3*d^4 - 48*A*b^2*c^3*d^4 - 64*A*a*c^4*d^4 - 36*B*b^4*c*d^3*e - 96*B *a*b^2*c^2*d^3*e + 96*A*b^3*c^2*d^3*e - 64*B*a^2*c^3*d^3*e + 128*A*a*b*c^3 *d^3*e + 9*B*b^5*d^2*e^2 + 120*B*a*b^3*c*d^2*e^2 - 54*A*b^4*c*d^2*e^2 + 14 4*B*a^2*b*c^2*d^2*e^2 - 144*A*a*b^2*c^2*d^2*e^2 - 96*A*a^2*c^3*d^2*e^2 - 1 6*B*a*b^4*d*e^3 + 6*A*b^5*d*e^3 - 192*B*a^2*b^2*c*d*e^3 + 80*A*a*b^3*c*d*e ^3 + 96*A*a^2*b*c^2*d*e^3 + 8*B*a^2*b^3*e^4 - 4*A*a*b^4*e^4 + 96*B*a^3*b*c *e^4 - 48*A*a^2*b^2*c*e^4)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*...
Time = 14.60 (sec) , antiderivative size = 7972, normalized size of antiderivative = 37.96 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
((a*((b*((16*c*e^3*(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(24*A *b*c^2*e^4 + 48*B*a*c^2*e^4 - 30*B*b^2*c*e^4 - 96*A*c^3*d*e^3 - 144*B*c^3* d^2*e^2 + 96*B*b*c^2*d*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b *e^3*(A*c*e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16* B*a*c*e^4)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - x*((a*((16*c*e^3*(A*c *e - B*b*e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4) /(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (b*((b*((16*c*e^3*(A*c*e - B*b* e + 4*B*c*d))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*B*b*c*e^4)/(5*(4*a* c^2 - b^2*c)*(4*a*c - b^2))))/c + (2*(24*A*b*c^2*e^4 + 48*B*a*c^2*e^4 - 30 *B*b^2*c*e^4 - 96*A*c^3*d*e^3 - 144*B*c^3*d^2*e^2 + 96*B*b*c^2*d*e^3))/(15 *c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*e^3*(A*c*e - B*b*e + 4*B*c*d))/ (5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*B*a*c*e^4)/(5*(4*a*c^2 - b^2*c)* (4*a*c - b^2))))/c - (2*(4*B*b^3*e^4 - 16*A*a*c^2*e^4 - 2*A*b^2*c*e^4 + 32 *B*c^3*d^3*e + 48*A*c^3*d^2*e^2 + 16*B*a*b*c*e^4 - 64*B*a*c^2*d*e^3 - 8*B* b^2*c*d*e^3))/(15*c*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (b*(24*A*b*c^2*e^4 + 48*B*a*c^2*e^4 - 30*B*b^2*c*e^4 - 96*A*c^3*d*e^3 - 144*B*c^3*d^2*e^2 + 9 6*B*b*c^2*d*e^3))/(15*c^2*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (b*(4*B*b^3* e^4 - 16*A*a*c^2*e^4 - 2*A*b^2*c*e^4 + 32*B*c^3*d^3*e + 48*A*c^3*d^2*e^2 + 16*B*a*b*c*e^4 - 64*B*a*c^2*d*e^3 - 8*B*b^2*c*d*e^3))/(15*c^2*(4*a*c^2...