Integrand size = 44, antiderivative size = 285 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (2 c d-b e) (d+e x)^6}-\frac {2 (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 e^2 (2 c d-b e)^2 (d+e x)^5}-\frac {8 c (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 e^2 (2 c d-b e)^3 (d+e x)^4}-\frac {16 c^2 (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{315 e^2 (2 c d-b e)^4 (d+e x)^3} \] Output:
-2/9*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)/(e *x+d)^6-2/21*(-3*b*e*g+4*c*d*g+2*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^( 3/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^5-8/105*c*(-3*b*e*g+4*c*d*g+2*c*e*f)*(d*(- b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)^3/(e*x+d)^4-16/315*c^2* (-3*b*e*g+4*c*d*g+2*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b* e+2*c*d)^4/(e*x+d)^3
Time = 0.39 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.86 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx=\frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} \left (-5 b^3 e^3 (7 e f+2 d g+9 e g x)+6 b^2 c e^2 \left (11 d^2 g+e^2 x (5 f+6 g x)+d e (40 f+52 g x)\right )-12 b c^2 e \left (12 d^3 g+2 e^3 x^2 (f+g x)+2 d e^2 x (7 f+8 g x)+d^2 e (47 f+61 g x)\right )+8 c^3 \left (11 d^4 g+2 e^4 f x^3+4 d e^3 x^2 (3 f+g x)+3 d^2 e^2 x (11 f+8 g x)+d^3 e (58 f+66 g x)\right )\right )}{315 e^2 (-2 c d+b e)^4 (d+e x)^5} \] Input:
Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^ 6,x]
Output:
(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-5*b^3*e ^3*(7*e*f + 2*d*g + 9*e*g*x) + 6*b^2*c*e^2*(11*d^2*g + e^2*x*(5*f + 6*g*x) + d*e*(40*f + 52*g*x)) - 12*b*c^2*e*(12*d^3*g + 2*e^3*x^2*(f + g*x) + 2*d *e^2*x*(7*f + 8*g*x) + d^2*e*(47*f + 61*g*x)) + 8*c^3*(11*d^4*g + 2*e^4*f* x^3 + 4*d*e^3*x^2*(3*f + g*x) + 3*d^2*e^2*x*(11*f + 8*g*x) + d^3*e*(58*f + 66*g*x))))/(315*e^2*(-2*c*d + b*e)^4*(d + e*x)^5)
Time = 1.02 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1216, 1218, 1129, 1129, 1123}
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.
| \(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 1216 |
| \(\displaystyle \int \frac {(f+g x) \left (\frac {c d^2-b d e}{d}-c e x\right )^6}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle -\frac {(3 b e g-2 c (2 d g+e f)) \int \frac {(c d-b e-c e x)^5}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{9/2}}dx}{3 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^6}{9 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {(3 b e g-2 c (2 d g+e f)) \left (-\frac {4 \int \frac {(c d-b e-c e x)^6}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{9/2}}dx}{3 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^5}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^6}{9 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {(3 b e g-2 c (2 d g+e f)) \left (-\frac {4 \left (-\frac {2 \int \frac {(c d-b e-c e x)^7}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{9/2}}dx}{5 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^6}{5 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}\right )}{3 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^5}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^6}{9 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle -\frac {2 (e f-d g) (-b e+c d-c e x)^6}{9 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}-\frac {\left (-\frac {2 (-b e+c d-c e x)^5}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}-\frac {4 \left (\frac {4 (-b e+c d-c e x)^7}{35 e (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}-\frac {2 (-b e+c d-c e x)^6}{5 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}\right )}{3 (2 c d-b e)}\right ) (3 b e g-2 c (2 d g+e f))}{3 e (2 c d-b e)}\) |
Input:
Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^6,x]
Output:
(-2*(e*f - d*g)*(c*d - b*e - c*e*x)^6)/(9*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(9/2)) - ((3*b*e*g - 2*c*(e*f + 2*d*g))*((-2*(c*d - b*e - c*e*x)^5)/(3*e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2) ^(7/2)) - (4*((-2*(c*d - b*e - c*e*x)^6)/(5*e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2)) + (4*(c*d - b*e - c*e*x)^7)/(35*e*(2*c*d - b*e)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))))/(3*(2*c*d - b*e))))/ (3*e*(2*c*d - b*e))
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) 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] && ILtQ[Simplify[m + 2*p + 2], 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_.)*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2], x_Symbol] :> Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + 1/2))/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c* d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && IntegerQ[n]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(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[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 6.72 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.34
| method | result | size |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (24 b \,c^{2} e^{4} g \,x^{3}-32 c^{3} d \,e^{3} g \,x^{3}-16 c^{3} e^{4} f \,x^{3}-36 b^{2} c \,e^{4} g \,x^{2}+192 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}-192 c^{3} d^{2} e^{2} g \,x^{2}-96 c^{3} d \,e^{3} f \,x^{2}+45 b^{3} e^{4} g x -312 b^{2} c d \,e^{3} g x -30 b^{2} c \,e^{4} f x +732 b \,c^{2} d^{2} e^{2} g x +168 b \,c^{2} d \,e^{3} f x -528 c^{3} d^{3} e g x -264 c^{3} d^{2} e^{2} f x +10 b^{3} d \,e^{3} g +35 b^{3} e^{4} f -66 b^{2} c \,d^{2} e^{2} g -240 b^{2} c d \,e^{3} f +144 b \,c^{2} d^{3} e g +564 b \,c^{2} d^{2} e^{2} f -88 c^{3} d^{4} g -464 c^{3} d^{3} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{315 \left (e x +d \right )^{5} e^{2} \left (b^{4} e^{4}-8 d \,e^{3} b^{3} c +24 d^{2} e^{2} b^{2} c^{2}-32 d^{3} e b \,c^{3}+16 d^{4} c^{4}\right )}\) | \(382\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (24 b \,c^{2} e^{4} g \,x^{3}-32 c^{3} d \,e^{3} g \,x^{3}-16 c^{3} e^{4} f \,x^{3}-36 b^{2} c \,e^{4} g \,x^{2}+192 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}-192 c^{3} d^{2} e^{2} g \,x^{2}-96 c^{3} d \,e^{3} f \,x^{2}+45 b^{3} e^{4} g x -312 b^{2} c d \,e^{3} g x -30 b^{2} c \,e^{4} f x +732 b \,c^{2} d^{2} e^{2} g x +168 b \,c^{2} d \,e^{3} f x -528 c^{3} d^{3} e g x -264 c^{3} d^{2} e^{2} f x +10 b^{3} d \,e^{3} g +35 b^{3} e^{4} f -66 b^{2} c \,d^{2} e^{2} g -240 b^{2} c d \,e^{3} f +144 b \,c^{2} d^{3} e g +564 b \,c^{2} d^{2} e^{2} f -88 c^{3} d^{4} g -464 c^{3} d^{3} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{315 \left (e x +d \right )^{5} e^{2} \left (b^{4} e^{4}-8 d \,e^{3} b^{3} c +24 d^{2} e^{2} b^{2} c^{2}-32 d^{3} e b \,c^{3}+16 d^{4} c^{4}\right )}\) | \(382\) |
| default | \(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{5}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{6}}-\frac {\left (d g -e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {2 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{5}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (-b \,e^{2}+2 d e c \right )}\right )}{3 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{7}}\) | \(534\) |
| trager | \(-\frac {2 \left (24 b \,c^{3} e^{5} g \,x^{4}-32 c^{4} d \,e^{4} g \,x^{4}-16 c^{4} e^{5} f \,x^{4}-12 b^{2} c^{2} e^{5} g \,x^{3}+136 b \,c^{3} d \,e^{4} g \,x^{3}+8 b \,c^{3} e^{5} f \,x^{3}-160 c^{4} d^{2} e^{3} g \,x^{3}-80 c^{4} d \,e^{4} f \,x^{3}+9 b^{3} c \,e^{5} g \,x^{2}-84 b^{2} c^{2} d \,e^{4} g \,x^{2}-6 b^{2} c^{2} e^{5} f \,x^{2}+348 b \,c^{3} d^{2} e^{3} g \,x^{2}+48 b \,c^{3} d \,e^{4} f \,x^{2}-336 c^{4} d^{3} e^{2} g \,x^{2}-168 c^{4} d^{2} e^{3} f \,x^{2}+45 b^{4} e^{5} g x -347 b^{3} c d \,e^{4} g x +5 b^{3} c \,e^{5} f x +978 b^{2} c^{2} d^{2} e^{3} g x -42 b^{2} c^{2} d \,e^{4} f x -1116 b \,c^{3} d^{3} e^{2} g x +132 b \,c^{3} d^{2} e^{3} f x +440 c^{4} d^{4} e g x -200 c^{4} d^{3} e^{2} f x +10 b^{4} d \,e^{4} g +35 b^{4} e^{5} f -76 b^{3} c \,d^{2} e^{3} g -275 b^{3} c d \,e^{4} f +210 b^{2} c^{2} d^{3} e^{2} g +804 b^{2} c^{2} d^{2} e^{3} f -232 b \,c^{3} d^{4} e g -1028 b \,c^{3} d^{3} e^{2} f +88 c^{4} d^{5} g +464 c^{4} d^{4} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{315 \left (b^{4} e^{4}-8 d \,e^{3} b^{3} c +24 d^{2} e^{2} b^{2} c^{2}-32 d^{3} e b \,c^{3}+16 d^{4} c^{4}\right ) e^{2} \left (e x +d \right )^{5}}\) | \(538\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,x,method=_RET URNVERBOSE)
Output:
-2/315*(c*e*x+b*e-c*d)*(24*b*c^2*e^4*g*x^3-32*c^3*d*e^3*g*x^3-16*c^3*e^4*f *x^3-36*b^2*c*e^4*g*x^2+192*b*c^2*d*e^3*g*x^2+24*b*c^2*e^4*f*x^2-192*c^3*d ^2*e^2*g*x^2-96*c^3*d*e^3*f*x^2+45*b^3*e^4*g*x-312*b^2*c*d*e^3*g*x-30*b^2* c*e^4*f*x+732*b*c^2*d^2*e^2*g*x+168*b*c^2*d*e^3*f*x-528*c^3*d^3*e*g*x-264* c^3*d^2*e^2*f*x+10*b^3*d*e^3*g+35*b^3*e^4*f-66*b^2*c*d^2*e^2*g-240*b^2*c*d *e^3*f+144*b*c^2*d^3*e*g+564*b*c^2*d^2*e^2*f-88*c^3*d^4*g-464*c^3*d^3*e*f) *(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^5/e^2/(b^4*e^4-8*b^3*c*d*e ^3+24*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (269) = 538\).
Time = 107.64 (sec) , antiderivative size = 817, normalized size of antiderivative = 2.87 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,x, algo rithm="fricas")
Output:
2/315*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(2*c^4*e^5*f + (4*c^4* d*e^4 - 3*b*c^3*e^5)*g)*x^4 + 4*(2*(10*c^4*d*e^4 - b*c^3*e^5)*f + (40*c^4* d^2*e^3 - 34*b*c^3*d*e^4 + 3*b^2*c^2*e^5)*g)*x^3 + 3*(2*(28*c^4*d^2*e^3 - 8*b*c^3*d*e^4 + b^2*c^2*e^5)*f + (112*c^4*d^3*e^2 - 116*b*c^3*d^2*e^3 + 28 *b^2*c^2*d*e^4 - 3*b^3*c*e^5)*g)*x^2 - (464*c^4*d^4*e - 1028*b*c^3*d^3*e^2 + 804*b^2*c^2*d^2*e^3 - 275*b^3*c*d*e^4 + 35*b^4*e^5)*f - 2*(44*c^4*d^5 - 116*b*c^3*d^4*e + 105*b^2*c^2*d^3*e^2 - 38*b^3*c*d^2*e^3 + 5*b^4*d*e^4)*g + ((200*c^4*d^3*e^2 - 132*b*c^3*d^2*e^3 + 42*b^2*c^2*d*e^4 - 5*b^3*c*e^5) *f - (440*c^4*d^4*e - 1116*b*c^3*d^3*e^2 + 978*b^2*c^2*d^2*e^3 - 347*b^3*c *d*e^4 + 45*b^4*e^5)*g)*x)/(16*c^4*d^9*e^2 - 32*b*c^3*d^8*e^3 + 24*b^2*c^2 *d^7*e^4 - 8*b^3*c*d^6*e^5 + b^4*d^5*e^6 + (16*c^4*d^4*e^7 - 32*b*c^3*d^3* e^8 + 24*b^2*c^2*d^2*e^9 - 8*b^3*c*d*e^10 + b^4*e^11)*x^5 + 5*(16*c^4*d^5* e^6 - 32*b*c^3*d^4*e^7 + 24*b^2*c^2*d^3*e^8 - 8*b^3*c*d^2*e^9 + b^4*d*e^10 )*x^4 + 10*(16*c^4*d^6*e^5 - 32*b*c^3*d^5*e^6 + 24*b^2*c^2*d^4*e^7 - 8*b^3 *c*d^3*e^8 + b^4*d^2*e^9)*x^3 + 10*(16*c^4*d^7*e^4 - 32*b*c^3*d^6*e^5 + 24 *b^2*c^2*d^5*e^6 - 8*b^3*c*d^4*e^7 + b^4*d^3*e^8)*x^2 + 5*(16*c^4*d^8*e^3 - 32*b*c^3*d^7*e^4 + 24*b^2*c^2*d^6*e^5 - 8*b^3*c*d^5*e^6 + b^4*d^4*e^7)*x )
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{6}}\, dx \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**6,x )
Output:
Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**6, x)
Exception generated. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,x, algo rithm="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(b*e-2*c*d>0)', see `assume?` for more deta
Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,x, algo rithm="giac")
Output:
Timed out
Time = 13.06 (sec) , antiderivative size = 4962, normalized size of antiderivative = 17.41 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^6,x)
Output:
(((d*((32*c^5*e*f - 160*c^5*d*g + 96*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (8*b*c^3*(5*b*e*g - 10*c*d*g + 2*c*e*f))/(945*e*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^( 1/2))/(d + e*x) - (((d*((32*c^5*e*f - 320*c^5*d*g + 176*b*c^4*e*g)/(945*e* (b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (16*b*c^3*(5 *b*e*g - 10*c*d*g + c*e*f))/(945*e*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((32*c^5*e*f - 384*c^5*d*g + 208* b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)) )/e - (16*b*c^3*(6*b*e*g - 12*c*d*g + c*e*f))/(945*e*(b*e - 2*c*d)^5))*(c* d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((32*c^5*e*f - 448*c^5*d*g + 240*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e *(b*e - 2*c*d)^5)))/e - (16*b*c^3*(7*b*e*g - 14*c*d*g + c*e*f))/(945*e*(b* e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (( (d*((32*c^5*e*f - 544*c^5*d*g + 288*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - ( 32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (8*b*c^3*(17*b*e*g - 34*c*d*g + 2*c*e*f))/(945*e*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^( 1/2))/(d + e*x) + (((d*((32*c^5*e*f - 608*c^5*d*g + 320*b*c^4*e*g)/(945*e* (b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (8*b*c^3*(19 *b*e*g - 38*c*d*g + 2*c*e*f))/(945*e*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((32*c^5*e*f - 672*c^5*d*g +...
Time = 1.51 (sec) , antiderivative size = 2777, normalized size of antiderivative = 9.74 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,x)
Output:
(2*i*( - 10*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b *e + c*d - c*e*x)*b**4*d*e**4*g - 35*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*e**5*f - 45*sqrt(d + e*x)* sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*e** 5*g*x + 76*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b* e + c*d - c*e*x)*b**3*c*d**2*e**3*g + 275*sqrt(d + e*x)*sqrt(b*e - 2*c*d)* sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*d*e**4*f + 347*sqrt (d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x )*b**3*c*d*e**4*g*x - 5*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c* d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*e**5*f*x - 9*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*e**5*g*x**2 - 210*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**3*e**2*g - 804*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*s qrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**2*e**3*f - 978 *sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**2*e**3*g*x + 42*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d*e**4*f*x + 84*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)* b**2*c**2*d*e**4*g*x**2 + 6*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*e**5*f*x**2 + 12*sqrt(d + e...