Integrand size = 44, antiderivative size = 210 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e^2 (2 c d-b e) (d+e x)^7}-\frac {2 (4 c e f+14 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 e^2 (2 c d-b e)^2 (d+e x)^6}-\frac {4 c (4 c e f+14 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{315 e^2 (2 c d-b e)^3 (d+e x)^5} \] Output:
-2/9*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(-b*e+2*c*d)/(e *x+d)^7-2/63*(-9*b*e*g+14*c*d*g+4*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^ (5/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^6-4/315*c*(-9*b*e*g+14*c*d*g+4*c*e*f)*(d* (-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(-b*e+2*c*d)^3/(e*x+d)^5
Time = 0.37 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.80 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (5 b^2 e^2 (7 e f+2 d g+9 e g x)+4 c^2 \left (7 d^3 g+2 e^3 f x^2+7 d e^2 x (2 f+g x)+d^2 e (47 f+49 g x)\right )-2 b c e \left (19 d^2 g+e^2 x (10 f+9 g x)+d e (80 f+98 g x)\right )\right )}{315 e^2 (-2 c d+b e)^3 (d+e x)^5} \] Input:
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x )^7,x]
Output:
(2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(5*b^2* e^2*(7*e*f + 2*d*g + 9*e*g*x) + 4*c^2*(7*d^3*g + 2*e^3*f*x^2 + 7*d*e^2*x*( 2*f + g*x) + d^2*e*(47*f + 49*g*x)) - 2*b*c*e*(19*d^2*g + e^2*x*(10*f + 9* g*x) + d*e*(80*f + 98*g*x))))/(315*e^2*(-2*c*d + b*e)^3*(d + e*x)^5)
Time = 0.75 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1220, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-9 b e g+14 c d g+4 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^6}dx}{9 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e^2 (d+e x)^7 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-9 b e g+14 c d g+4 c e f) \left (\frac {2 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^5}dx}{7 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e (d+e x)^6 (2 c d-b e)}\right )}{9 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e^2 (d+e x)^7 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 e (d+e x)^5 (2 c d-b e)^2}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e (d+e x)^6 (2 c d-b e)}\right ) (-9 b e g+14 c d g+4 c e f)}{9 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e^2 (d+e x)^7 (2 c d-b e)}\) |
Input:
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^7,x]
Output:
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*e^2*(2*c*d - b*e)*(d + e*x)^7) + ((4*c*e*f + 14*c*d*g - 9*b*e*g)*((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*e*(2*c*d - b*e)*(d + e*x)^6) - (4*c*(d*( c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(35*e*(2*c*d - b*e)^2*(d + e*x)^5 )))/(9*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_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(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] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 8.71 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.12
| method | result | size |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-18 b c \,e^{3} g \,x^{2}+28 c^{2} d \,e^{2} g \,x^{2}+8 f \,c^{2} e^{3} x^{2}+45 b^{2} e^{3} g x -196 b c d \,e^{2} g x -20 b c \,e^{3} f x +196 c^{2} d^{2} e g x +56 c^{2} d \,e^{2} f x +10 b^{2} d \,e^{2} g +35 b^{2} e^{3} f -38 b c \,d^{2} e g -160 b c d \,e^{2} f +28 c^{2} d^{3} g +188 c^{2} d^{2} e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{6} \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2}}\) | \(236\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-18 b c \,e^{3} g \,x^{2}+28 c^{2} d \,e^{2} g \,x^{2}+8 f \,c^{2} e^{3} x^{2}+45 b^{2} e^{3} g x -196 b c d \,e^{2} g x -20 b c \,e^{3} f x +196 c^{2} d^{2} e g x +56 c^{2} d \,e^{2} f x +10 b^{2} d \,e^{2} g +35 b^{2} e^{3} f -38 b c \,d^{2} e g -160 b c d \,e^{2} f +28 c^{2} d^{3} g +188 c^{2} d^{2} e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{6} \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2}}\) | \(236\) |
| default | \(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{e^{7}}-\frac {\left (d g -e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{8}}\) | \(366\) |
| trager | \(\frac {2 \left (-18 b \,c^{3} e^{5} g \,x^{4}+28 c^{4} d \,e^{4} g \,x^{4}+8 c^{4} e^{5} f \,x^{4}+9 b^{2} c^{2} e^{5} g \,x^{3}-104 b \,c^{3} d \,e^{4} g \,x^{3}-4 b \,c^{3} e^{5} f \,x^{3}+140 c^{4} d^{2} e^{3} g \,x^{3}+40 c^{4} d \,e^{4} f \,x^{3}+72 b^{3} c \,e^{5} g \,x^{2}-408 b^{2} c^{2} d \,e^{4} g \,x^{2}+3 b^{2} c^{2} e^{5} f \,x^{2}+672 b \,c^{3} d^{2} e^{3} g \,x^{2}-24 b \,c^{3} d \,e^{4} f \,x^{2}-336 c^{4} d^{3} e^{2} g \,x^{2}+84 c^{4} d^{2} e^{3} f \,x^{2}+45 b^{4} e^{5} g x -266 b^{3} c d \,e^{4} g x +50 b^{3} c \,e^{5} f x +537 b^{2} c^{2} d^{2} e^{3} g x -294 b^{2} c^{2} d \,e^{4} f x -456 b \,c^{3} d^{3} e^{2} g x +564 b \,c^{3} d^{2} e^{3} f x +140 c^{4} d^{4} e g x -320 c^{4} d^{3} e^{2} f x +10 b^{4} d \,e^{4} g +35 b^{4} e^{5} f -58 b^{3} c \,d^{2} e^{3} g -230 b^{3} c d \,e^{4} f +114 b^{2} c^{2} d^{3} e^{2} g +543 b^{2} c^{2} d^{2} e^{3} f -94 b \,c^{3} d^{4} e g -536 b \,c^{3} d^{3} e^{2} f +28 c^{4} d^{5} g +188 c^{4} d^{4} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{315 \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2} \left (e x +d \right )^{5}}\) | \(524\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7,x,method=_RET URNVERBOSE)
Output:
-2/315*(c*e*x+b*e-c*d)*(-18*b*c*e^3*g*x^2+28*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x ^2+45*b^2*e^3*g*x-196*b*c*d*e^2*g*x-20*b*c*e^3*f*x+196*c^2*d^2*e*g*x+56*c^ 2*d*e^2*f*x+10*b^2*d*e^2*g+35*b^2*e^3*f-38*b*c*d^2*e*g-160*b*c*d*e^2*f+28* c^2*d^3*g+188*c^2*d^2*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^ 6/(b^3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/e^2
Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (198) = 396\).
Time = 102.83 (sec) , antiderivative size = 739, normalized size of antiderivative = 3.52 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{4} e^{5} f + {\left (14 \, c^{4} d e^{4} - 9 \, b c^{3} e^{5}\right )} g\right )} x^{4} + {\left (4 \, {\left (10 \, c^{4} d e^{4} - b c^{3} e^{5}\right )} f + {\left (140 \, c^{4} d^{2} e^{3} - 104 \, b c^{3} d e^{4} + 9 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} + 3 \, {\left ({\left (28 \, c^{4} d^{2} e^{3} - 8 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}\right )} f - 8 \, {\left (14 \, c^{4} d^{3} e^{2} - 28 \, b c^{3} d^{2} e^{3} + 17 \, b^{2} c^{2} d e^{4} - 3 \, b^{3} c e^{5}\right )} g\right )} x^{2} + {\left (188 \, c^{4} d^{4} e - 536 \, b c^{3} d^{3} e^{2} + 543 \, b^{2} c^{2} d^{2} e^{3} - 230 \, b^{3} c d e^{4} + 35 \, b^{4} e^{5}\right )} f + 2 \, {\left (14 \, c^{4} d^{5} - 47 \, b c^{3} d^{4} e + 57 \, b^{2} c^{2} d^{3} e^{2} - 29 \, b^{3} c d^{2} e^{3} + 5 \, b^{4} d e^{4}\right )} g - {\left (2 \, {\left (160 \, c^{4} d^{3} e^{2} - 282 \, b c^{3} d^{2} e^{3} + 147 \, b^{2} c^{2} d e^{4} - 25 \, b^{3} c e^{5}\right )} f - {\left (140 \, c^{4} d^{4} e - 456 \, b c^{3} d^{3} e^{2} + 537 \, b^{2} c^{2} d^{2} e^{3} - 266 \, b^{3} c d e^{4} + 45 \, b^{4} e^{5}\right )} g\right )} x\right )}}{315 \, {\left (8 \, c^{3} d^{8} e^{2} - 12 \, b c^{2} d^{7} e^{3} + 6 \, b^{2} c d^{6} e^{4} - b^{3} d^{5} e^{5} + {\left (8 \, c^{3} d^{3} e^{7} - 12 \, b c^{2} d^{2} e^{8} + 6 \, b^{2} c d e^{9} - b^{3} e^{10}\right )} x^{5} + 5 \, {\left (8 \, c^{3} d^{4} e^{6} - 12 \, b c^{2} d^{3} e^{7} + 6 \, b^{2} c d^{2} e^{8} - b^{3} d e^{9}\right )} x^{4} + 10 \, {\left (8 \, c^{3} d^{5} e^{5} - 12 \, b c^{2} d^{4} e^{6} + 6 \, b^{2} c d^{3} e^{7} - b^{3} d^{2} e^{8}\right )} x^{3} + 10 \, {\left (8 \, c^{3} d^{6} e^{4} - 12 \, b c^{2} d^{5} e^{5} + 6 \, b^{2} c d^{4} e^{6} - b^{3} d^{3} e^{7}\right )} x^{2} + 5 \, {\left (8 \, c^{3} d^{7} e^{3} - 12 \, b c^{2} d^{6} e^{4} + 6 \, b^{2} c d^{5} e^{5} - b^{3} d^{4} e^{6}\right )} x\right )}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7,x, algo rithm="fricas")
Output:
-2/315*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^4*e^5*f + (14*c^ 4*d*e^4 - 9*b*c^3*e^5)*g)*x^4 + (4*(10*c^4*d*e^4 - b*c^3*e^5)*f + (140*c^4 *d^2*e^3 - 104*b*c^3*d*e^4 + 9*b^2*c^2*e^5)*g)*x^3 + 3*((28*c^4*d^2*e^3 - 8*b*c^3*d*e^4 + b^2*c^2*e^5)*f - 8*(14*c^4*d^3*e^2 - 28*b*c^3*d^2*e^3 + 17 *b^2*c^2*d*e^4 - 3*b^3*c*e^5)*g)*x^2 + (188*c^4*d^4*e - 536*b*c^3*d^3*e^2 + 543*b^2*c^2*d^2*e^3 - 230*b^3*c*d*e^4 + 35*b^4*e^5)*f + 2*(14*c^4*d^5 - 47*b*c^3*d^4*e + 57*b^2*c^2*d^3*e^2 - 29*b^3*c*d^2*e^3 + 5*b^4*d*e^4)*g - (2*(160*c^4*d^3*e^2 - 282*b*c^3*d^2*e^3 + 147*b^2*c^2*d*e^4 - 25*b^3*c*e^5 )*f - (140*c^4*d^4*e - 456*b*c^3*d^3*e^2 + 537*b^2*c^2*d^2*e^3 - 266*b^3*c *d*e^4 + 45*b^4*e^5)*g)*x)/(8*c^3*d^8*e^2 - 12*b*c^2*d^7*e^3 + 6*b^2*c*d^6 *e^4 - b^3*d^5*e^5 + (8*c^3*d^3*e^7 - 12*b*c^2*d^2*e^8 + 6*b^2*c*d*e^9 - b ^3*e^10)*x^5 + 5*(8*c^3*d^4*e^6 - 12*b*c^2*d^3*e^7 + 6*b^2*c*d^2*e^8 - b^3 *d*e^9)*x^4 + 10*(8*c^3*d^5*e^5 - 12*b*c^2*d^4*e^6 + 6*b^2*c*d^3*e^7 - b^3 *d^2*e^8)*x^3 + 10*(8*c^3*d^6*e^4 - 12*b*c^2*d^5*e^5 + 6*b^2*c*d^4*e^6 - b ^3*d^3*e^7)*x^2 + 5*(8*c^3*d^7*e^3 - 12*b*c^2*d^6*e^4 + 6*b^2*c*d^5*e^5 - b^3*d^4*e^6)*x)
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{7}}\, dx \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**7,x )
Output:
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**7, x )
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7,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. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7,x, algo rithm="giac")
Output:
Timed out
Time = 17.59 (sec) , antiderivative size = 8039, normalized size of antiderivative = 38.28 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^7,x)
Output:
(((d*((d*((32*c^5*(4*b*e*g - 6*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32 *c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (608*c^6*d^2*g + 208*b^2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2*f - 704*b*c^5*d*e*g)/(945*e*(b*e - 2*c*d)^5 )))/e + (4*b*c^3*(19*b^2*e^2*g + 76*c^2*d^2*g + 14*b*c*e^2*f - 24*c^2*d*e* f - 76*b*c*d*e*g))/(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*((d*((16*c^5*(13*b*e*g - 22*c*d*g + 2*c*e* f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (1568 *c^6*d^2*g + 488*b^2*c^4*e^2*g - 352*c^6*d*e*f + 208*b*c^5*e^2*f - 1744*b* c^5*d*e*g)/(945*e*(b*e - 2*c*d)^5)))/e + (4*b*c^3*(49*b^2*e^2*g + 196*c^2* d^2*g + 24*b*c*e^2*f - 44*c^2*d*e*f - 196*b*c*d*e*g))/(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*((d*((1 6*c^5*(15*b*e*g - 26*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g )/(945*(b*e - 2*c*d)^5)))/e - (1952*c^6*d^2*g + 600*b^2*c^4*e^2*g - 416*c^ 6*d*e*f + 240*b*c^5*e^2*f - 2160*b*c^5*d*e*g)/(945*e*(b*e - 2*c*d)^5)))/e + (4*b*c^3*(61*b^2*e^2*g + 244*c^2*d^2*g + 28*b*c*e^2*f - 52*c^2*d*e*f - 2 44*b*c*d*e*g))/(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*((d*((16*c^5*(17*b*e*g - 30*c*d*g + 2*c*e*f))/ (945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (2336*c^6 *d^2*g + 712*b^2*c^4*e^2*g - 480*c^6*d*e*f + 272*b*c^5*e^2*f - 2576*b*c^5* d*e*g)/(945*e*(b*e - 2*c*d)^5)))/e + (4*b*c^3*(73*b^2*e^2*g + 292*c^2*d...
Time = 1.70 (sec) , antiderivative size = 2680, normalized size of antiderivative = 12.76 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7,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)*sqr t(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 - 58*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 - 230*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqr t( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*d*e**4*f - 266*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 + 50*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 + 72*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 + 114*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 + 543*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq rt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**2*e**3*f + 537* 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 - 294*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 - 408*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 + 3*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 + 9*sqrt(d + e...