Integrand size = 44, antiderivative size = 210 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (4 c e f+10 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{35 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac {4 c (4 c e f+10 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3} \] Output:
-2/7*(-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)^5-2/35*(-7*b*e*g+10*c*d*g+4*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)^4-4/105*c*(-7*b*e*g+10*c*d*g+4*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)^3
Time = 0.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.79 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx=-\frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} \left (3 b^2 e^2 (5 e f+2 d g+7 e g x)+4 c^2 \left (5 d^3 g+2 e^3 f x^2+5 d e^2 x (2 f+g x)+d^2 e (23 f+25 g x)\right )-2 b c e \left (13 d^2 g+e^2 x (6 f+7 g x)+d e (36 f+50 g x)\right )\right )}{105 e^2 (-2 c d+b e)^3 (d+e x)^4} \] Input:
Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^ 5,x]
Output:
(-2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(3*b^2*e ^2*(5*e*f + 2*d*g + 7*e*g*x) + 4*c^2*(5*d^3*g + 2*e^3*f*x^2 + 5*d*e^2*x*(2 *f + g*x) + d^2*e*(23*f + 25*g*x)) - 2*b*c*e*(13*d^2*g + e^2*x*(6*f + 7*g* x) + d*e*(36*f + 50*g*x))))/(105*e^2*(-2*c*d + b*e)^3*(d + e*x)^4)
Time = 0.84 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1216, 1218, 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.
| \(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 1216 |
| \(\displaystyle \int \frac {(f+g x) \left (\frac {c d^2-b d e}{d}-c e x\right )^5}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {(-7 b e g+10 c d g+4 c e f) \int \frac {(c d-b e-c e x)^4}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{7/2}}dx}{7 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^5}{7 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-7 b e g+10 c d g+4 c e f) \left (-\frac {2 \int \frac {(c d-b e-c e x)^5}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{7/2}}dx}{3 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^4}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}\right )}{7 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^5}{7 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (\frac {4 (-b e+c d-c e x)^5}{15 e (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}-\frac {2 (-b e+c d-c e x)^4}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}\right ) (-7 b e g+10 c d g+4 c e f)}{7 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^5}{7 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}\) |
Input:
Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^5,x]
Output:
(-2*(e*f - d*g)*(c*d - b*e - c*e*x)^5)/(7*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2)) + ((4*c*e*f + 10*c*d*g - 7*b*e*g)*((-2*(c*d - b*e - c*e*x)^4)/(3*e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2 )^(5/2)) + (4*(c*d - b*e - c*e*x)^5)/(15*e*(2*c*d - b*e)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))))/(7*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 = 5.19 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.12
| method | result | size |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-14 b c \,e^{3} g \,x^{2}+20 c^{2} d \,e^{2} g \,x^{2}+8 f \,c^{2} e^{3} x^{2}+21 b^{2} e^{3} g x -100 b c d \,e^{2} g x -12 b c \,e^{3} f x +100 c^{2} d^{2} e g x +40 c^{2} d \,e^{2} f x +6 b^{2} d \,e^{2} g +15 b^{2} e^{3} f -26 b c \,d^{2} e g -72 b c d \,e^{2} f +20 c^{2} d^{3} g +92 c^{2} d^{2} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{105 \left (e x +d \right )^{4} \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2}}\) | \(236\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-14 b c \,e^{3} g \,x^{2}+20 c^{2} d \,e^{2} g \,x^{2}+8 f \,c^{2} e^{3} x^{2}+21 b^{2} e^{3} g x -100 b c d \,e^{2} g x -12 b c \,e^{3} f x +100 c^{2} d^{2} e g x +40 c^{2} d \,e^{2} f x +6 b^{2} d \,e^{2} g +15 b^{2} e^{3} f -26 b c \,d^{2} e g -72 b c d \,e^{2} f +20 c^{2} d^{3} g +92 c^{2} d^{2} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{105 \left (e x +d \right )^{4} \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2}}\) | \(236\) |
| trager | \(-\frac {2 \left (-14 b \,c^{2} e^{4} g \,x^{3}+20 c^{3} d \,e^{3} g \,x^{3}+8 c^{3} e^{4} f \,x^{3}+7 b^{2} c \,e^{4} g \,x^{2}-66 b \,c^{2} d \,e^{3} g \,x^{2}-4 b \,c^{2} e^{4} f \,x^{2}+80 c^{3} d^{2} e^{2} g \,x^{2}+32 c^{3} d \,e^{3} f \,x^{2}+21 b^{3} e^{4} g x -115 b^{2} c d \,e^{3} g x +3 b^{2} c \,e^{4} f x +174 b \,c^{2} d^{2} e^{2} g x -20 b \,c^{2} d \,e^{3} f x -80 c^{3} d^{3} e g x +52 c^{3} d^{2} e^{2} f x +6 b^{3} d \,e^{3} g +15 b^{3} e^{4} f -32 b^{2} c \,d^{2} e^{2} g -87 b^{2} c d \,e^{3} f +46 b \,c^{2} d^{3} e g +164 b \,c^{2} d^{2} e^{2} f -20 c^{3} d^{4} g -92 c^{3} d^{3} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{105 \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2} \left (e x +d \right )^{4}}\) | \(356\) |
| default | \(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{e^{5}}-\frac {\left (d g -e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{5}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{6}}\) | \(366\) |
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)^5,x,method=_RET URNVERBOSE)
Output:
-2/105*(c*e*x+b*e-c*d)*(-14*b*c*e^3*g*x^2+20*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x ^2+21*b^2*e^3*g*x-100*b*c*d*e^2*g*x-12*b*c*e^3*f*x+100*c^2*d^2*e*g*x+40*c^ 2*d*e^2*f*x+6*b^2*d*e^2*g+15*b^2*e^3*f-26*b*c*d^2*e*g-72*b*c*d*e^2*f+20*c^ 2*d^3*g+92*c^2*d^2*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4/( 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. 540 vs. \(2 (198) = 396\).
Time = 26.50 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.57 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{3} e^{4} f + {\left (10 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} + {\left (4 \, {\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f + {\left (80 \, c^{3} d^{2} e^{2} - 66 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} - {\left (92 \, c^{3} d^{3} e - 164 \, b c^{2} d^{2} e^{2} + 87 \, b^{2} c d e^{3} - 15 \, b^{3} e^{4}\right )} f - 2 \, {\left (10 \, c^{3} d^{4} - 23 \, b c^{2} d^{3} e + 16 \, b^{2} c d^{2} e^{2} - 3 \, b^{3} d e^{3}\right )} g + {\left ({\left (52 \, c^{3} d^{2} e^{2} - 20 \, b c^{2} d e^{3} + 3 \, b^{2} c e^{4}\right )} f - {\left (80 \, c^{3} d^{3} e - 174 \, b c^{2} d^{2} e^{2} + 115 \, b^{2} c d e^{3} - 21 \, b^{3} e^{4}\right )} g\right )} x\right )}}{105 \, {\left (8 \, c^{3} d^{7} e^{2} - 12 \, b c^{2} d^{6} e^{3} + 6 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5} + {\left (8 \, c^{3} d^{3} e^{6} - 12 \, b c^{2} d^{2} e^{7} + 6 \, b^{2} c d e^{8} - b^{3} e^{9}\right )} x^{4} + 4 \, {\left (8 \, c^{3} d^{4} e^{5} - 12 \, b c^{2} d^{3} e^{6} + 6 \, b^{2} c d^{2} e^{7} - b^{3} d e^{8}\right )} x^{3} + 6 \, {\left (8 \, c^{3} d^{5} e^{4} - 12 \, b c^{2} d^{4} e^{5} + 6 \, b^{2} c d^{3} e^{6} - b^{3} d^{2} e^{7}\right )} x^{2} + 4 \, {\left (8 \, c^{3} d^{6} e^{3} - 12 \, b c^{2} d^{5} e^{4} + 6 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x\right )}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^5,x, algo rithm="fricas")
Output:
2/105*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^3*e^4*f + (10*c^3 *d*e^3 - 7*b*c^2*e^4)*g)*x^3 + (4*(8*c^3*d*e^3 - b*c^2*e^4)*f + (80*c^3*d^ 2*e^2 - 66*b*c^2*d*e^3 + 7*b^2*c*e^4)*g)*x^2 - (92*c^3*d^3*e - 164*b*c^2*d ^2*e^2 + 87*b^2*c*d*e^3 - 15*b^3*e^4)*f - 2*(10*c^3*d^4 - 23*b*c^2*d^3*e + 16*b^2*c*d^2*e^2 - 3*b^3*d*e^3)*g + ((52*c^3*d^2*e^2 - 20*b*c^2*d*e^3 + 3 *b^2*c*e^4)*f - (80*c^3*d^3*e - 174*b*c^2*d^2*e^2 + 115*b^2*c*d*e^3 - 21*b ^3*e^4)*g)*x)/(8*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^ 4*e^5 + (8*c^3*d^3*e^6 - 12*b*c^2*d^2*e^7 + 6*b^2*c*d*e^8 - b^3*e^9)*x^4 + 4*(8*c^3*d^4*e^5 - 12*b*c^2*d^3*e^6 + 6*b^2*c*d^2*e^7 - b^3*d*e^8)*x^3 + 6*(8*c^3*d^5*e^4 - 12*b*c^2*d^4*e^5 + 6*b^2*c*d^3*e^6 - b^3*d^2*e^7)*x^2 + 4*(8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x)
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, dx \] 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)**5,x )
Output:
Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**5, x)
Exception generated. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \] 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)^5,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
Leaf count of result is larger than twice the leaf count of optimal. 1619 vs. \(2 (198) = 396\).
Time = 0.17 (sec) , antiderivative size = 1619, normalized size of antiderivative = 7.71 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx=\text {Too large to display} \] 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)^5,x, algo rithm="giac")
Output:
-2/105*(2*(4*sqrt(-c)*c^3*e*f + 10*sqrt(-c)*c^3*d*g - 7*b*sqrt(-c)*c^2*e*g )*sgn(1/(e*x + d))*sgn(e)/(8*c^3*d^3*e^3 - 12*b*c^2*d^2*e^4 + 6*b^2*c*d*e^ 5 - b^3*e^6) - (6*(5*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^3*sqrt(-c + 2*c *d/(e*x + d) - b*e/(e*x + d)) - 21*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2 *c*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) - 35*c^3*sqrt(-c + 2*c*d/(e* x + d) - b*e/(e*x + d)) - 35*c^2*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3 /2))*c*d*e^3*f*sgn(1/(e*x + d))*sgn(e)/(8*c^3*d^3*e^3 - 12*b*c^2*d^2*e^4 + 6*b^2*c*d*e^5 - b^3*e^6) - 3*(5*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^3*s qrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) - 21*(c - 2*c*d/(e*x + d) + b*e/ (e*x + d))^2*c*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) - 35*c^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) - 35*c^2*(-c + 2*c*d/(e*x + d) - b*e/( e*x + d))^(3/2))*b*e^4*f*sgn(1/(e*x + d))*sgn(e)/(8*c^3*d^3*e^3 - 12*b*c^2 *d^2*e^4 + 6*b^2*c*d*e^5 - b^3*e^6) - 6*(5*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) - 21*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*c*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) - 35*c ^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) - 35*c^2*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2))*c*d^2*e^2*g*sgn(1/(e*x + d))*sgn(e)/(8*c^3*d^3* e^3 - 12*b*c^2*d^2*e^4 + 6*b^2*c*d*e^5 - b^3*e^6) + 3*(5*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) - 21*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*c*sqrt(-c + 2*c*d/(e*x + d) - b*e/...
Time = 8.89 (sec) , antiderivative size = 2325, normalized size of antiderivative = 11.07 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx=\text {Too large to display} \] 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)^5,x)
Output:
(((d*((16*c^4*e*f - 144*c^4*d*g + 80*b*c^3*e*g)/(105*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)))/e - (4*b*c^2*(9*b*e*g - 18*c*d*g + 2*c*e*f))/(105*e*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^( 1/2))/(d + e*x) - (((d*((16*c^4*e*f - 64*c^4*d*g + 40*b*c^3*e*g)/(105*e*(b *e - 2*c*d)^4) - (16*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)))/e - (8*b*c^2*(2*b* e*g - 4*c*d*g + c*e*f))/(105*e*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d* e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((16*c^4*e*f - 176*c^4*d*g + 96*b*c^3 *e*g)/(105*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)))/e - (4*b*c^2*(11*b*e*g - 22*c*d*g + 2*c*e*f))/(105*e*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((16*c^4*e*f - 256 *c^4*d*g + 136*b*c^3*e*g)/(105*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105*e*(b *e - 2*c*d)^4)))/e - (8*b*c^2*(8*b*e*g - 16*c*d*g + c*e*f))/(105*e*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*( (4*c^2*e*f - 8*c^2*d*g + 6*b*c*e*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d))))/e - (2*b*(b*e*g - 2*c*d*g + c*e*f))/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x ^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((2*f*(b*e - c*d))/(7*b*e^2 - 14*c*d*e) - (d*((2*b*e*g - 2*c*d*g + 2*c*e*f)/(7*b*e^2 - 14*c*d*e) - (2*c* d*g)/(7*b*e^2 - 14*c*d*e)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) )/(d + e*x)^4 - (((d*((4*c^2*(9*b*e*g - 16*c*d*g + 2*c*e*f))/(35*(3*b*e...
Time = 0.66 (sec) , antiderivative size = 1918, normalized size of antiderivative = 9.13 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx =\text {Too large to display} \] 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)^5,x)
Output:
(2*i*( - 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**3*d*e**3*g - 15*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*e**4*f - 21*sqrt(d + e*x)*s qrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*e**4 *g*x + 32*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*d**2*e**2*g + 87*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*d*e**3*f + 115*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*d*e**3*g*x - 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*e**4*f*x - 7*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*e**4*g*x**2 - 46*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*c**2*d**3*e*g - 164*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*c**2*d**2*e**2*f - 174*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*c** 2*d**2*e**2*g*x + 20*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*c**2*d*e**3*f*x + 66*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*c**2*d*e**3*g*x* *2 + 4*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*c**2*e**4*f*x**2 + 14*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq...