Integrand size = 44, antiderivative size = 284 \[ \int \frac {f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {16 c^2 (8 c e f+6 c d g-7 b e g) (b+2 c x)}{35 e (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (8 c e f+6 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {4 c (8 c e f+6 c d g-7 b e g)}{35 e^2 (2 c d-b e)^3 (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \] Output:
16/35*c^2*(-7*b*e*g+6*c*d*g+8*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^5/(d*(-b*e+c *d)-b*e^2*x-c*e^2*x^2)^(1/2)-2/7*(-d*g+e*f)/e^2/(-b*e+2*c*d)/(e*x+d)^3/(d* (-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-2/35*(-7*b*e*g+6*c*d*g+8*c*e*f)/e^2/(- b*e+2*c*d)^2/(e*x+d)^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-4/35*c*(-7*b *e*g+6*c*d*g+8*c*e*f)/e^2/(-b*e+2*c*d)^3/(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e ^2*x^2)^(1/2)
Time = 0.50 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.17 \[ \int \frac {f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \left (b^4 e^4 (5 e f+2 d g+7 e g x)+16 c^4 \left (d^5 g-8 e^5 f x^4+d^3 e^2 x (4 f-15 g x)-6 d e^4 x^3 (4 f+g x)+d^4 e (13 f+3 g x)-2 d^2 e^3 x^2 (10 f+9 g x)\right )-2 b^3 c e^3 \left (11 d^2 g+e^2 x (4 f+7 g x)+d e (24 f+38 g x)\right )-8 b c^3 e \left (15 d^4 g+d^2 e^2 x (52 f-11 g x)+4 d e^3 x^2 (8 f-9 g x)+2 e^4 x^3 (4 f-7 g x)+d^3 e (48 f+46 g x)\right )+4 b^2 c^2 e^2 \left (31 d^3 g+2 e^3 x^2 (2 f+7 g x)+2 d^2 e (23 f+53 g x)+d e^2 x (20 f+59 g x)\right )\right )}{35 e^2 (-2 c d+b e)^5 (d+e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[(f + g*x)/((d + e*x)^3*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/ 2)),x]
Output:
(2*(b^4*e^4*(5*e*f + 2*d*g + 7*e*g*x) + 16*c^4*(d^5*g - 8*e^5*f*x^4 + d^3* e^2*x*(4*f - 15*g*x) - 6*d*e^4*x^3*(4*f + g*x) + d^4*e*(13*f + 3*g*x) - 2* d^2*e^3*x^2*(10*f + 9*g*x)) - 2*b^3*c*e^3*(11*d^2*g + e^2*x*(4*f + 7*g*x) + d*e*(24*f + 38*g*x)) - 8*b*c^3*e*(15*d^4*g + d^2*e^2*x*(52*f - 11*g*x) + 4*d*e^3*x^2*(8*f - 9*g*x) + 2*e^4*x^3*(4*f - 7*g*x) + d^3*e*(48*f + 46*g* x)) + 4*b^2*c^2*e^2*(31*d^3*g + 2*e^3*x^2*(2*f + 7*g*x) + 2*d^2*e*(23*f + 53*g*x) + d*e^2*x*(20*f + 59*g*x))))/(35*e^2*(-2*c*d + b*e)^5*(d + e*x)^3* Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.79 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1220, 1129, 1129, 1088}
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.
| \(\displaystyle \int \frac {f+g x}{(d+e x)^3 \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-7 b e g+6 c d g+8 c e f) \int \frac {1}{(d+e x)^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{7 e (2 c d-b e)}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^3 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-7 b e g+6 c d g+8 c e f) \left (\frac {6 c \int \frac {1}{(d+e x) \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{5 (2 c d-b e)}-\frac {2}{5 e (d+e x)^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{7 e (2 c d-b e)}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^3 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-7 b e g+6 c d g+8 c e f) \left (\frac {6 c \left (\frac {4 c \int \frac {1}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 (2 c d-b e)}-\frac {2}{3 e (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 (2 c d-b e)}-\frac {2}{5 e (d+e x)^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{7 e (2 c d-b e)}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^3 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {\left (\frac {6 c \left (\frac {8 c (b+2 c x)}{3 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2}{3 e (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 (2 c d-b e)}-\frac {2}{5 e (d+e x)^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-7 b e g+6 c d g+8 c e f)}{7 e (2 c d-b e)}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^3 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
Input:
Int[(f + g*x)/((d + e*x)^3*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]
Output:
(-2*(e*f - d*g))/(7*e^2*(2*c*d - b*e)*(d + e*x)^3*Sqrt[d*(c*d - b*e) - b*e ^2*x - c*e^2*x^2]) + ((8*c*e*f + 6*c*d*g - 7*b*e*g)*(-2/(5*e*(2*c*d - b*e) *(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (6*c*((8*c*(b + 2*c*x))/(3*(2*c*d - b*e)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - 2/ (3*e*(2*c*d - b*e)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])))/ (5*(2*c*d - b*e))))/(7*e*(2*c*d - b*e))
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 ]
Leaf count of result is larger than twice the leaf count of optimal. \(561\) vs. \(2(268)=536\).
Time = 2.88 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.98
| method | result | size |
| trager | \(-\frac {2 \left (112 b \,c^{3} e^{5} g \,x^{4}-96 c^{4} d \,e^{4} g \,x^{4}-128 c^{4} e^{5} f \,x^{4}+56 b^{2} c^{2} e^{5} g \,x^{3}+288 b \,c^{3} d \,e^{4} g \,x^{3}-64 b \,c^{3} e^{5} f \,x^{3}-288 c^{4} d^{2} e^{3} g \,x^{3}-384 c^{4} d \,e^{4} f \,x^{3}-14 b^{3} c \,e^{5} g \,x^{2}+236 b^{2} c^{2} d \,e^{4} g \,x^{2}+16 b^{2} c^{2} e^{5} f \,x^{2}+88 b \,c^{3} d^{2} e^{3} g \,x^{2}-256 b \,c^{3} d \,e^{4} f \,x^{2}-240 c^{4} d^{3} e^{2} g \,x^{2}-320 c^{4} d^{2} e^{3} f \,x^{2}+7 b^{4} e^{5} g x -76 b^{3} c d \,e^{4} g x -8 b^{3} c \,e^{5} f x +424 b^{2} c^{2} d^{2} e^{3} g x +80 b^{2} c^{2} d \,e^{4} f x -368 b \,c^{3} d^{3} e^{2} g x -416 b \,c^{3} d^{2} e^{3} f x +48 c^{4} d^{4} e g x +64 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +5 b^{4} e^{5} f -22 b^{3} c \,d^{2} e^{3} g -48 b^{3} c d \,e^{4} f +124 b^{2} c^{2} d^{3} e^{2} g +184 b^{2} c^{2} d^{2} e^{3} f -120 b \,c^{3} d^{4} e g -384 b \,c^{3} d^{3} e^{2} f +16 c^{4} d^{5} g +208 c^{4} d^{4} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{35 \left (c e x +b e -c d \right ) \left (b e -2 c d \right ) \left (b^{4} e^{4}-8 d \,e^{3} b^{3} c +24 d^{2} e^{2} b^{2} c^{2}-32 d^{3} e b \,c^{3}+16 d^{4} c^{4}\right ) e^{2} \left (e x +d \right )^{4}}\) | \(562\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (112 b \,c^{3} e^{5} g \,x^{4}-96 c^{4} d \,e^{4} g \,x^{4}-128 c^{4} e^{5} f \,x^{4}+56 b^{2} c^{2} e^{5} g \,x^{3}+288 b \,c^{3} d \,e^{4} g \,x^{3}-64 b \,c^{3} e^{5} f \,x^{3}-288 c^{4} d^{2} e^{3} g \,x^{3}-384 c^{4} d \,e^{4} f \,x^{3}-14 b^{3} c \,e^{5} g \,x^{2}+236 b^{2} c^{2} d \,e^{4} g \,x^{2}+16 b^{2} c^{2} e^{5} f \,x^{2}+88 b \,c^{3} d^{2} e^{3} g \,x^{2}-256 b \,c^{3} d \,e^{4} f \,x^{2}-240 c^{4} d^{3} e^{2} g \,x^{2}-320 c^{4} d^{2} e^{3} f \,x^{2}+7 b^{4} e^{5} g x -76 b^{3} c d \,e^{4} g x -8 b^{3} c \,e^{5} f x +424 b^{2} c^{2} d^{2} e^{3} g x +80 b^{2} c^{2} d \,e^{4} f x -368 b \,c^{3} d^{3} e^{2} g x -416 b \,c^{3} d^{2} e^{3} f x +48 c^{4} d^{4} e g x +64 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +5 b^{4} e^{5} f -22 b^{3} c \,d^{2} e^{3} g -48 b^{3} c d \,e^{4} f +124 b^{2} c^{2} d^{3} e^{2} g +184 b^{2} c^{2} d^{2} e^{3} f -120 b \,c^{3} d^{4} e g -384 b \,c^{3} d^{3} e^{2} f +16 c^{4} d^{5} g +208 c^{4} d^{4} e f \right )}{35 \left (e x +d \right )^{2} \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} d^{2} e^{3} c^{2}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 d^{5} c^{5}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(564\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (112 b \,c^{3} e^{5} g \,x^{4}-96 c^{4} d \,e^{4} g \,x^{4}-128 c^{4} e^{5} f \,x^{4}+56 b^{2} c^{2} e^{5} g \,x^{3}+288 b \,c^{3} d \,e^{4} g \,x^{3}-64 b \,c^{3} e^{5} f \,x^{3}-288 c^{4} d^{2} e^{3} g \,x^{3}-384 c^{4} d \,e^{4} f \,x^{3}-14 b^{3} c \,e^{5} g \,x^{2}+236 b^{2} c^{2} d \,e^{4} g \,x^{2}+16 b^{2} c^{2} e^{5} f \,x^{2}+88 b \,c^{3} d^{2} e^{3} g \,x^{2}-256 b \,c^{3} d \,e^{4} f \,x^{2}-240 c^{4} d^{3} e^{2} g \,x^{2}-320 c^{4} d^{2} e^{3} f \,x^{2}+7 b^{4} e^{5} g x -76 b^{3} c d \,e^{4} g x -8 b^{3} c \,e^{5} f x +424 b^{2} c^{2} d^{2} e^{3} g x +80 b^{2} c^{2} d \,e^{4} f x -368 b \,c^{3} d^{3} e^{2} g x -416 b \,c^{3} d^{2} e^{3} f x +48 c^{4} d^{4} e g x +64 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +5 b^{4} e^{5} f -22 b^{3} c \,d^{2} e^{3} g -48 b^{3} c d \,e^{4} f +124 b^{2} c^{2} d^{3} e^{2} g +184 b^{2} c^{2} d^{2} e^{3} f -120 b \,c^{3} d^{4} e g -384 b \,c^{3} d^{3} e^{2} f +16 c^{4} d^{5} g +208 c^{4} d^{4} e f \right )}{35 \left (e x +d \right )^{2} \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} d^{2} e^{3} c^{2}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 d^{5} c^{5}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(564\) |
| default | \(\frac {g \left (-\frac {2}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}+\frac {6 c \,e^{2} \left (-\frac {2}{3 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c \,e^{2} \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 d e c \right )}{3 \left (-b \,e^{2}+2 d e c \right )^{3} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{3}}-\frac {\left (d g -e f \right ) \left (-\frac {2}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{3} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c \,e^{2} \left (-\frac {2}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}+\frac {6 c \,e^{2} \left (-\frac {2}{3 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c \,e^{2} \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 d e c \right )}{3 \left (-b \,e^{2}+2 d e c \right )^{3} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (-b \,e^{2}+2 d e c \right )}\right )}{7 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{4}}\) | \(566\) |
Input:
int((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method=_RET URNVERBOSE)
Output:
-2/35*(112*b*c^3*e^5*g*x^4-96*c^4*d*e^4*g*x^4-128*c^4*e^5*f*x^4+56*b^2*c^2 *e^5*g*x^3+288*b*c^3*d*e^4*g*x^3-64*b*c^3*e^5*f*x^3-288*c^4*d^2*e^3*g*x^3- 384*c^4*d*e^4*f*x^3-14*b^3*c*e^5*g*x^2+236*b^2*c^2*d*e^4*g*x^2+16*b^2*c^2* e^5*f*x^2+88*b*c^3*d^2*e^3*g*x^2-256*b*c^3*d*e^4*f*x^2-240*c^4*d^3*e^2*g*x ^2-320*c^4*d^2*e^3*f*x^2+7*b^4*e^5*g*x-76*b^3*c*d*e^4*g*x-8*b^3*c*e^5*f*x+ 424*b^2*c^2*d^2*e^3*g*x+80*b^2*c^2*d*e^4*f*x-368*b*c^3*d^3*e^2*g*x-416*b*c ^3*d^2*e^3*f*x+48*c^4*d^4*e*g*x+64*c^4*d^3*e^2*f*x+2*b^4*d*e^4*g+5*b^4*e^5 *f-22*b^3*c*d^2*e^3*g-48*b^3*c*d*e^4*f+124*b^2*c^2*d^3*e^2*g+184*b^2*c^2*d ^2*e^3*f-120*b*c^3*d^4*e*g-384*b*c^3*d^3*e^2*f+16*c^4*d^5*g+208*c^4*d^4*e* f)/(c*e*x+b*e-c*d)/(b*e-2*c*d)/(b^4*e^4-8*b^3*c*d*e^3+24*b^2*c^2*d^2*e^2-3 2*b*c^3*d^3*e+16*c^4*d^4)/e^2/(e*x+d)^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^( 1/2)
Leaf count of result is larger than twice the leaf count of optimal. 974 vs. \(2 (268) = 536\).
Time = 74.11 (sec) , antiderivative size = 974, normalized size of antiderivative = 3.43 \[ \int \frac {f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algo rithm="fricas")
Output:
2/35*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(16*(8*c^4*e^5*f + (6*c^4* d*e^4 - 7*b*c^3*e^5)*g)*x^4 + 8*(8*(6*c^4*d*e^4 + b*c^3*e^5)*f + (36*c^4*d ^2*e^3 - 36*b*c^3*d*e^4 - 7*b^2*c^2*e^5)*g)*x^3 + 2*(8*(20*c^4*d^2*e^3 + 1 6*b*c^3*d*e^4 - b^2*c^2*e^5)*f + (120*c^4*d^3*e^2 - 44*b*c^3*d^2*e^3 - 118 *b^2*c^2*d*e^4 + 7*b^3*c*e^5)*g)*x^2 - (208*c^4*d^4*e - 384*b*c^3*d^3*e^2 + 184*b^2*c^2*d^2*e^3 - 48*b^3*c*d*e^4 + 5*b^4*e^5)*f - 2*(8*c^4*d^5 - 60* b*c^3*d^4*e + 62*b^2*c^2*d^3*e^2 - 11*b^3*c*d^2*e^3 + b^4*d*e^4)*g - (8*(8 *c^4*d^3*e^2 - 52*b*c^3*d^2*e^3 + 10*b^2*c^2*d*e^4 - b^3*c*e^5)*f + (48*c^ 4*d^4*e - 368*b*c^3*d^3*e^2 + 424*b^2*c^2*d^2*e^3 - 76*b^3*c*d*e^4 + 7*b^4 *e^5)*g)*x)/(32*c^6*d^10*e^2 - 112*b*c^5*d^9*e^3 + 160*b^2*c^4*d^8*e^4 - 1 20*b^3*c^3*d^7*e^5 + 50*b^4*c^2*d^6*e^6 - 11*b^5*c*d^5*e^7 + b^6*d^4*e^8 - (32*c^6*d^5*e^7 - 80*b*c^5*d^4*e^8 + 80*b^2*c^4*d^3*e^9 - 40*b^3*c^3*d^2* e^10 + 10*b^4*c^2*d*e^11 - b^5*c*e^12)*x^5 - (96*c^6*d^6*e^6 - 208*b*c^5*d ^5*e^7 + 160*b^2*c^4*d^4*e^8 - 40*b^3*c^3*d^3*e^9 - 10*b^4*c^2*d^2*e^10 + 7*b^5*c*d*e^11 - b^6*e^12)*x^4 - 2*(32*c^6*d^7*e^5 - 16*b*c^5*d^6*e^6 - 80 *b^2*c^4*d^5*e^7 + 120*b^3*c^3*d^4*e^8 - 70*b^4*c^2*d^3*e^9 + 19*b^5*c*d^2 *e^10 - 2*b^6*d*e^11)*x^3 + 2*(32*c^6*d^8*e^4 - 176*b*c^5*d^7*e^5 + 320*b^ 2*c^4*d^6*e^6 - 280*b^3*c^3*d^5*e^7 + 130*b^4*c^2*d^4*e^8 - 31*b^5*c*d^3*e ^9 + 3*b^6*d^2*e^10)*x^2 + (96*c^6*d^9*e^3 - 368*b*c^5*d^8*e^4 + 560*b^2*c ^4*d^7*e^5 - 440*b^3*c^3*d^6*e^6 + 190*b^4*c^2*d^5*e^7 - 43*b^5*c*d^4*e...
\[ \int \frac {f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \] Input:
integrate((g*x+f)/(e*x+d)**3/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x )
Output:
Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)**3), x)
Exception generated. \[ \int \frac {f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),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
\[ \int \frac {f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algo rithm="giac")
Output:
integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d )^3), x)
Time = 16.42 (sec) , antiderivative size = 4339, normalized size of antiderivative = 15.28 \[ \int \frac {f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
int((f + g*x)/((d + e*x)^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2)),x)
Output:
(((8*c*g*(2*b*e - 3*c*d))/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (8* c^2*d*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((d*((8*c^4*(2*c*d*g - 7*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^7) + (16*c^5*d*g)/(105*(b*e - 2*c*d)^7)))/e + (64*c^5*d^2*g - 20*b^2*c^3*e^2*g - 320*c^5*d*e*f + 112*b*c^4*e^2*f + 80 *b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)^7)))/e - (2*b*c^2*(16*c^2*d^2*g - 11*b^ 2*e^2*g + 34*b*c*e^2*f - 80*c^2*d*e*f + 22*b*c*d*e*g))/(105*e*(b*e - 2*c*d )^7))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((4*b*c*g) /(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (8*c^2*d*g)/(35*e*(3*b*e^2 - 6 *c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((8*c^2*g*(3*b*e - 4*c*d))/(105*e^2*(b*e - 2*c*d)^5) - (16*c^3 *d*g)/(105*e^2*(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*((48*c^4*f - 40*b*c^3*g)/(105*(b*e - 2*c*d)^6) + (16* c^4*d*g)/(105*e*(b*e - 2*c*d)^6)))/e + (8*b*c^2*(2*b*g - 3*c*f))/(105*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + ((( d*((48*c^4*d*g + 48*c^4*e*f - 64*b*c^3*e*g)/(105*e*(b*e - 2*c*d)^6) + (16* c^4*d*g)/(105*e*(b*e - 2*c*d)^6)))/e + (96*c^4*d*f - 72*b*c^3*d*g - 72*b*c ^3*e*f + 52*b^2*c^2*e*g)/(105*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d *e - b*e^2*x)^(1/2))/(d + e*x) + (((2*b*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (4*c*d*g)/(7*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2))*(c*d^...
Time = 0.68 (sec) , antiderivative size = 2565, normalized size of antiderivative = 9.03 \[ \int \frac {f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
Output:
(2*i*( - 112*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**4*e**2*g - 44 8*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**3*e**3*g*x - 672*sqrt(c) *sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**2*e**4*g*x**2 - 448*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d*e**5*g*x**3 - 112*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*e**6*g*x**4 + 320*sqrt(c)*sqrt( - b*e + c*d - c*e*x )*b*c**3*d**5*e*g + 128*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*d**4*e** 2*f + 1280*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*d**4*e**2*g*x + 512*s qrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*d**3*e**3*f*x + 1920*sqrt(c)*sqrt ( - b*e + c*d - c*e*x)*b*c**3*d**3*e**3*g*x**2 + 768*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*d**2*e**4*f*x**2 + 1280*sqrt(c)*sqrt( - b*e + c*d - c *e*x)*b*c**3*d**2*e**4*g*x**3 + 512*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c **3*d*e**5*f*x**3 + 320*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*d*e**5*g *x**4 + 128*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*e**6*f*x**4 - 192*sq rt(c)*sqrt( - b*e + c*d - c*e*x)*c**4*d**6*g - 256*sqrt(c)*sqrt( - b*e + c *d - c*e*x)*c**4*d**5*e*f - 768*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**4*d* *5*e*g*x - 1024*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**4*d**4*e**2*f*x - 11 52*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**4*d**4*e**2*g*x**2 - 1536*sqrt(c) *sqrt( - b*e + c*d - c*e*x)*c**4*d**3*e**3*f*x**2 - 768*sqrt(c)*sqrt( - b* e + c*d - c*e*x)*c**4*d**3*e**3*g*x**3 - 1024*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**4*d**2*e**4*f*x**3 - 192*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c...