Integrand size = 44, antiderivative size = 208 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (8 c e f+2 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {16 c (8 c e f+2 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \] Output:
2/15*(-5*b*e*g+2*c*d*g+8*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^3/(d*(-b*e+c*d)-b *e^2*x-c*e^2*x^2)^(3/2)-2/5*(-d*g+e*f)/e^2/(-b*e+2*c*d)/(e*x+d)/(d*(-b*e+c *d)-b*e^2*x-c*e^2*x^2)^(3/2)+16/15*c*(-5*b*e*g+2*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)
Time = 0.50 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.66 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \left (b^4 e^4 (3 e f+2 d g+5 e g x)+12 b^2 c^2 e^2 \left (d^3 g+d e^2 x (12 f-19 g x)+2 e^3 x^2 (2 f-5 g x)+2 d^2 e (7 f-g x)\right )-16 c^4 \left (3 d^5 g-8 e^5 f x^4-3 d^4 e (f-g x)-2 d^2 e^3 x^2 (-6 f+g x)+3 d^3 e^2 x (4 f+g x)-2 d e^4 x^3 (4 f+g x)\right )+8 b c^3 e \left (9 d^4 g+2 e^4 x^3 (12 f-5 g x)-6 d^3 e (4 f-3 g x)-4 d e^3 x^2 (-12 f+g x)+3 d^2 e^2 x (4 f+9 g x)\right )-2 b^3 c e^3 \left (19 d^2 g+e^2 x (4 f+15 g x)+2 d e (8 f+23 g x)\right )\right )}{15 e^2 (-2 c d+b e)^5 (d+e x)^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2) ),x]
Output:
(-2*(b^4*e^4*(3*e*f + 2*d*g + 5*e*g*x) + 12*b^2*c^2*e^2*(d^3*g + d*e^2*x*( 12*f - 19*g*x) + 2*e^3*x^2*(2*f - 5*g*x) + 2*d^2*e*(7*f - g*x)) - 16*c^4*( 3*d^5*g - 8*e^5*f*x^4 - 3*d^4*e*(f - g*x) - 2*d^2*e^3*x^2*(-6*f + g*x) + 3 *d^3*e^2*x*(4*f + g*x) - 2*d*e^4*x^3*(4*f + g*x)) + 8*b*c^3*e*(9*d^4*g + 2 *e^4*x^3*(12*f - 5*g*x) - 6*d^3*e*(4*f - 3*g*x) - 4*d*e^3*x^2*(-12*f + g*x ) + 3*d^2*e^2*x*(4*f + 9*g*x)) - 2*b^3*c*e^3*(19*d^2*g + e^2*x*(4*f + 15*g *x) + 2*d*e*(8*f + 23*g*x))))/(15*e^2*(-2*c*d + b*e)^5*(d + e*x)^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.64 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1220, 1089, 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.
| \(\displaystyle \int \frac {f+g x}{(d+e x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-5 b e g+2 c d g+8 c e f) \int \frac {1}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}dx}{5 e (2 c d-b e)}-\frac {2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1089 |
| \(\displaystyle \frac {(-5 b e g+2 c d g+8 c e f) \left (\frac {8 c \int \frac {1}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 (2 c d-b e)^2}+\frac {2 (b+2 c x)}{3 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{5 e (2 c d-b e)}-\frac {2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {\left (\frac {16 c (b+2 c x)}{3 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (b+2 c x)}{3 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right ) (-5 b e g+2 c d g+8 c e f)}{5 e (2 c d-b e)}-\frac {2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
Input:
Int[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
Output:
(-2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + ((8*c*e*f + 2*c*d*g - 5*b*e*g)*((2*(b + 2*c*x))/(3*(2 *c*d - b*e)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (16*c*(b + 2* c*x))/(3*(2*c*d - b*e)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])))/(5*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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
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. \(417\) vs. \(2(196)=392\).
Time = 2.40 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.01
| method | result | size |
| default | \(\frac {g \left (\frac {-\frac {4}{3} c \,e^{2} x -\frac {2}{3} b \,e^{2}}{\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}-\frac {16 c \,e^{2} \left (-2 c \,e^{2} x -b \,e^{2}\right )}{3 \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right )^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{e}-\frac {\left (d g -e f \right ) \left (-\frac {2}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right ) \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {8 c \,e^{2} \left (-\frac {2 \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 d e c \right )}{3 \left (-b \,e^{2}+2 d e c \right )^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {16 c \,e^{2} \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 d e c \right )}{3 \left (-b \,e^{2}+2 d e c \right )^{4} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{2}}\) | \(418\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-80 b \,c^{3} e^{5} g \,x^{4}+32 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}-120 b^{2} c^{2} e^{5} g \,x^{3}-32 b \,c^{3} d \,e^{4} g \,x^{3}+192 b \,c^{3} e^{5} f \,x^{3}+32 c^{4} d^{2} e^{3} g \,x^{3}+128 c^{4} d \,e^{4} f \,x^{3}-30 b^{3} c \,e^{5} g \,x^{2}-228 b^{2} c^{2} d \,e^{4} g \,x^{2}+48 b^{2} c^{2} e^{5} f \,x^{2}+216 b \,c^{3} d^{2} e^{3} g \,x^{2}+384 b \,c^{3} d \,e^{4} f \,x^{2}-48 c^{4} d^{3} e^{2} g \,x^{2}-192 c^{4} d^{2} e^{3} f \,x^{2}+5 b^{4} e^{5} g x -92 b^{3} c d \,e^{4} g x -8 b^{3} c \,e^{5} f x -24 b^{2} c^{2} d^{2} e^{3} g x +144 b^{2} c^{2} d \,e^{4} f x +144 b \,c^{3} d^{3} e^{2} g x +96 b \,c^{3} d^{2} e^{3} f x -48 c^{4} d^{4} e g x -192 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +3 b^{4} e^{5} f -38 b^{3} c \,d^{2} e^{3} g -32 b^{3} c d \,e^{4} f +12 b^{2} c^{2} d^{3} e^{2} g +168 b^{2} c^{2} d^{2} e^{3} f +72 b \,c^{3} d^{4} e g -192 b \,c^{3} d^{3} e^{2} f -48 c^{4} d^{5} g +48 c^{4} d^{4} e f \right )}{15 \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} d^{2} e^{3} c^{2}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 d^{5} c^{5}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(557\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-80 b \,c^{3} e^{5} g \,x^{4}+32 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}-120 b^{2} c^{2} e^{5} g \,x^{3}-32 b \,c^{3} d \,e^{4} g \,x^{3}+192 b \,c^{3} e^{5} f \,x^{3}+32 c^{4} d^{2} e^{3} g \,x^{3}+128 c^{4} d \,e^{4} f \,x^{3}-30 b^{3} c \,e^{5} g \,x^{2}-228 b^{2} c^{2} d \,e^{4} g \,x^{2}+48 b^{2} c^{2} e^{5} f \,x^{2}+216 b \,c^{3} d^{2} e^{3} g \,x^{2}+384 b \,c^{3} d \,e^{4} f \,x^{2}-48 c^{4} d^{3} e^{2} g \,x^{2}-192 c^{4} d^{2} e^{3} f \,x^{2}+5 b^{4} e^{5} g x -92 b^{3} c d \,e^{4} g x -8 b^{3} c \,e^{5} f x -24 b^{2} c^{2} d^{2} e^{3} g x +144 b^{2} c^{2} d \,e^{4} f x +144 b \,c^{3} d^{3} e^{2} g x +96 b \,c^{3} d^{2} e^{3} f x -48 c^{4} d^{4} e g x -192 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +3 b^{4} e^{5} f -38 b^{3} c \,d^{2} e^{3} g -32 b^{3} c d \,e^{4} f +12 b^{2} c^{2} d^{3} e^{2} g +168 b^{2} c^{2} d^{2} e^{3} f +72 b \,c^{3} d^{4} e g -192 b \,c^{3} d^{3} e^{2} f -48 c^{4} d^{5} g +48 c^{4} d^{4} e f \right )}{15 \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} d^{2} e^{3} c^{2}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 d^{5} c^{5}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(557\) |
| trager | \(\frac {2 \left (-80 b \,c^{3} e^{5} g \,x^{4}+32 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}-120 b^{2} c^{2} e^{5} g \,x^{3}-32 b \,c^{3} d \,e^{4} g \,x^{3}+192 b \,c^{3} e^{5} f \,x^{3}+32 c^{4} d^{2} e^{3} g \,x^{3}+128 c^{4} d \,e^{4} f \,x^{3}-30 b^{3} c \,e^{5} g \,x^{2}-228 b^{2} c^{2} d \,e^{4} g \,x^{2}+48 b^{2} c^{2} e^{5} f \,x^{2}+216 b \,c^{3} d^{2} e^{3} g \,x^{2}+384 b \,c^{3} d \,e^{4} f \,x^{2}-48 c^{4} d^{3} e^{2} g \,x^{2}-192 c^{4} d^{2} e^{3} f \,x^{2}+5 b^{4} e^{5} g x -92 b^{3} c d \,e^{4} g x -8 b^{3} c \,e^{5} f x -24 b^{2} c^{2} d^{2} e^{3} g x +144 b^{2} c^{2} d \,e^{4} f x +144 b \,c^{3} d^{3} e^{2} g x +96 b \,c^{3} d^{2} e^{3} f x -48 c^{4} d^{4} e g x -192 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +3 b^{4} e^{5} f -38 b^{3} c \,d^{2} e^{3} g -32 b^{3} c d \,e^{4} f +12 b^{2} c^{2} d^{3} e^{2} g +168 b^{2} c^{2} d^{2} e^{3} f +72 b \,c^{3} d^{4} e g -192 b \,c^{3} d^{3} e^{2} f -48 c^{4} d^{5} g +48 c^{4} d^{4} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{15 \left (b^{4} e^{4}-8 d \,e^{3} b^{3} c +24 d^{2} e^{2} b^{2} c^{2}-32 d^{3} e b \,c^{3}+16 d^{4} c^{4}\right ) \left (b e -2 c d \right ) e^{2} \left (c e x +b e -c d \right )^{2} \left (e x +d \right )^{3}}\) | \(562\) |
Input:
int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RETUR NVERBOSE)
Output:
g/e*(2/3*(-2*c*e^2*x-b*e^2)/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c*e^2*x^2- b*e^2*x-b*d*e+c*d^2)^(3/2)-16/3*c*e^2/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)^2* (-2*c*e^2*x-b*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-(d*g-e*f)/e^2*( -2/5/(-b*e^2+2*c*d*e)/(x+d/e)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^ (3/2)+8/5*c*e^2/(-b*e^2+2*c*d*e)*(-2/3*(-2*c*e^2*(x+d/e)-b*e^2+2*d*e*c)/(- b*e^2+2*c*d*e)^2/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)-16/3*c* e^2/(-b*e^2+2*c*d*e)^4*(-2*c*e^2*(x+d/e)-b*e^2+2*d*e*c)/(-c*e^2*(x+d/e)^2+ (-b*e^2+2*c*d*e)*(x+d/e))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (196) = 392\).
Time = 52.10 (sec) , antiderivative size = 1028, normalized size of antiderivative = 4.94 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori thm="fricas")
Output:
-2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(16*(8*c^4*e^5*f + (2*c^4 *d*e^4 - 5*b*c^3*e^5)*g)*x^4 + 8*(8*(2*c^4*d*e^4 + 3*b*c^3*e^5)*f + (4*c^4 *d^2*e^3 - 4*b*c^3*d*e^4 - 15*b^2*c^2*e^5)*g)*x^3 - 6*(8*(4*c^4*d^2*e^3 - 8*b*c^3*d*e^4 - b^2*c^2*e^5)*f + (8*c^4*d^3*e^2 - 36*b*c^3*d^2*e^3 + 38*b^ 2*c^2*d*e^4 + 5*b^3*c*e^5)*g)*x^2 + (48*c^4*d^4*e - 192*b*c^3*d^3*e^2 + 16 8*b^2*c^2*d^2*e^3 - 32*b^3*c*d*e^4 + 3*b^4*e^5)*f - 2*(24*c^4*d^5 - 36*b*c ^3*d^4*e - 6*b^2*c^2*d^3*e^2 + 19*b^3*c*d^2*e^3 - b^4*d*e^4)*g - (8*(24*c^ 4*d^3*e^2 - 12*b*c^3*d^2*e^3 - 18*b^2*c^2*d*e^4 + b^3*c*e^5)*f + (48*c^4*d ^4*e - 144*b*c^3*d^3*e^2 + 24*b^2*c^2*d^2*e^3 + 92*b^3*c*d*e^4 - 5*b^4*e^5 )*g)*x)/(32*c^7*d^10*e^2 - 144*b*c^6*d^9*e^3 + 272*b^2*c^5*d^8*e^4 - 280*b ^3*c^4*d^7*e^5 + 170*b^4*c^3*d^6*e^6 - 61*b^5*c^2*d^5*e^7 + 12*b^6*c*d^4*e ^8 - b^7*d^3*e^9 + (32*c^7*d^5*e^7 - 80*b*c^6*d^4*e^8 + 80*b^2*c^5*d^3*e^9 - 40*b^3*c^4*d^2*e^10 + 10*b^4*c^3*d*e^11 - b^5*c^2*e^12)*x^5 + (32*c^7*d ^6*e^6 - 16*b*c^6*d^5*e^7 - 80*b^2*c^5*d^4*e^8 + 120*b^3*c^4*d^3*e^9 - 70* b^4*c^3*d^2*e^10 + 19*b^5*c^2*d*e^11 - 2*b^6*c*e^12)*x^4 - (64*c^7*d^7*e^5 - 288*b*c^6*d^6*e^6 + 448*b^2*c^5*d^5*e^7 - 320*b^3*c^4*d^4*e^8 + 100*b^4 *c^3*d^3*e^9 - 2*b^5*c^2*d^2*e^10 - 6*b^6*c*d*e^11 + b^7*e^12)*x^3 - (64*c ^7*d^8*e^4 - 160*b*c^6*d^7*e^5 + 64*b^2*c^5*d^6*e^6 + 160*b^3*c^4*d^5*e^7 - 220*b^4*c^3*d^4*e^8 + 118*b^5*c^2*d^3*e^9 - 30*b^6*c*d^2*e^10 + 3*b^7*d* e^11)*x^2 + (32*c^7*d^9*e^3 - 208*b*c^6*d^8*e^4 + 496*b^2*c^5*d^7*e^5 -...
\[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
Output:
Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(d + e*x)), x)
Exception generated. \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori thm="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
\[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (e x + d\right )}} \,d x } \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori thm="giac")
Output:
integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d )), x)
Time = 13.64 (sec) , antiderivative size = 3326, normalized size of antiderivative = 15.99 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
int((f + g*x)/((d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2)),x)
Output:
(x*((16*c^2*(b*g - c*f))/(15*(b*e - 2*c*d)^5) - (8*b*c^2*g)/(15*(b*e - 2*c *d)^5)) + (72*c^3*d*e*f - 56*c^3*d^2*g - 44*b*c^2*e^2*f + 10*b^2*c*e^2*g + 20*b*c^2*d*e*g)/(15*e^2*(b*e - 2*c*d)^5) + (8*c^2*g*(c*d^2 - b*d*e))/(15* 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) + (((4*b *c*g)/(15*e*(b*e - 2*c*d)^5) - (8*c^2*d*g)/(15*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) + (((2*e^2*f)/(5*b^3*e^6 - 40*c^3*d^3*e^3 + 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5) - (2*d*e*g)/(5*b^3* e^6 - 40*c^3*d^3*e^3 + 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5))*(c*d^2 - c*e^2* x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((2*b*g)/(5*(3*b*e^2 - 6*c*d* e)*(b*e - 2*c*d)^3) - (4*c*d*g)/(5*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 - (((4*c*g*(3*b* e - 4*c*d))/(15*e^2*(b*e - 2*c*d)^5) - (8*c^2*d*g)/(15*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) + ((x*(((e*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*((4*c^4*e^2*(5*b*g - 4*c*f))/(15 *(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*c^4*g*(e*(b*e - c*d) + c*d*e))/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e )) + (4*b*c^4*e^2*g)/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2* d*e))))/(c*e^2) - (2*c^2*(8*b^2*c*e^3*g - 26*b*c^2*e^3*f + 36*c^3*d*e^2*f - 32*c^3*d^2*e*g + 14*b*c^2*d*e^2*g))/(15*e*(b*e - 2*c*d)^3*(4*c^3*d^2 + b ^2*c*e^2 - 4*b*c^2*d*e)) - (2*b*c^3*e^2*(5*b*g - 4*c*f))/(15*(b*e - 2*c...
Time = 0.64 (sec) , antiderivative size = 2763, normalized size of antiderivative = 13.28 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*i*( - 80*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*c*d**3*e**3*g - 240*sq rt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*c*d**2*e**4*g*x - 240*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*c*d*e**5*g*x**2 - 80*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*c*e**6*g*x**3 + 272*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c **2*d**4*e**2*g + 128*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**3*e* *3*f + 736*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**3*e**3*g*x + 38 4*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**2*e**4*f*x + 576*sqrt(c) *sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d**2*e**4*g*x**2 + 384*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*d*e**5*f*x**2 + 32*sqrt(c)*sqrt( - b*e + c *d - c*e*x)*b**2*c**2*d*e**5*g*x**3 + 128*sqrt(c)*sqrt( - b*e + c*d - c*e* x)*b**2*c**2*e**6*f*x**3 - 80*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2 *e**6*g*x**4 - 256*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*d**5*e*g - 38 4*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*d**4*e**2*f - 576*sqrt(c)*sqrt ( - b*e + c*d - c*e*x)*b*c**3*d**4*e**2*g*x - 1024*sqrt(c)*sqrt( - b*e + c *d - c*e*x)*b*c**3*d**3*e**3*f*x - 192*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 + 320*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**3*d**2*e**4*g *x**3 + 192*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 + 64*sqrt(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...