Integrand size = 44, antiderivative size = 290 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {5 (6 c e f-8 c d g+b e g) (8 c d-5 b e-2 c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 e^2}-\frac {(6 c e f-8 c d g+b e g) (c d-b e-c e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac {5 (2 c d-b e)^2 (6 c e f-8 c d g+b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 \sqrt {c} e^2} \] Output:
-5/24*(b*e*g-8*c*d*g+6*c*e*f)*(-2*c*e*x-5*b*e+8*c*d)*(d*(-b*e+c*d)-b*e^2*x -c*e^2*x^2)^(1/2)/e^2-1/3*(b*e*g-8*c*d*g+6*c*e*f)*(-c*e*x-b*e+c*d)^2*(d*(- b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)-2*(-d*g+e*f)*(d*(-b*e+c *d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)/(e*x+d)^4-5/8*(-b*e+2*c*d)^2 *(b*e*g-8*c*d*g+6*c*e*f)*arctan(c^(1/2)*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^ 2*x^2)^(1/2))/c^(1/2)/e^2
Time = 1.52 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.89 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {3 b^2 e^2 (-16 e f+27 d g+11 e g x)+2 b c e \left (-176 d^2 g+d e (123 f-67 g x)+e^2 x (27 f+13 g x)\right )+4 c^2 \left (94 d^3 g+e^3 x^2 (3 f+2 g x)-d e^2 x (21 f+10 g x)+d^2 e (-72 f+34 g x)\right )}{(d+e x)^3 (-c d+b e+c e x)^2}+\frac {15 (-2 c d+b e)^2 (6 c e f-8 c d g+b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {c} (d+e x)^{5/2} (-b e+c (d-e x))^{5/2}}\right )}{24 e^2} \] Input:
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x )^4,x]
Output:
(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((3*b^2*e^2*(-16*e*f + 27*d*g + 11*e*g*x) + 2*b*c*e*(-176*d^2*g + d*e*(123*f - 67*g*x) + e^2*x*(27*f + 13* g*x)) + 4*c^2*(94*d^3*g + e^3*x^2*(3*f + 2*g*x) - d*e^2*x*(21*f + 10*g*x) + d^2*e*(-72*f + 34*g*x)))/((d + e*x)^3*(-(c*d) + b*e + c*e*x)^2) + (15*(- 2*c*d + b*e)^2*(6*c*e*f - 8*c*d*g + b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/ (Sqrt[c]*Sqrt[d + e*x])])/(Sqrt[c]*(d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^ (5/2))))/(24*e^2)
Time = 1.01 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1220, 1130, 1131, 1087, 1092, 217}
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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {(b e g-8 c d g+6 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^3}dx}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1130 |
| \(\displaystyle -\frac {(b e g-8 c d g+6 c e f) \left (5 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{d+e x}dx+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^2}\right )}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(b e g-8 c d g+6 c e f) \left (5 c \left (\frac {1}{2} (2 c d-b e) \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^2}\right )}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle -\frac {(b e g-8 c d g+6 c e f) \left (5 c \left (\frac {1}{2} (2 c d-b e) \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^2}\right )}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {(b e g-8 c d g+6 c e f) \left (5 c \left (\frac {1}{2} (2 c d-b e) \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^2}\right )}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\left (5 c \left (\frac {1}{2} (2 c d-b e) \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^2}\right ) (b e g-8 c d g+6 c e f)}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}\) |
Input:
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^4,x]
Output:
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^4) - ((6*c*e*f - 8*c*d*g + b*e*g)*((2*(d*(c*d - b*e) - b*e ^2*x - c*e^2*x^2)^(5/2))/(e*(d + e*x)^2) + 5*c*((d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)/(3*e) + ((2*c*d - b*e)*(((b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c) + ((2*c*d - b*e)^2*ArcTan[(e*(b + 2*c*x))/(2 *Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(8*c^(3/2)*e)))/2))) /(e*(2*c*d - b*e))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + p + 1))), x] - Simp[c*(p/(e^2*(m + p + 1))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] & & IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b *d*e + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne Q[m + 2*p + 1, 0] && IntegerQ[2*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. \(1087\) vs. \(2(272)=544\).
Time = 4.23 (sec) , antiderivative size = 1088, normalized size of antiderivative = 3.75
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x,method=_RET URNVERBOSE)
Output:
g/e^4*(2/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+ d/e))^(7/2)+8*c*e^2/(-b*e^2+2*c*d*e)*(2/3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e ^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+10/3*c*e^2/(-b*e^2+2*c*d*e)*( 1/5*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+1/2*(-b*e^2+2*c*d*e) *(-1/8*(-2*c*e^2*(x+d/e)-b*e^2+2*d*e*c)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2* c*d*e)*(x+d/e))^(3/2)+3/16*(-b*e^2+2*c*d*e)^2/c/e^2*(-1/4*(-2*c*e^2*(x+d/e )-b*e^2+2*d*e*c)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+1 /8*(-b*e^2+2*c*d*e)^2/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2* (-b*e^2+2*c*d*e)/c/e^2)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)) )))))-(d*g-e*f)/e^5*(-2/(-b*e^2+2*c*d*e)/(x+d/e)^4*(-c*e^2*(x+d/e)^2+(-b*e ^2+2*c*d*e)*(x+d/e))^(7/2)-6*c*e^2/(-b*e^2+2*c*d*e)*(2/(-b*e^2+2*c*d*e)/(x +d/e)^3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+8*c*e^2/(-b*e^2+ 2*c*d*e)*(2/3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e )*(x+d/e))^(7/2)+10/3*c*e^2/(-b*e^2+2*c*d*e)*(1/5*(-c*e^2*(x+d/e)^2+(-b*e^ 2+2*c*d*e)*(x+d/e))^(5/2)+1/2*(-b*e^2+2*c*d*e)*(-1/8*(-2*c*e^2*(x+d/e)-b*e ^2+2*d*e*c)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+3/16*( -b*e^2+2*c*d*e)^2/c/e^2*(-1/4*(-2*c*e^2*(x+d/e)-b*e^2+2*d*e*c)/c/e^2*(-c*e ^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+1/8*(-b*e^2+2*c*d*e)^2/c/e^2/ (c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c* e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))))))))
Time = 1.04 (sec) , antiderivative size = 931, normalized size of antiderivative = 3.21 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x, algo rithm="fricas")
Output:
[-1/96*(15*(6*(4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (32*c^3*d^ 4 - 36*b*c^2*d^3*e + 12*b^2*c*d^2*e^2 - b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (32*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 12*b^2* c*d*e^3 - b^3*e^4)*g)*x)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2* d^2 + 4*b*c*d*e + b^2*e^2 + 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*( 2*c*e*x + b*e)*sqrt(-c)) - 4*(8*c^3*e^3*g*x^3 + 2*(6*c^3*e^3*f - (20*c^3*d *e^2 - 13*b*c^2*e^3)*g)*x^2 - 6*(48*c^3*d^2*e - 41*b*c^2*d*e^2 + 8*b^2*c*e ^3)*f + (376*c^3*d^3 - 352*b*c^2*d^2*e + 81*b^2*c*d*e^2)*g - (6*(14*c^3*d* e^2 - 9*b*c^2*e^3)*f - (136*c^3*d^2*e - 134*b*c^2*d*e^2 + 33*b^2*c*e^3)*g) *x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2), 1/48* (15*(6*(4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (32*c^3*d^4 - 36* b*c^2*d^3*e + 12*b^2*c*d^2*e^2 - b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2 - 4*b*c^ 2*d*e^3 + b^2*c*e^4)*f - (32*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 12*b^2*c*d*e^3 - b^3*e^4)*g)*x)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b *d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e )) + 2*(8*c^3*e^3*g*x^3 + 2*(6*c^3*e^3*f - (20*c^3*d*e^2 - 13*b*c^2*e^3)*g )*x^2 - 6*(48*c^3*d^2*e - 41*b*c^2*d*e^2 + 8*b^2*c*e^3)*f + (376*c^3*d^3 - 352*b*c^2*d^2*e + 81*b^2*c*d*e^2)*g - (6*(14*c^3*d*e^2 - 9*b*c^2*e^3)*f - (136*c^3*d^2*e - 134*b*c^2*d*e^2 + 33*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2)]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{4}}\, dx \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**4,x )
Output:
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**4, x )
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,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
Time = 0.43 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.73 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {1}{24} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} g x + \frac {6 \, c^{4} e^{4} f - 24 \, c^{4} d e^{3} g + 13 \, b c^{3} e^{4} g}{c^{2} e^{4}}\right )} x - \frac {96 \, c^{4} d e^{3} f - 54 \, b c^{3} e^{4} f - 184 \, c^{4} d^{2} e^{2} g + 160 \, b c^{3} d e^{3} g - 33 \, b^{2} c^{2} e^{4} g}{c^{2} e^{4}}\right )} + \frac {5 \, {\left (24 \, c^{3} d^{2} e f - 24 \, b c^{2} d e^{2} f + 6 \, b^{2} c e^{3} f - 32 \, c^{3} d^{3} g + 36 \, b c^{2} d^{2} e g - 12 \, b^{2} c d e^{2} g + b^{3} e^{3} g\right )} \log \left ({\left | b c d^{2} e^{2} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} c d^{2} {\left | e \right |} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b \sqrt {-c} d e {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} c d e - {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} \sqrt {-c} {\left | e \right |} \right |}\right )}{48 \, \sqrt {-c} e {\left | e \right |}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x, algo rithm="giac")
Output:
1/24*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*g*x + (6*c^4*e^4 *f - 24*c^4*d*e^3*g + 13*b*c^3*e^4*g)/(c^2*e^4))*x - (96*c^4*d*e^3*f - 54* b*c^3*e^4*f - 184*c^4*d^2*e^2*g + 160*b*c^3*d*e^3*g - 33*b^2*c^2*e^4*g)/(c ^2*e^4)) + 5/48*(24*c^3*d^2*e*f - 24*b*c^2*d*e^2*f + 6*b^2*c*e^3*f - 32*c^ 3*d^3*g + 36*b*c^2*d^2*e*g - 12*b^2*c*d*e^2*g + b^3*e^3*g)*log(abs(b*c*d^2 *e^2 - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqr t(-c)*c*d^2*abs(e) - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b*sqrt(-c)*d*e*abs(e) - 4*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b *e^2*x + c*d^2 - b*d*e))^2*c*d*e - (sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e ^2*x + c*d^2 - b*d*e))^2*b*e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e ^2*x + c*d^2 - b*d*e))^3*sqrt(-c)*abs(e)))/(sqrt(-c)*e*abs(e))
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^4,x)
Output:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^4, x )
Time = 0.41 (sec) , antiderivative size = 1826, normalized size of antiderivative = 6.30 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x)
Output:
(i*( - 120*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c* d))*b**4*d*e**4*g - 120*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*e**5*g*x + 1680*sqrt(c)*asinh((sqrt( - b*e + c*d - c *e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c*d**2*e**3*g - 720*sqrt(c)*asinh((sqr t( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c*d*e**4*f + 1680*sq rt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c*d* e**4*g*x - 720*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c*e**5*f*x - 7200*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i )/sqrt( - b*e + 2*c*d))*b**2*c**2*d**3*e**2*g + 4320*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**2*d**2*e**3*f - 7200 *sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c **2*d**2*e**3*g*x + 4320*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt ( - b*e + 2*c*d))*b**2*c**2*d*e**4*f*x + 12480*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**3*d**4*e*g - 8640*sqrt(c)*asi nh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**3*d**3*e**2*f + 12480*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d) )*b*c**3*d**3*e**2*g*x - 8640*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i) /sqrt( - b*e + 2*c*d))*b*c**3*d**2*e**3*f*x - 7680*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**4*d**5*g + 5760*sqrt(c)*asi nh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**4*d**4*e*f -...