Integrand size = 44, antiderivative size = 354 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {(2 c d-b e)^2 (10 c e f-4 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^2 e}+\frac {(10 c e f-4 c d g-3 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c e}+\frac {(10 c e f-4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^2}+\frac {(2 c d-b e)^4 (10 c e f-4 c d g-3 b e g) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{5/2} e^2} \]
1/48*(-3*b*e*g-4*c*d*g+10*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2 )^(3/2)/c/e+1/15*(-3*b*e*g-4*c*d*g+10*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)+2/3*(-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)^2+1/256*(-b*e+2*c*d)^4*(-3*b*e*g-4*c*d*g+1 0*c*e*f)*arctan(1/2*e*(2*c*x+b)/c^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^( 1/2))/c^(5/2)/e^2+1/128*(-b*e+2*c*d)^2*(-3*b*e*g-4*c*d*g+10*c*e*f)*(2*c*x+ b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^2/e
Time = 1.08 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.06 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {(-2 c d+b e)^4 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {\sqrt {c} \left (-45 b^4 e^4 g+30 b^3 c e^3 (5 e f+8 d g+e g x)-16 c^4 \left (56 d^4 g+20 d e^3 x^2 (4 f+3 g x)-10 d^3 e (8 f+3 g x)-6 e^4 x^3 (5 f+4 g x)-d^2 e^2 x (45 f+32 g x)\right )+8 b c^3 e \left (174 d^3 g+2 e^3 x^2 (85 f+63 g x)-d^2 e (195 f+71 g x)-2 d e^2 x (125 f+82 g x)\right )+4 b^2 c^2 e^2 \left (-199 d^2 g+d e (70 f+32 g x)+e^2 x (295 f+186 g x)\right )\right )}{(-2 c d+b e)^4 (-c d+b e+c e x)^2}-\frac {15 (10 c e f-4 c d g-3 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {d+e x} (-b e+c (d-e x))^{5/2}}\right )}{1920 c^{5/2} e^2} \]
((-2*c*d + b*e)^4*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((Sqrt[c]*(-45*b^4*e^4*g + 30*b^3*c*e^3*(5*e*f + 8*d*g + e*g*x) - 16*c^4*(56*d^4*g + 20*d*e^3*x^2*(4*f + 3*g*x) - 10*d^3*e*(8*f + 3*g*x) - 6 *e^4*x^3*(5*f + 4*g*x) - d^2*e^2*x*(45*f + 32*g*x)) + 8*b*c^3*e*(174*d^3*g + 2*e^3*x^2*(85*f + 63*g*x) - d^2*e*(195*f + 71*g*x) - 2*d*e^2*x*(125*f + 82*g*x)) + 4*b^2*c^2*e^2*(-199*d^2*g + d*e*(70*f + 32*g*x) + e^2*x*(295*f + 186*g*x))))/((-2*c*d + b*e)^4*(-(c*d) + b*e + c*e*x)^2) - (15*(10*c*e*f - 4*c*d*g - 3*b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x ])])/(Sqrt[d + e*x]*(-(b*e) + c*(d - e*x))^(5/2))))/(1920*c^(5/2)*e^2)
Time = 0.58 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1220, 1131, 1087, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-3 b e g-4 c d g+10 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{d+e x}dx}{3 e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle \frac {(-3 b e g-4 c d g+10 c e f) \left (\frac {1}{2} (2 c d-b e) \int \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e}\right )}{3 e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(-3 b e g-4 c d g+10 c e f) \left (\frac {1}{2} (2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e}\right )}{3 e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(-3 b e g-4 c d g+10 c e f) \left (\frac {1}{2} (2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e}\right )}{3 e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {(-3 b e g-4 c d g+10 c e f) \left (\frac {1}{2} (2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e}\right )}{3 e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (\frac {1}{2} (2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e}\right ) (-3 b e g-4 c d g+10 c e f)}{3 e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}\) |
(2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^2) + ((10*c*e*f - 4*c*d*g - 3*b*e*g)*((d*(c*d - b*e) - b* e^2*x - c*e^2*x^2)^(5/2)/(5*e) + ((2*c*d - b*e)*(((b + 2*c*x)*(d*(c*d - b* e) - b*e^2*x - c*e^2*x^2)^(3/2))/(8*c) + (3*(2*c*d - b*e)^2*(((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)))/(16*c)))/2))/(3*e*(2*c*d - b*e))
3.22.99.3.1 Defintions of rubi rules used
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 + 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. \(750\) vs. \(2(328)=656\).
Time = 0.75 (sec) , antiderivative size = 751, normalized size of antiderivative = 2.12
| method | result | size |
| default | \(\frac {g \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (-b \,e^{2}+2 c d e \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c \,e^{2}}+\frac {3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 c d e \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{16 c \,e^{2}}\right )}{2}\right )}{e^{2}}+\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 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {10 c \,e^{2} \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (-b \,e^{2}+2 c d e \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c \,e^{2}}+\frac {3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 c d e \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{16 c \,e^{2}}\right )}{2}\right )}{3 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{3}}\) | \(751\) |
g/e^2*(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*c*d*e)/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*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)+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^3*(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*c*d*e)/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*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)+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 = 0.91 (sec) , antiderivative size = 1097, normalized size of antiderivative = 3.10 \[ \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)^2} \, dx=\left [\frac {15 \, {\left (10 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 24 \, b^{2} c^{3} d^{2} e^{3} - 8 \, b^{3} c^{2} d e^{4} + b^{4} c e^{5}\right )} f - {\left (64 \, c^{5} d^{5} - 80 \, b c^{4} d^{4} e + 40 \, b^{3} c^{2} d^{2} e^{3} - 20 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} g\right )} \sqrt {-c} \log \left (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} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, {\left (384 \, c^{5} e^{4} g x^{4} + 48 \, {\left (10 \, c^{5} e^{4} f - {\left (20 \, c^{5} d e^{3} - 21 \, b c^{4} e^{4}\right )} g\right )} x^{3} - 8 \, {\left (10 \, {\left (16 \, c^{5} d e^{3} - 17 \, b c^{4} e^{4}\right )} f - {\left (64 \, c^{5} d^{2} e^{2} - 164 \, b c^{4} d e^{3} + 93 \, b^{2} c^{3} e^{4}\right )} g\right )} x^{2} + 10 \, {\left (128 \, c^{5} d^{3} e - 156 \, b c^{4} d^{2} e^{2} + 28 \, b^{2} c^{3} d e^{3} + 15 \, b^{3} c^{2} e^{4}\right )} f - {\left (896 \, c^{5} d^{4} - 1392 \, b c^{4} d^{3} e + 796 \, b^{2} c^{3} d^{2} e^{2} - 240 \, b^{3} c^{2} d e^{3} + 45 \, b^{4} c e^{4}\right )} g + 2 \, {\left (10 \, {\left (36 \, c^{5} d^{2} e^{2} - 100 \, b c^{4} d e^{3} + 59 \, b^{2} c^{3} e^{4}\right )} f + {\left (240 \, c^{5} d^{3} e - 284 \, b c^{4} d^{2} e^{2} + 64 \, b^{2} c^{3} d e^{3} + 15 \, b^{3} c^{2} e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{7680 \, c^{3} e^{2}}, -\frac {15 \, {\left (10 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 24 \, b^{2} c^{3} d^{2} e^{3} - 8 \, b^{3} c^{2} d e^{4} + b^{4} c e^{5}\right )} f - {\left (64 \, c^{5} d^{5} - 80 \, b c^{4} d^{4} e + 40 \, b^{3} c^{2} d^{2} e^{3} - 20 \, b^{4} c d e^{4} + 3 \, b^{5} e^{5}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (384 \, c^{5} e^{4} g x^{4} + 48 \, {\left (10 \, c^{5} e^{4} f - {\left (20 \, c^{5} d e^{3} - 21 \, b c^{4} e^{4}\right )} g\right )} x^{3} - 8 \, {\left (10 \, {\left (16 \, c^{5} d e^{3} - 17 \, b c^{4} e^{4}\right )} f - {\left (64 \, c^{5} d^{2} e^{2} - 164 \, b c^{4} d e^{3} + 93 \, b^{2} c^{3} e^{4}\right )} g\right )} x^{2} + 10 \, {\left (128 \, c^{5} d^{3} e - 156 \, b c^{4} d^{2} e^{2} + 28 \, b^{2} c^{3} d e^{3} + 15 \, b^{3} c^{2} e^{4}\right )} f - {\left (896 \, c^{5} d^{4} - 1392 \, b c^{4} d^{3} e + 796 \, b^{2} c^{3} d^{2} e^{2} - 240 \, b^{3} c^{2} d e^{3} + 45 \, b^{4} c e^{4}\right )} g + 2 \, {\left (10 \, {\left (36 \, c^{5} d^{2} e^{2} - 100 \, b c^{4} d e^{3} + 59 \, b^{2} c^{3} e^{4}\right )} f + {\left (240 \, c^{5} d^{3} e - 284 \, b c^{4} d^{2} e^{2} + 64 \, b^{2} c^{3} d e^{3} + 15 \, b^{3} c^{2} e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{3840 \, c^{3} e^{2}}\right ] \]
[1/7680*(15*(10*(16*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8* b^3*c^2*d*e^4 + b^4*c*e^5)*f - (64*c^5*d^5 - 80*b*c^4*d^4*e + 40*b^3*c^2*d ^2*e^3 - 20*b^4*c*d*e^4 + 3*b^5*e^5)*g)*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*(384*c^5*e^4*g*x^4 + 48*(10*c^ 5*e^4*f - (20*c^5*d*e^3 - 21*b*c^4*e^4)*g)*x^3 - 8*(10*(16*c^5*d*e^3 - 17* b*c^4*e^4)*f - (64*c^5*d^2*e^2 - 164*b*c^4*d*e^3 + 93*b^2*c^3*e^4)*g)*x^2 + 10*(128*c^5*d^3*e - 156*b*c^4*d^2*e^2 + 28*b^2*c^3*d*e^3 + 15*b^3*c^2*e^ 4)*f - (896*c^5*d^4 - 1392*b*c^4*d^3*e + 796*b^2*c^3*d^2*e^2 - 240*b^3*c^2 *d*e^3 + 45*b^4*c*e^4)*g + 2*(10*(36*c^5*d^2*e^2 - 100*b*c^4*d*e^3 + 59*b^ 2*c^3*e^4)*f + (240*c^5*d^3*e - 284*b*c^4*d^2*e^2 + 64*b^2*c^3*d*e^3 + 15* b^3*c^2*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^3*e^2), -1/3840*(15*(10*(16*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8* b^3*c^2*d*e^4 + b^4*c*e^5)*f - (64*c^5*d^5 - 80*b*c^4*d^4*e + 40*b^3*c^2*d ^2*e^3 - 20*b^4*c*d*e^4 + 3*b^5*e^5)*g)*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*(384*c^5*e^4*g*x^4 + 48*(10*c^5*e^4*f - (20*c ^5*d*e^3 - 21*b*c^4*e^4)*g)*x^3 - 8*(10*(16*c^5*d*e^3 - 17*b*c^4*e^4)*f - (64*c^5*d^2*e^2 - 164*b*c^4*d*e^3 + 93*b^2*c^3*e^4)*g)*x^2 + 10*(128*c^5*d ^3*e - 156*b*c^4*d^2*e^2 + 28*b^2*c^3*d*e^3 + 15*b^3*c^2*e^4)*f - (896*...
Time = 9.31 (sec) , antiderivative size = 4675, normalized size of antiderivative = 13.21 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \]
b**2*e**2*f*Piecewise(((b/(4*c) + x/2)*sqrt(-b*d*e - b*e**2*x + c*d**2 - c *e**2*x**2) + (b**2*e**2/(8*c) - b*d*e/2 + c*d**2/2)*Piecewise((log(-b*e** 2 - 2*c*e**2*x + 2*sqrt(-c*e**2)*sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2* x**2))/sqrt(-c*e**2), Ne(b**2*e**2/(4*c) - b*d*e + c*d**2, 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(-c*e**2*(b/(2*c) + x)**2), True)), Ne(c*e**2, 0 )), (-2*(-b*d*e - b*e**2*x + c*d**2)**(3/2)/(3*b*e**2), Ne(b*e**2, 0)), (x *sqrt(-b*d*e + c*d**2), True)) + b**2*e**2*g*Piecewise(((-b*(-b*d*e + c*d* *2)/(12*c) - b*(b**2*e**2/(8*c) - b*d*e/3 + c*d**2/3)/(2*c))*Piecewise((lo g(-b*e**2 - 2*c*e**2*x + 2*sqrt(-c*e**2)*sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2))/sqrt(-c*e**2), Ne(b**2*e**2/(4*c) - b*d*e + c*d**2, 0)), (( b/(2*c) + x)*log(b/(2*c) + x)/sqrt(-c*e**2*(b/(2*c) + x)**2), True)) + (b* x/(12*c) + x**2/3 - (b**2*e**2/(8*c) - b*d*e/3 + c*d**2/3)/(c*e**2))*sqrt( -b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2), Ne(c*e**2, 0)), (2*((b*d*e - c* d**2)*(-b*d*e - b*e**2*x + c*d**2)**(3/2)/3 + (-b*d*e - b*e**2*x + c*d**2) **(5/2)/5)/(b**2*e**4), Ne(b*e**2, 0)), (x**2*sqrt(-b*d*e + c*d**2)/2, Tru e)) - 2*b*c*d*e*f*Piecewise(((b/(4*c) + x/2)*sqrt(-b*d*e - b*e**2*x + c*d* *2 - c*e**2*x**2) + (b**2*e**2/(8*c) - b*d*e/2 + c*d**2/2)*Piecewise((log( -b*e**2 - 2*c*e**2*x + 2*sqrt(-c*e**2)*sqrt(-b*d*e - b*e**2*x + c*d**2 - c *e**2*x**2))/sqrt(-c*e**2), Ne(b**2*e**2/(4*c) - b*d*e + c*d**2, 0)), ((b/ (2*c) + x)*log(b/(2*c) + x)/sqrt(-c*e**2*(b/(2*c) + x)**2), True)), Ne(...
Leaf count of result is larger than twice the leaf count of optimal. 1751 vs. \(2 (328) = 656\).
Time = 0.33 (sec) , antiderivative size = 1751, normalized size of antiderivative = 4.95 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \]
5/4*b*c^3*d^3*f*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/( 2*c*d - b*e))/(-c)^(3/2) - 5/8*c^4*d^4*f*arcsin(2*c*e*x/(2*c*d - b*e) + 4* c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/((-c)^(3/2)*e) - 15/16*b^2*c^2*d^2* e*f*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e) )/(-c)^(3/2) + 5/16*b^3*c*d*e^2*f*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2* c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) - 5/128*b^4*e^3*f*arcsin(2*c*e* x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) + 1/ 4*c^4*d^5*g*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c* d - b*e))/((-c)^(3/2)*e^2) - 5/16*b*c^3*d^4*g*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/((-c)^(3/2)*e) + 5/32*b^3*c*d^ 2*e*g*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b* e))/(-c)^(3/2) - 5/64*b^4*d*e^2*g*arcsin(2*c*e*x/(2*c*d - b*e) + 4*c*d/(2* c*d - b*e) - b*e/(2*c*d - b*e))/(-c)^(3/2) + 3/256*b^5*e^3*g*arcsin(2*c*e* x/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))/((-c)^(3/2)*c) + 5/8*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*c^2*d^2*f*x - 5/8*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b*c*d*e*f*x + 5/32*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b^2*e^2*f*x + 1/16*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b*c*d^2*g* x - 1/4*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*c^2*d^3*g* x/e + 1/8*sqrt(c*e^2*x^2 + 4*c*d*e*x - b*e^2*x + 3*c*d^2 - b*d*e)*b^2*d...
Leaf count of result is larger than twice the leaf count of optimal. 3630 vs. \(2 (328) = 656\).
Time = 1.28 (sec) , antiderivative size = 3630, normalized size of antiderivative = 10.25 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \]
-1/1920*(15*(160*c^5*d^4*e*f*sgn(1/(e*x + d))*sgn(e) - 320*b*c^4*d^3*e^2*f *sgn(1/(e*x + d))*sgn(e) + 240*b^2*c^3*d^2*e^3*f*sgn(1/(e*x + d))*sgn(e) - 80*b^3*c^2*d*e^4*f*sgn(1/(e*x + d))*sgn(e) + 10*b^4*c*e^5*f*sgn(1/(e*x + d))*sgn(e) - 64*c^5*d^5*g*sgn(1/(e*x + d))*sgn(e) + 80*b*c^4*d^4*e*g*sgn(1 /(e*x + d))*sgn(e) - 40*b^3*c^2*d^2*e^3*g*sgn(1/(e*x + d))*sgn(e) + 20*b^4 *c*d*e^4*g*sgn(1/(e*x + d))*sgn(e) - 3*b^5*e^5*g*sgn(1/(e*x + d))*sgn(e))* arctan(sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))/sqrt(c))/(c^(5/2)*e^3) + (2400*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^4*c^5*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^4*e*f*sgn(1/(e*x + d))*sgn(e) + 9280*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^3*c^6*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^ 4*e*f*sgn(1/(e*x + d))*sgn(e) - 20480*(c - 2*c*d/(e*x + d) + b*e/(e*x + d) )^2*c^7*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^4*e*f*sgn(1/(e*x + d) )*sgn(e) - 2400*c^9*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^4*e*f*sgn (1/(e*x + d))*sgn(e) - 11200*c^8*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3 /2)*d^4*e*f*sgn(1/(e*x + d))*sgn(e) - 4800*b*(c - 2*c*d/(e*x + d) + b*e/(e *x + d))^4*c^4*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^3*e^2*f*sgn(1/ (e*x + d))*sgn(e) - 18560*b*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^3*c^5*sq rt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^3*e^2*f*sgn(1/(e*x + d))*sgn(e) + 40960*b*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*c^6*sqrt(-c + 2*c*d/(e* x + d) - b*e/(e*x + d))*d^3*e^2*f*sgn(1/(e*x + d))*sgn(e) + 4800*b*c^8*...
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]