Integrand size = 44, antiderivative size = 350 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {5 \sqrt {c} (2 c d-b e) (4 c e f-10 c d g+3 b e g) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]
5/6*c*(3*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)/(e*x+d)+2/3*(3*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)^(5/2)/e^2/(-b*e+2*c*d)/(e*x+d)^3-2/3*(-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)^5+5/8*(-b*e+2*c*d)*(3*b *e*g-10*c*d*g+4*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^(1/2)/e^2+5/4*c*(3*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)^(1/2)/e^2
Time = 2.24 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.76 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (-\sqrt {-b e+c (d-e x)} \left (b c e \left (-147 d^2 g+d e (24 f-206 g x)+e^2 x (56 f-27 g x)\right )+8 b^2 e^2 (2 d g+e (f+3 g x))+2 c^2 \left (118 d^3 g-23 d^2 e (2 f-7 g x)-3 e^3 x^2 (2 f+g x)+4 d e^2 x (-17 f+6 g x)\right )\right )+15 \sqrt {-c} (2 c d-b e) (4 c e f-10 c d g+3 b e g) (d+e x)^{3/2} \log \left (-\sqrt {-c} \sqrt {d+e x}+\sqrt {c d-b e-c e x}\right )\right )}{12 e^2 (d+e x)^4 (-b e+c (d-e x))^{5/2}} \]
(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(-(Sqrt[-(b*e) + c*(d - e*x)]*(b *c*e*(-147*d^2*g + d*e*(24*f - 206*g*x) + e^2*x*(56*f - 27*g*x)) + 8*b^2*e ^2*(2*d*g + e*(f + 3*g*x)) + 2*c^2*(118*d^3*g - 23*d^2*e*(2*f - 7*g*x) - 3 *e^3*x^2*(2*f + g*x) + 4*d*e^2*x*(-17*f + 6*g*x)))) + 15*Sqrt[-c]*(2*c*d - b*e)*(4*c*e*f - 10*c*d*g + 3*b*e*g)*(d + e*x)^(3/2)*Log[-(Sqrt[-c]*Sqrt[d + e*x]) + Sqrt[c*d - b*e - c*e*x]]))/(12*e^2*(d + e*x)^4*(-(b*e) + c*(d - e*x))^(5/2))
Time = 0.76 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1220, 1125, 2192, 27, 1160, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {(3 b e g-10 c d g+4 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^4}dx}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1125 |
| \(\displaystyle -\frac {(3 b e g-10 c d g+4 c e f) \left (-\frac {\int \frac {c^3 x^2 e^8-c^2 (4 c d-3 b e) x e^7+c \left (7 c^2 d^2-9 b c e d+3 b^2 e^2\right ) e^6}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{e^6}-\frac {2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle -\frac {(3 b e g-10 c d g+4 c e f) \left (-\frac {-\frac {\int -\frac {c^2 e^8 (2 (5 c d-3 b e) (3 c d-2 b e)-c e (16 c d-9 b e) x)}{2 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c e^2}-\frac {1}{2} c^2 e^6 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^6}-\frac {2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(3 b e g-10 c d g+4 c e f) \left (-\frac {\frac {1}{4} c e^6 \int \frac {2 (5 c d-3 b e) (3 c d-2 b e)-c e (16 c d-9 b e) x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {1}{2} c^2 e^6 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^6}-\frac {2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle -\frac {(3 b e g-10 c d g+4 c e f) \left (-\frac {\frac {1}{4} c e^6 \left (\frac {15}{2} (2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {(16 c d-9 b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {1}{2} c^2 e^6 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^6}-\frac {2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {(3 b e g-10 c d g+4 c e f) \left (-\frac {\frac {1}{4} c e^6 \left (15 (2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )+\frac {(16 c d-9 b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {1}{2} c^2 e^6 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^6}-\frac {2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\left (-\frac {\frac {1}{4} c e^6 \left (\frac {15 (2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt {c} e}+\frac {(16 c d-9 b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {1}{2} c^2 e^6 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^6}-\frac {2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right ) (3 b e g-10 c d g+4 c e f)}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}\) |
(-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)^5) - ((4*c*e*f - 10*c*d*g + 3*b*e*g)*((-2*(2*c*d - b*e)^ 2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e*(d + e*x)) - (-1/2*(c^2*e^ 6*x*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (c*e^6*(((16*c*d - 9*b*e) *Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/e + (15*(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])])/( 2*Sqrt[c]*e)))/4)/e^6))/(3*e*(2*c*d - b*e))
3.23.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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_ Symbol] :> Simp[-2*e^(2*m + 3)*(Sqrt[a + b*x + c*x^2]/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m + 2) Int[(1/Sqrt[a + b*x + c*x^2])*Expan dToSum[((-2*c*d + b*e)^(-m - 1) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x ), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && EqQ[m + p, -3/2]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
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 ]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1254\) vs. \(2(324)=648\).
Time = 2.27 (sec) , antiderivative size = 1255, normalized size of antiderivative = 3.59
g/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*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^6*(-2/3/(-b*e^2+2*c*d* e)/(x+d/e)^5*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-4/3*c*e^2/( -b*e^2+2*c*d*e)*(-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 *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^...
Time = 2.40 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.72 \[ \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)^5} \, dx=\left [-\frac {15 \, {\left ({\left (4 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (20 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} g\right )} x^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (20 \, c^{2} d^{4} - 16 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (20 \, c^{2} d^{3} e - 16 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} g\right )} x\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 (6 \, c^{2} e^{3} g x^{3} + 3 \, {\left (4 \, c^{2} e^{3} f - {\left (16 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} g\right )} x^{2} + 4 \, {\left (23 \, c^{2} d^{2} e - 6 \, b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (236 \, c^{2} d^{3} - 147 \, b c d^{2} e + 16 \, b^{2} d e^{2}\right )} g + 2 \, {\left (4 \, {\left (17 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (161 \, c^{2} d^{2} e - 103 \, b c d e^{2} + 12 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, -\frac {15 \, {\left ({\left (4 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (20 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} g\right )} x^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (20 \, c^{2} d^{4} - 16 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (20 \, c^{2} d^{3} e - 16 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} g\right )} x\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 (6 \, c^{2} e^{3} g x^{3} + 3 \, {\left (4 \, c^{2} e^{3} f - {\left (16 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} g\right )} x^{2} + 4 \, {\left (23 \, c^{2} d^{2} e - 6 \, b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (236 \, c^{2} d^{3} - 147 \, b c d^{2} e + 16 \, b^{2} d e^{2}\right )} g + 2 \, {\left (4 \, {\left (17 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (161 \, c^{2} d^{2} e - 103 \, b c d e^{2} + 12 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{24 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]
[-1/48*(15*((4*(2*c^2*d*e^3 - b*c*e^4)*f - (20*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*g)*x^2 + 4*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (20*c^2*d^4 - 16*b *c*d^3*e + 3*b^2*d^2*e^2)*g + 2*(4*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (20*c^2 *d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*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*(6*c^2*e^3*g*x^3 + 3*(4*c ^2*e^3*f - (16*c^2*d*e^2 - 9*b*c*e^3)*g)*x^2 + 4*(23*c^2*d^2*e - 6*b*c*d*e ^2 - 2*b^2*e^3)*f - (236*c^2*d^3 - 147*b*c*d^2*e + 16*b^2*d*e^2)*g + 2*(4* (17*c^2*d*e^2 - 7*b*c*e^3)*f - (161*c^2*d^2*e - 103*b*c*d*e^2 + 12*b^2*e^3 )*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2), -1/24*(15*((4*(2*c^2*d*e^3 - b*c*e^4)*f - (20*c^2*d^2*e^2 - 16* b*c*d*e^3 + 3*b^2*e^4)*g)*x^2 + 4*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (20*c^2* d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2)*g + 2*(4*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (20*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*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*(6*c^2*e^3*g*x^3 + 3*(4*c^2*e^ 3*f - (16*c^2*d*e^2 - 9*b*c*e^3)*g)*x^2 + 4*(23*c^2*d^2*e - 6*b*c*d*e^2 - 2*b^2*e^3)*f - (236*c^2*d^3 - 147*b*c*d^2*e + 16*b^2*d*e^2)*g + 2*(4*(17*c ^2*d*e^2 - 7*b*c*e^3)*f - (161*c^2*d^2*e - 103*b*c*d*e^2 + 12*b^2*e^3)*g)* x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3*x + d...
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, dx \]
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]
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. 1076 vs. \(2 (324) = 648\).
Time = 0.69 (sec) , antiderivative size = 1076, normalized size of antiderivative = 3.07 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Too large to display} \]
-1/12*(15*(8*c^3*d*e*f*sgn(1/(e*x + d))*sgn(e) - 4*b*c^2*e^2*f*sgn(1/(e*x + d))*sgn(e) - 20*c^3*d^2*g*sgn(1/(e*x + d))*sgn(e) + 16*b*c^2*d*e*g*sgn(1 /(e*x + d))*sgn(e) - 3*b^2*c*e^2*g*sgn(1/(e*x + d))*sgn(e))*arctan(sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))/sqrt(c))/(sqrt(c)*e^3) - 3*(8*c^4*sqrt (-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d*e*f*sgn(1/(e*x + d))*sgn(e) + 8*c ^3*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d*e*f*sgn(1/(e*x + d))*sgn (e) - 4*b*c^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*e^2*f*sgn(1/(e*x + d))*sgn(e) - 4*b*c^2*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*e^2*f* sgn(1/(e*x + d))*sgn(e) - 36*c^4*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d) )*d^2*g*sgn(1/(e*x + d))*sgn(e) - 44*c^3*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d^2*g*sgn(1/(e*x + d))*sgn(e) + 32*b*c^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d*e*g*sgn(1/(e*x + d))*sgn(e) + 40*b*c^2*(-c + 2*c*d /(e*x + d) - b*e/(e*x + d))^(3/2)*d*e*g*sgn(1/(e*x + d))*sgn(e) - 7*b^2*c^ 2*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*e^2*g*sgn(1/(e*x + d))*sgn(e) - 9*b^2*c*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*e^2*g*sgn(1/(e*x + d))*sgn(e))/((2*c*d/(e*x + d) - b*e/(e*x + d))^2*e^3) - 8*(12*c^2*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d*e^7*f*sgn(1/(e*x + d))*sgn(e) - 2*c* (-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d*e^7*f*sgn(1/(e*x + d))*sgn( e) - 6*b*c*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*e^8*f*sgn(1/(e*x + d ))*sgn(e) + b*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*e^8*f*sgn(1/...
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^5} \,d x \]