Integrand size = 44, antiderivative size = 354 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {5 (2 c d-b e) (8 c e f-6 c d g-b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c e}+\frac {5 (8 c e f-6 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2}+\frac {(8 c e f-6 c d g-b e g) (c d-b e-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^3}+\frac {5 (2 c d-b e)^3 (8 c e f-6 c d g-b e g) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{3/2} e^2} \]
5/24*(-b*e*g-6*c*d*g+8*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2+1 /4*(-b*e*g-6*c*d*g+8*c*e*f)*(-c*e*x-b*e+c*d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x ^2)^(3/2)/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)^3+5/128*(-b*e+2*c*d)^3*(-b*e*g-6*c*d*g+8*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^(3/2)/e^2+5/64*(-b*e+2*c*d)*(-b*e*g-6*c*d*g+8*c*e*f)*(2*c*x+b)*(d*(-b*e +c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c/e
Time = 0.89 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.82 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {(2 c d-b e)^3 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {\sqrt {c} \left (15 b^3 e^3 g+2 b^2 c e^2 (132 e f-118 d g+59 e g x)-8 c^3 \left (72 d^3 g+12 d e^2 x (3 f+2 g x)-2 e^3 x^2 (4 f+3 g x)-d^2 e (88 f+45 g x)\right )+4 b c^2 e \left (173 d^2 g+2 e^2 x (26 f+17 g x)-2 d e (106 f+51 g x)\right )\right )}{(2 c d-b e)^3 (d+e x)^2 (-c d+b e+c e x)^2}-\frac {15 (8 c e f-6 c d g-b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{5/2} (-b e+c (d-e x))^{5/2}}\right )}{192 c^{3/2} e^2} \]
((2*c*d - b*e)^3*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((Sqrt[c]*(15*b^ 3*e^3*g + 2*b^2*c*e^2*(132*e*f - 118*d*g + 59*e*g*x) - 8*c^3*(72*d^3*g + 1 2*d*e^2*x*(3*f + 2*g*x) - 2*e^3*x^2*(4*f + 3*g*x) - d^2*e*(88*f + 45*g*x)) + 4*b*c^2*e*(173*d^2*g + 2*e^2*x*(26*f + 17*g*x) - 2*d*e*(106*f + 51*g*x) )))/((2*c*d - b*e)^3*(d + e*x)^2*(-(c*d) + b*e + c*e*x)^2) - (15*(8*c*e*f - 6*c*d*g - b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])] )/((d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(5/2))))/(192*c^(3/2)*e^2)
Time = 0.60 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1220, 1127, 1134, 1160, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-b e g-6 c d g+8 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^2}dx}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1127 |
| \(\displaystyle \frac {(-b e g-6 c d g+8 c e f) \int (c d-b e-c e x)^2 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {(-b e g-6 c d g+8 c e f) \left (\frac {5}{8} (2 c d-b e) \int (c d-b e-c e x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx+\frac {(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e}\right )}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {(-b e g-6 c d g+8 c e f) \left (\frac {5}{8} (2 c d-b e) \left (\frac {1}{2} (2 c d-b e) \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e}\right )}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(-b e g-6 c d g+8 c e f) \left (\frac {5}{8} (2 c d-b e) \left (\frac {1}{2} (2 c d-b e) \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e}\right )}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {(-b e g-6 c d g+8 c e f) \left (\frac {5}{8} (2 c d-b e) \left (\frac {1}{2} (2 c d-b e) \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e}\right )}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (\frac {5}{8} (2 c d-b e) \left (\frac {1}{2} (2 c d-b e) \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e}\right ) (-b e g-6 c d g+8 c e f)}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
(2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^3) + ((8*c*e*f - 6*c*d*g - b*e*g)*(((c*d - b*e - c*e*x)*(d* (c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(4*e) + (5*(2*c*d - b*e)*((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*ArcTa n[(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))/8))/(e*(2*c*d - b*e))
3.22.100.3.1 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_Sy mbol] :> Int[(a + b*x + c*x^2)^(m + p)/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b , c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m] && RationalQ [p] && (LtQ[0, -m, p] || LtQ[p, -m, 0]) && NeQ[m, 2] && NeQ[m, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^ (m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 *p]
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 ]
Leaf count of result is larger than twice the leaf count of optimal. \(918\) vs. \(2(330)=660\).
Time = 1.05 (sec) , antiderivative size = 919, normalized size of antiderivative = 2.60
| method | result | size |
| default | \(\frac {g \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}}+\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}}}{\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {8 c \,e^{2} \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 )}{-b \,e^{2}+2 c d e}\right )}{e^{4}}\) | \(919\) |
g/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))))))+(-d*g+e*f)/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*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 = 0.85 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.30 \[ \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)^3} \, dx=\left [-\frac {15 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f - {\left (48 \, c^{4} d^{4} - 64 \, b c^{3} d^{3} e + 24 \, b^{2} c^{2} d^{2} e^{2} - b^{4} e^{4}\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 (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f - {\left (24 \, c^{4} d e^{2} - 17 \, b c^{3} e^{3}\right )} g\right )} x^{2} + 8 \, {\left (88 \, c^{4} d^{2} e - 106 \, b c^{3} d e^{2} + 33 \, b^{2} c^{2} e^{3}\right )} f - {\left (576 \, c^{4} d^{3} - 692 \, b c^{3} d^{2} e + 236 \, b^{2} c^{2} d e^{2} - 15 \, b^{3} c e^{3}\right )} g - 2 \, {\left (8 \, {\left (18 \, c^{4} d e^{2} - 13 \, b c^{3} e^{3}\right )} f - {\left (180 \, c^{4} d^{2} e - 204 \, b c^{3} d e^{2} + 59 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{768 \, c^{2} e^{2}}, -\frac {15 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f - {\left (48 \, c^{4} d^{4} - 64 \, b c^{3} d^{3} e + 24 \, b^{2} c^{2} d^{2} e^{2} - b^{4} e^{4}\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 (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f - {\left (24 \, c^{4} d e^{2} - 17 \, b c^{3} e^{3}\right )} g\right )} x^{2} + 8 \, {\left (88 \, c^{4} d^{2} e - 106 \, b c^{3} d e^{2} + 33 \, b^{2} c^{2} e^{3}\right )} f - {\left (576 \, c^{4} d^{3} - 692 \, b c^{3} d^{2} e + 236 \, b^{2} c^{2} d e^{2} - 15 \, b^{3} c e^{3}\right )} g - 2 \, {\left (8 \, {\left (18 \, c^{4} d e^{2} - 13 \, b c^{3} e^{3}\right )} f - {\left (180 \, c^{4} d^{2} e - 204 \, b c^{3} d e^{2} + 59 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{384 \, c^{2} e^{2}}\right ] \]
[-1/768*(15*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e ^4)*f - (48*c^4*d^4 - 64*b*c^3*d^3*e + 24*b^2*c^2*d^2*e^2 - b^4*e^4)*g)*sq rt(-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*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^3*f - (24*c^4*d*e^2 - 17*b*c^3*e^3)*g)*x^ 2 + 8*(88*c^4*d^2*e - 106*b*c^3*d*e^2 + 33*b^2*c^2*e^3)*f - (576*c^4*d^3 - 692*b*c^3*d^2*e + 236*b^2*c^2*d*e^2 - 15*b^3*c*e^3)*g - 2*(8*(18*c^4*d*e^ 2 - 13*b*c^3*e^3)*f - (180*c^4*d^2*e - 204*b*c^3*d*e^2 + 59*b^2*c^2*e^3)*g )*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^2*e^2), -1/384*(15*(8* (8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f - (48*c^4 *d^4 - 64*b*c^3*d^3*e + 24*b^2*c^2*d^2*e^2 - b^4*e^4)*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*(48*c^4*e^3*g*x^3 + 8*(8*c^4 *e^3*f - (24*c^4*d*e^2 - 17*b*c^3*e^3)*g)*x^2 + 8*(88*c^4*d^2*e - 106*b*c^ 3*d*e^2 + 33*b^2*c^2*e^3)*f - (576*c^4*d^3 - 692*b*c^3*d^2*e + 236*b^2*c^2 *d*e^2 - 15*b^3*c*e^3)*g - 2*(8*(18*c^4*d*e^2 - 13*b*c^3*e^3)*f - (180*c^4 *d^2*e - 204*b*c^3*d*e^2 + 59*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^2*e^2)]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.38 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.16 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {1}{192} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, {\left (6 \, c^{2} e g x + \frac {8 \, c^{5} e^{5} f - 24 \, c^{5} d e^{4} g + 17 \, b c^{4} e^{5} g}{c^{3} e^{4}}\right )} x - \frac {144 \, c^{5} d e^{4} f - 104 \, b c^{4} e^{5} f - 180 \, c^{5} d^{2} e^{3} g + 204 \, b c^{4} d e^{4} g - 59 \, b^{2} c^{3} e^{5} g}{c^{3} e^{4}}\right )} x + \frac {704 \, c^{5} d^{2} e^{3} f - 848 \, b c^{4} d e^{4} f + 264 \, b^{2} c^{3} e^{5} f - 576 \, c^{5} d^{3} e^{2} g + 692 \, b c^{4} d^{2} e^{3} g - 236 \, b^{2} c^{3} d e^{4} g + 15 \, b^{3} c^{2} e^{5} g}{c^{3} e^{4}}\right )} - \frac {5 \, {\left (64 \, c^{4} d^{3} e f - 96 \, b c^{3} d^{2} e^{2} f + 48 \, b^{2} c^{2} d e^{3} f - 8 \, b^{3} c e^{4} f - 48 \, c^{4} d^{4} g + 64 \, b c^{3} d^{3} e g - 24 \, b^{2} c^{2} d^{2} e^{2} g + b^{4} e^{4} g\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{128 \, \sqrt {-c} c e {\left | e \right |}} \]
1/192*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(6*c^2*e*g*x + (8*c ^5*e^5*f - 24*c^5*d*e^4*g + 17*b*c^4*e^5*g)/(c^3*e^4))*x - (144*c^5*d*e^4* f - 104*b*c^4*e^5*f - 180*c^5*d^2*e^3*g + 204*b*c^4*d*e^4*g - 59*b^2*c^3*e ^5*g)/(c^3*e^4))*x + (704*c^5*d^2*e^3*f - 848*b*c^4*d*e^4*f + 264*b^2*c^3* e^5*f - 576*c^5*d^3*e^2*g + 692*b*c^4*d^2*e^3*g - 236*b^2*c^3*d*e^4*g + 15 *b^3*c^2*e^5*g)/(c^3*e^4)) - 5/128*(64*c^4*d^3*e*f - 96*b*c^3*d^2*e^2*f + 48*b^2*c^2*d*e^3*f - 8*b^3*c*e^4*f - 48*c^4*d^4*g + 64*b*c^3*d^3*e*g - 24* b^2*c^2*d^2*e^2*g + b^4*e^4*g)*log(abs(-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))*sqrt(-c)*abs(e)))/(sqrt(-c)*c*e*abs( e))
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]