Integrand size = 46, antiderivative size = 231 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=-\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (c e f+7 c d g-4 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \]
-1/2*(-d*g+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)^(7/2)+1/4*c*(-4*b*e*g+7*c*d*g+c*e*f)*arctanh((d*(-b*e+c*d)-b*e^2*x-c *e^2*x^2)^(1/2)/(-b*e+2*c*d)^(1/2)/(e*x+d)^(1/2))/e^2/(-b*e+2*c*d)^(3/2)-1 /4*(-4*b*e*g+7*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b *e+2*c*d)/(e*x+d)^(3/2)
Time = 0.46 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.78 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\frac {c \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {2 b e (d g+e (f+2 g x))+c \left (-5 d^2 g+e^2 f x-3 d e (f+3 g x)\right )}{c (2 c d-b e) (d+e x)^2}+\frac {(c e f+7 c d g-4 b e g) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{3/2} \sqrt {-b e+c (d-e x)}}\right )}{4 e^2 \sqrt {d+e x}} \]
(c*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((2*b*e*(d*g + e*(f + 2*g*x)) + c*(-5*d^2*g + e^2*f*x - 3*d*e*(f + 3*g*x)))/(c*(2*c*d - b*e)*(d + e*x)^2) + ((c*e*f + 7*c*d*g - 4*b*e*g)*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c *d + b*e]])/((-2*c*d + b*e)^(3/2)*Sqrt[-(b*e) + c*(d - e*x)])))/(4*e^2*Sqr t[d + e*x])
Time = 0.45 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1220, 1130, 1136, 221}
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.
| \(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-4 b e g+7 c d g+c e f) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{5/2}}dx}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1130 |
| \(\displaystyle \frac {(-4 b e g+7 c d g+c e f) \left (-\frac {1}{2} c \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right )}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {(-4 b e g+7 c d g+c e f) \left (-c e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right )}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {c \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e \sqrt {2 c d-b e}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right ) (-4 b e g+7 c d g+c e f)}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
-1/2*((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) + ((c*e*f + 7*c*d*g - 4*b*e*g)*(-(Sqrt[d*(c*d - b *e) - b*e^2*x - c*e^2*x^2]/(e*(d + e*x)^(3/2))) + (c*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*x])])/(e*Sqrt[2 *c*d - b*e])))/(4*e*(2*c*d - b*e))
3.23.37.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
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. \(621\) vs. \(2(209)=418\).
Time = 0.36 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.69
| method | result | size |
| default | \(-\frac {\left (4 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}-7 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}-\arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}+8 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x -14 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -2 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x +4 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -7 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -\arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +4 b \,e^{2} g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-9 c d e g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+c \,e^{2} f x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+2 b d e g \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+2 b \,e^{2} f \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-5 c \,d^{2} g \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-3 c d e f \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\right ) \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}}{4 \left (b e -2 c d \right )^{\frac {3}{2}} e^{2} \sqrt {-x c e -b e +c d}\, \left (e x +d \right )^{\frac {5}{2}}}\) | \(622\) |
-1/4*(4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*e^3*g*x^2-7*a rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e^2*g*x^2-arctan((-c *e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*e^3*f*x^2+8*arctan((-c*e*x-b*e+ c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d*e^2*g*x-14*arctan((-c*e*x-b*e+c*d)^(1/ 2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*g*x-2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2 *c*d)^(1/2))*c^2*d*e^2*f*x+4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/ 2))*b*c*d^2*e*g-7*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3 *g-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f+4*b*e^2*g* x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-9*c*d*e*g*x*(-c*e*x-b*e+c*d)^(1 /2)*(b*e-2*c*d)^(1/2)+c*e^2*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+2 *b*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+2*b*e^2*f*(-c*e*x-b*e+c* d)^(1/2)*(b*e-2*c*d)^(1/2)-5*c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1 /2)-3*c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))*(-(e*x+d)*(c*e*x+b *e-c*d))^(1/2)/(b*e-2*c*d)^(3/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)/(e*x+d)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (209) = 418\).
Time = 0.46 (sec) , antiderivative size = 1043, normalized size of antiderivative = 4.52 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\left [\frac {{\left (c^{2} d^{3} e f + {\left (c^{2} e^{4} f + {\left (7 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} f + {\left (7 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} + {\left (7 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (c^{2} d^{2} e^{2} f + {\left (7 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (6 \, c^{2} d^{2} e - 7 \, b c d e^{2} + 2 \, b^{2} e^{3}\right )} f + {\left (10 \, c^{2} d^{3} - 9 \, b c d^{2} e + 2 \, b^{2} d e^{2}\right )} g - {\left ({\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (18 \, c^{2} d^{2} e - 17 \, b c d e^{2} + 4 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (4 \, c^{2} d^{5} e^{2} - 4 \, b c d^{4} e^{3} + b^{2} d^{3} e^{4} + {\left (4 \, c^{2} d^{2} e^{5} - 4 \, b c d e^{6} + b^{2} e^{7}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{3} e^{4} - 4 \, b c d^{2} e^{5} + b^{2} d e^{6}\right )} x^{2} + 3 \, {\left (4 \, c^{2} d^{4} e^{3} - 4 \, b c d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} x\right )}}, \frac {{\left (c^{2} d^{3} e f + {\left (c^{2} e^{4} f + {\left (7 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} f + {\left (7 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} + {\left (7 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (c^{2} d^{2} e^{2} f + {\left (7 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (6 \, c^{2} d^{2} e - 7 \, b c d e^{2} + 2 \, b^{2} e^{3}\right )} f + {\left (10 \, c^{2} d^{3} - 9 \, b c d^{2} e + 2 \, b^{2} d e^{2}\right )} g - {\left ({\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (18 \, c^{2} d^{2} e - 17 \, b c d e^{2} + 4 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (4 \, c^{2} d^{5} e^{2} - 4 \, b c d^{4} e^{3} + b^{2} d^{3} e^{4} + {\left (4 \, c^{2} d^{2} e^{5} - 4 \, b c d e^{6} + b^{2} e^{7}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{3} e^{4} - 4 \, b c d^{2} e^{5} + b^{2} d e^{6}\right )} x^{2} + 3 \, {\left (4 \, c^{2} d^{4} e^{3} - 4 \, b c d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} x\right )}}\right ] \]
[1/8*((c^2*d^3*e*f + (c^2*e^4*f + (7*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(c^ 2*d*e^3*f + (7*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (7*c^2*d^4 - 4*b*c*d^3* e)*g + 3*(c^2*d^2*e^2*f + (7*c^2*d^3*e - 4*b*c*d^2*e^2)*g)*x)*sqrt(2*c*d - b*e)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x - 2*sqrt(- c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2 *x^2 + 2*d*e*x + d^2)) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((6* c^2*d^2*e - 7*b*c*d*e^2 + 2*b^2*e^3)*f + (10*c^2*d^3 - 9*b*c*d^2*e + 2*b^2 *d*e^2)*g - ((2*c^2*d*e^2 - b*c*e^3)*f - (18*c^2*d^2*e - 17*b*c*d*e^2 + 4* b^2*e^3)*g)*x)*sqrt(e*x + d))/(4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*c*d*e^6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^4 - 4*b*c* d^2*e^5 + b^2*d*e^6)*x^2 + 3*(4*c^2*d^4*e^3 - 4*b*c*d^3*e^4 + b^2*d^2*e^5) *x), 1/4*((c^2*d^3*e*f + (c^2*e^4*f + (7*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3 *(c^2*d*e^3*f + (7*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (7*c^2*d^4 - 4*b*c* d^3*e)*g + 3*(c^2*d^2*e^2*f + (7*c^2*d^3*e - 4*b*c*d^2*e^2)*g)*x)*sqrt(-2* c*d + b*e)*arctan(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)/(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)) - sqrt(-c*e^2*x ^2 - b*e^2*x + c*d^2 - b*d*e)*((6*c^2*d^2*e - 7*b*c*d*e^2 + 2*b^2*e^3)*f + (10*c^2*d^3 - 9*b*c*d^2*e + 2*b^2*d*e^2)*g - ((2*c^2*d*e^2 - b*c*e^3)*f - (18*c^2*d^2*e - 17*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(4*c^2*d^5 *e^2 - 4*b*c*d^4*e^3 + b^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*c*d*e^6 + b^2...
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Time = 0.43 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.42 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=-\frac {\frac {{\left (c^{3} e f + 7 \, c^{3} d g - 4 \, b c^{2} e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (2 \, c d - b e\right )} \sqrt {-2 \, c d + b e}} + \frac {2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d e f - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} e^{2} f + 14 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{2} g - 15 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d e g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} e^{2} g + {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} e f - 9 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} d g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{2} e g}{{\left (2 \, c d - b e\right )} {\left (e x + d\right )}^{2} c^{2}}}{4 \, c e^{2}} \]
-1/4*((c^3*e*f + 7*c^3*d*g - 4*b*c^2*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((2*c*d - b*e)*sqrt(-2*c*d + b*e)) + (2*sqrt(- (e*x + d)*c + 2*c*d - b*e)*c^4*d*e*f - sqrt(-(e*x + d)*c + 2*c*d - b*e)*b* c^3*e^2*f + 14*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^2*g - 15*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^3*d*e*g + 4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2 *c^2*e^2*g + (-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^3*e*f - 9*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^3*d*g + 4*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^2* e*g)/((2*c*d - b*e)*(e*x + d)^2*c^2))/(c*e^2)
Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]