Integrand size = 46, antiderivative size = 372 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {5 c (3 c e f-11 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt {d+e x}}+\frac {5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac {5 c \sqrt {2 c d-b e} (3 c e f-11 c d g+4 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2} \]
5/12*c*(4*b*e*g-11*c*d*g+3*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)^(3/2)+1/4*(4*b*e*g-11*c*d*g+3*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)^(7/2)-1/2*(-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)^(11/2)-5/4*c *(4*b*e*g-11*c*d*g+3*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))*(-b*e+2*c*d)^(1/2)/e^2+5/4*c*(4*b*e*g-1 1*c*d*g+3*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)^(1/2)
Time = 0.74 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.68 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {c ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {c^2 \left (-206 d^3 g+d^2 e (54 f-350 g x)+2 d e^2 x (51 f-56 g x)+8 e^3 x^2 (3 f+g x)\right )-6 b^2 e^2 (d g+e (f+2 g x))+b c e \left (107 d^2 g+e^2 x (-27 f+56 g x)+d e (-3 f+187 g x)\right )}{c (d+e x)^2 (-c d+b e+c e x)^2}+\frac {15 \sqrt {-2 c d+b e} (-3 c e f+11 c d g-4 b e g) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-b e+c (d-e x))^{5/2}}\right )}{12 e^2 (d+e x)^{5/2}} \]
(c*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((c^2*(-206*d^3*g + d^2*e*(54* f - 350*g*x) + 2*d*e^2*x*(51*f - 56*g*x) + 8*e^3*x^2*(3*f + g*x)) - 6*b^2* e^2*(d*g + e*(f + 2*g*x)) + b*c*e*(107*d^2*g + e^2*x*(-27*f + 56*g*x) + d* e*(-3*f + 187*g*x)))/(c*(d + e*x)^2*(-(c*d) + b*e + c*e*x)^2) + (15*Sqrt[- 2*c*d + b*e]*(-3*c*e*f + 11*c*d*g - 4*b*e*g)*ArcTan[Sqrt[-(b*e) + c*(d - e *x)]/Sqrt[-2*c*d + b*e]])/(-(b*e) + c*(d - e*x))^(5/2)))/(12*e^2*(d + e*x) ^(5/2))
Time = 0.61 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1220, 1130, 1131, 1131, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {(4 b e g-11 c d g+3 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{9/2}}dx}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1130 |
| \(\displaystyle -\frac {(4 b e g-11 c d g+3 c e f) \left (-\frac {5}{2} c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{5/2}}dx-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^{7/2}}\right )}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(4 b e g-11 c d g+3 c e f) \left (-\frac {5}{2} c \left ((2 c d-b e) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{3/2}}dx+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^{7/2}}\right )}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(4 b e g-11 c d g+3 c e f) \left (-\frac {5}{2} c \left ((2 c d-b e) \left ((2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^{7/2}}\right )}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle -\frac {(4 b e g-11 c d g+3 c e f) \left (-\frac {5}{2} c \left ((2 c d-b e) \left (2 e (2 c d-b e) \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^{7/2}}\right )}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\left (-\frac {5}{2} c \left ((2 c d-b e) \left (\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2 c d-b e} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^{7/2}}\right ) (4 b e g-11 c d g+3 c e f)}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}\) |
-1/2*((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) - ((3*c*e*f - 11*c*d*g + 4*b*e*g)*(-((d*(c*d - b *e) - b*e^2*x - c*e^2*x^2)^(5/2)/(e*(d + e*x)^(7/2))) - (5*c*((2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) + (2*c*d - b*e)* ((2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e*Sqrt[d + e*x]) - (2*Sqrt [2*c*d - b*e]*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)))/2))/(4*e*(2*c*d - b*e))
3.23.58.3.1 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[((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[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. \(1180\) vs. \(2(338)=676\).
Time = 0.36 (sec) , antiderivative size = 1181, normalized size of antiderivative = 3.17
-1/12*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-8*c^2*e^3*g*x^3*(-c*e*x-b*e+c*d)^ (1/2)*(b*e-2*c*d)^(1/2)+6*b^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/ 2)+206*c^2*d^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-90*arctan((-c*e* x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f-24*c^2*e^3*f*x^2*(-c*e*x-b *e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+12*b^2*e^3*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e -2*c*d)^(1/2)+6*b^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-54*c^ 2*d^2*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+330*arctan((-c*e*x-b*e+ c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^2*e^2*g*x^2-90*arctan((-c*e*x-b*e+c*d) ^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d*e^3*f*x^2+660*arctan((-c*e*x-b*e+c*d)^(1/2 )/(b*e-2*c*d)^(1/2))*c^3*d^3*e*g*x-180*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e- 2*c*d)^(1/2))*c^3*d^2*e^2*f*x-285*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d )^(1/2))*b*c^2*d^3*e*g+45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)) *b*c^2*d^2*e^2*f+330*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3* d^4*g+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*e^4*g*x^2+ 45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*e^4*f*x^2+350*c^ 2*d^2*e*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-102*c^2*d*e^2*f*x*(-c *e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-107*b*c*d^2*e*g*(-c*e*x-b*e+c*d)^(1/ 2)*(b*e-2*c*d)^(1/2)+3*b*c*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2 )-285*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d*e^3*g*x^2-5 70*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*g*x-5...
Time = 0.34 (sec) , antiderivative size = 959, normalized size of antiderivative = 2.58 \[ \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)^{11/2}} \, dx=\left [\frac {15 \, {\left (3 \, c^{2} d^{3} e f + {\left (3 \, c^{2} e^{4} f - {\left (11 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (3 \, c^{2} d e^{3} f - {\left (11 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} - {\left (11 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (3 \, c^{2} d^{2} e^{2} f - {\left (11 \, 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 \, {\left (8 \, c^{2} e^{3} g x^{3} + 8 \, {\left (3 \, c^{2} e^{3} f - 7 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} e - b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (206 \, c^{2} d^{3} - 107 \, b c d^{2} e + 6 \, b^{2} d e^{2}\right )} g + {\left (3 \, {\left (34 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} f - {\left (350 \, c^{2} d^{2} e - 187 \, 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} \sqrt {e x + d}}{24 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}}, -\frac {15 \, {\left (3 \, c^{2} d^{3} e f + {\left (3 \, c^{2} e^{4} f - {\left (11 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (3 \, c^{2} d e^{3} f - {\left (11 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} - {\left (11 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (3 \, c^{2} d^{2} e^{2} f - {\left (11 \, 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 ) - {\left (8 \, c^{2} e^{3} g x^{3} + 8 \, {\left (3 \, c^{2} e^{3} f - 7 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} e - b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (206 \, c^{2} d^{3} - 107 \, b c d^{2} e + 6 \, b^{2} d e^{2}\right )} g + {\left (3 \, {\left (34 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} f - {\left (350 \, c^{2} d^{2} e - 187 \, 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} \sqrt {e x + d}}{12 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}}\right ] \]
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")
[1/24*(15*(3*c^2*d^3*e*f + (3*c^2*e^4*f - (11*c^2*d*e^3 - 4*b*c*e^4)*g)*x^ 3 + 3*(3*c^2*d*e^3*f - (11*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 - (11*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*e^2*f - (11*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)*sqr t(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(8*c^2*e^3*g*x^3 + 8*(3*c^2*e^3 *f - 7*(2*c^2*d*e^2 - b*c*e^3)*g)*x^2 + 3*(18*c^2*d^2*e - b*c*d*e^2 - 2*b^ 2*e^3)*f - (206*c^2*d^3 - 107*b*c*d^2*e + 6*b^2*d*e^2)*g + (3*(34*c^2*d*e^ 2 - 9*b*c*e^3)*f - (350*c^2*d^2*e - 187*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)*sqrt(e*x + d))/(e^5*x^3 + 3*d*e^4*x ^2 + 3*d^2*e^3*x + d^3*e^2), -1/12*(15*(3*c^2*d^3*e*f + (3*c^2*e^4*f - (11 *c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(3*c^2*d*e^3*f - (11*c^2*d^2*e^2 - 4*b* c*d*e^3)*g)*x^2 - (11*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*e^2*f - (11* 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)) - (8*c^2*e^3*g*x^3 + 8*(3*c^2*e^3*f - 7*(2*c^2*d *e^2 - b*c*e^3)*g)*x^2 + 3*(18*c^2*d^2*e - b*c*d*e^2 - 2*b^2*e^3)*f - (206 *c^2*d^3 - 107*b*c*d^2*e + 6*b^2*d*e^2)*g + (3*(34*c^2*d*e^2 - 9*b*c*e^3)* f - (350*c^2*d^2*e - 187*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)*sqrt(e*x + d))/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e...
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(11/2),x, algorithm="maxima")
Time = 0.59 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.54 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {24 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} e f - 120 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d g + 48 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e g - 8 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} g + \frac {15 \, {\left (6 \, c^{4} d e f - 3 \, b c^{3} e^{2} f - 22 \, c^{4} d^{2} g + 19 \, b c^{3} d e g - 4 \, b^{2} c^{2} e^{2} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, {\left (28 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{5} d^{2} e f - 28 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{4} d e^{2} f + 7 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{3} e^{3} f - 60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{5} d^{3} g + 76 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{4} d^{2} e g - 31 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{3} d e^{2} g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{3} c^{2} e^{3} g - 18 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{4} d e f + 9 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{3} e^{2} f + 34 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{4} d^{2} g - 25 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{3} d e g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{2} e^{2} g\right )}}{{\left (e x + d\right )}^{2} c^{2}}}{12 \, c e^{2}} \]
1/12*(24*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*e*f - 120*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d*g + 48*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*e*g - 8 *(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*g + 15*(6*c^4*d*e*f - 3*b*c^3*e^2* f - 22*c^4*d^2*g + 19*b*c^3*d*e*g - 4*b^2*c^2*e^2*g)*arctan(sqrt(-(e*x + d )*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/sqrt(-2*c*d + b*e) + 3*(28*sqrt(-(e *x + d)*c + 2*c*d - b*e)*c^5*d^2*e*f - 28*sqrt(-(e*x + d)*c + 2*c*d - b*e) *b*c^4*d*e^2*f + 7*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^3*e^3*f - 60*sqr t(-(e*x + d)*c + 2*c*d - b*e)*c^5*d^3*g + 76*sqrt(-(e*x + d)*c + 2*c*d - b *e)*b*c^4*d^2*e*g - 31*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^3*d*e^2*g + 4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^2*e^3*g - 18*(-(e*x + d)*c + 2*c* d - b*e)^(3/2)*c^4*d*e*f + 9*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^3*e^2* f + 34*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^4*d^2*g - 25*(-(e*x + d)*c + 2 *c*d - b*e)^(3/2)*b*c^3*d*e*g + 4*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c ^2*e^2*g)/((e*x + d)^2*c^2))/(c*e^2)
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]