Integrand size = 46, antiderivative size = 457 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {-e f+d g}{3 e^2 (2 c d-b e) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {3 c e f+c d g-2 b e g}{4 e^2 (2 c d-b e)^2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {7 c (3 c e f+c d g-2 b e g) \sqrt {d+e x}}{12 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {35 c (3 c e f+c d g-2 b e g)}{24 e^2 (2 c d-b e)^4 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {35 c^2 (3 c e f+c d g-2 b e g) \sqrt {d+e x}}{8 e^2 (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {35 c^2 (3 c e f+c d g-2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{8 e^2 (2 c d-b e)^{11/2}} \]
1/3*(d*g-e*f)/e^2/(-b*e+2*c*d)/(e*x+d)^(3/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x ^2)^(3/2)-35/8*c^2*(-2*b*e*g+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))/e^2/(-b*e+2*c*d)^(11/2) +1/4*(2*b*e*g-c*d*g-3*c*e*f)/e^2/(-b*e+2*c*d)^2/(d*(-b*e+c*d)-b*e^2*x-c*e^ 2*x^2)^(3/2)/(e*x+d)^(1/2)+7/12*c*(-2*b*e*g+c*d*g+3*c*e*f)*(e*x+d)^(1/2)/e ^2/(-b*e+2*c*d)^3/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)-35/24*c*(-2*b*e*g +c*d*g+3*c*e*f)/e^2/(-b*e+2*c*d)^4/(e*x+d)^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e ^2*x^2)^(1/2)+35/8*c^2*(-2*b*e*g+c*d*g+3*c*e*f)*(e*x+d)^(1/2)/e^2/(-b*e+2* c*d)^5/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)
Time = 1.79 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.96 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {c^2 (d+e x)^{5/2} \left (\frac {(-b e+c (d-e x)) \left (-4 b^4 e^4 (2 e f+d g+3 e g x)+2 b^3 c e^3 \left (28 d^2 g+3 e^2 x (3 f+7 g x)+d e (41 f+81 g x)\right )+c^4 \left (171 d^5 g-315 e^5 f x^4+14 d^2 e^3 x^2 (27 f-10 g x)-105 d e^4 x^3 (4 f+g x)+18 d^3 e^2 x (34 f+7 g x)+d^4 e (f+204 g x)\right )+b^2 c^2 e^2 \left (103 d^3 g+7 e^3 x^2 (-9 f+40 g x)+3 d e^2 x (-78 f+217 g x)+d^2 e (-363 f+282 g x)\right )-2 b c^3 e \left (163 d^4 g+14 d e^3 x^2 (36 f-5 g x)-105 e^4 x^3 (-2 f+g x)+6 d^2 e^2 x (45 f+49 g x)+d^3 e (-152 f+294 g x)\right )\right )}{c^2 (2 c d-b e)^5 (d+e x)^3}-\frac {105 (3 c e f+c d g-2 b e g) (-b e+c (d-e x))^{5/2} \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{11/2}}\right )}{24 e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}} \]
(c^2*(d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(-4*b^4*e^4*(2*e*f + d*g + 3 *e*g*x) + 2*b^3*c*e^3*(28*d^2*g + 3*e^2*x*(3*f + 7*g*x) + d*e*(41*f + 81*g *x)) + c^4*(171*d^5*g - 315*e^5*f*x^4 + 14*d^2*e^3*x^2*(27*f - 10*g*x) - 1 05*d*e^4*x^3*(4*f + g*x) + 18*d^3*e^2*x*(34*f + 7*g*x) + d^4*e*(f + 204*g* x)) + b^2*c^2*e^2*(103*d^3*g + 7*e^3*x^2*(-9*f + 40*g*x) + 3*d*e^2*x*(-78* f + 217*g*x) + d^2*e*(-363*f + 282*g*x)) - 2*b*c^3*e*(163*d^4*g + 14*d*e^3 *x^2*(36*f - 5*g*x) - 105*e^4*x^3*(-2*f + g*x) + 6*d^2*e^2*x*(45*f + 49*g* x) + d^3*e*(-152*f + 294*g*x))))/(c^2*(2*c*d - b*e)^5*(d + e*x)^3) - (105* (3*c*e*f + c*d*g - 2*b*e*g)*(-(b*e) + c*(d - e*x))^(5/2)*ArcTan[Sqrt[-(b*e ) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]])/(-2*c*d + b*e)^(11/2)))/(24*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))
Time = 0.69 (sec) , antiderivative size = 452, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1220, 1135, 1132, 1135, 1132, 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.
| \(\displaystyle \int \frac {f+g x}{(d+e x)^{3/2} \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \int \frac {1}{\sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}dx}{2 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {7 c \int \frac {\sqrt {d+e x}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}dx}{4 (2 c d-b e)}-\frac {1}{2 e \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {7 c \left (\frac {5 \int \frac {1}{\sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {7 c \left (\frac {5 \left (\frac {3 c \int \frac {\sqrt {d+e x}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {7 c \left (\frac {5 \left (\frac {3 c \left (\frac {\int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {(-2 b e g+c d g+3 c e f) \left (\frac {7 c \left (\frac {5 \left (\frac {3 c \left (\frac {2 e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {7 c \left (\frac {5 \left (\frac {3 c \left (\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 (2 c d-b e)}-\frac {1}{2 e \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right ) (-2 b e g+c d g+3 c e f)}{2 e (2 c d-b e)}-\frac {e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
-1/3*(e*f - d*g)/(e^2*(2*c*d - b*e)*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2 *x - c*e^2*x^2)^(3/2)) + ((3*c*e*f + c*d*g - 2*b*e*g)*(-1/2*1/(e*(2*c*d - b*e)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (7*c*((2 *Sqrt[d + e*x])/(3*e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^( 3/2)) + (5*(-(1/(e*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2* x - c*e^2*x^2])) + (3*c*((2*Sqrt[d + e*x])/(e*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*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*c*d - b*e)^(3/2))))/(2 *(2*c*d - b*e))))/(3*(2*c*d - b*e))))/(4*(2*c*d - b*e))))/(2*e*(2*c*d - b* e))
3.23.84.3.1 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[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ [0, m, 1] && IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))) 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] && LtQ[m, 0] && NeQ[m + p + 1, 0] && I ntegerQ[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. \(2002\) vs. \(2(418)=836\).
Time = 0.37 (sec) , antiderivative size = 2003, normalized size of antiderivative = 4.38
-1/24*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(651*(b*e-2*c*d)^(1/2)*b^2*c^2*d*e^ 4*g*x^2-588*(b*e-2*c*d)^(1/2)*b*c^3*d^2*e^3*g*x^2-1008*(b*e-2*c*d)^(1/2)*b *c^3*d*e^4*f*x^2+162*(b*e-2*c*d)^(1/2)*b^3*c*d*e^4*g*x+105*arctan((-c*e*x- b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^5*g*(-c*e*x-b*e+c*d)^(1/2)-8*(b*e- 2*c*d)^(1/2)*b^4*e^5*f+171*(b*e-2*c*d)^(1/2)*c^4*d^5*g-315*(b*e-2*c*d)^(1/ 2)*c^4*e^5*f*x^4+140*(b*e-2*c*d)^(1/2)*b*c^3*d*e^4*g*x^3-63*(b*e-2*c*d)^(1 /2)*b^2*c^2*e^5*f*x^2+126*(b*e-2*c*d)^(1/2)*c^4*d^3*e^2*g*x^2+378*(b*e-2*c *d)^(1/2)*c^4*d^2*e^3*f*x^2+18*(b*e-2*c*d)^(1/2)*b^3*c*e^5*f*x+204*(b*e-2* c*d)^(1/2)*c^4*d^4*e*g*x+612*(b*e-2*c*d)^(1/2)*c^4*d^3*e^2*f*x+56*(b*e-2*c *d)^(1/2)*b^3*c*d^2*e^3*g+82*(b*e-2*c*d)^(1/2)*b^3*c*d*e^4*f+103*(b*e-2*c* d)^(1/2)*b^2*c^2*d^3*e^2*g-363*(b*e-2*c*d)^(1/2)*b^2*c^2*d^2*e^3*f-326*(b* e-2*c*d)^(1/2)*b*c^3*d^4*e*g+304*(b*e-2*c*d)^(1/2)*b*c^3*d^3*e^2*f-12*(b*e -2*c*d)^(1/2)*b^4*e^5*g*x-4*(b*e-2*c*d)^(1/2)*b^4*d*e^4*g+(b*e-2*c*d)^(1/2 )*c^4*d^4*e*f+282*(b*e-2*c*d)^(1/2)*b^2*c^2*d^2*e^3*g*x-234*(b*e-2*c*d)^(1 /2)*b^2*c^2*d*e^4*f*x-588*(b*e-2*c*d)^(1/2)*b*c^3*d^3*e^2*g*x-540*(b*e-2*c *d)^(1/2)*b*c^3*d^2*e^3*f*x-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^ (1/2))*c^4*e^5*f*x^4*(-c*e*x-b*e+c*d)^(1/2)+315*arctan((-c*e*x-b*e+c*d)^(1 /2)/(b*e-2*c*d)^(1/2))*c^4*d^4*e*f*(-c*e*x-b*e+c*d)^(1/2)+210*(b*e-2*c*d)^ (1/2)*b*c^3*e^5*g*x^4-105*(b*e-2*c*d)^(1/2)*c^4*d*e^4*g*x^4+280*(b*e-2*c*d )^(1/2)*b^2*c^2*e^5*g*x^3-420*(b*e-2*c*d)^(1/2)*b*c^3*e^5*f*x^3-140*(b*...
Leaf count of result is larger than twice the leaf count of optimal. 2026 vs. \(2 (417) = 834\).
Time = 5.81 (sec) , antiderivative size = 4084, normalized size of antiderivative = 8.94 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/48*(105*((3*c^5*e^7*f + (c^5*d*e^6 - 2*b*c^4*e^7)*g)*x^6 + 2*(3*(c^5*d* e^6 + b*c^4*e^7)*f + (c^5*d^2*e^5 - b*c^4*d*e^6 - 2*b^2*c^3*e^7)*g)*x^5 - (3*(c^5*d^2*e^5 - 6*b*c^4*d*e^6 - b^2*c^3*e^7)*f + (c^5*d^3*e^4 - 8*b*c^4* d^2*e^5 + 11*b^2*c^3*d*e^6 + 2*b^3*c^2*e^7)*g)*x^4 - 4*(3*(c^5*d^3*e^4 - b *c^4*d^2*e^5 - b^2*c^3*d*e^6)*f + (c^5*d^4*e^3 - 3*b*c^4*d^3*e^4 + b^2*c^3 *d^2*e^5 + 2*b^3*c^2*d*e^6)*g)*x^3 - (3*(c^5*d^4*e^3 + 4*b*c^4*d^3*e^4 - 6 *b^2*c^3*d^2*e^5)*f + (c^5*d^5*e^2 + 2*b*c^4*d^4*e^3 - 14*b^2*c^3*d^3*e^4 + 12*b^3*c^2*d^2*e^5)*g)*x^2 + 3*(c^5*d^6*e - 2*b*c^4*d^5*e^2 + b^2*c^3*d^ 4*e^3)*f + (c^5*d^7 - 4*b*c^4*d^6*e + 5*b^2*c^3*d^5*e^2 - 2*b^3*c^2*d^4*e^ 3)*g + 2*(3*(c^5*d^5*e^2 - 3*b*c^4*d^4*e^3 + 2*b^2*c^3*d^3*e^4)*f + (c^5*d ^6*e - 5*b*c^4*d^5*e^2 + 8*b^2*c^3*d^4*e^3 - 4*b^3*c^2*d^3*e^4)*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)*(105*(3*(2*c^5*d*e^5 - b*c^4*e^6)*f + (2*c^5*d^2*e^4 - 5*b*c^4*d*e^5 + 2*b^2*c^3*e^6)*g)*x^4 + 140*(3*(2*c^5*d^2*e^4 + b*c^4*d*e^5 - b^2*c^3*e^6 )*f + (2*c^5*d^3*e^3 - 3*b*c^4*d^2*e^4 - 3*b^2*c^3*d*e^5 + 2*b^3*c^2*e^6)* g)*x^3 - 21*(3*(12*c^5*d^3*e^3 - 38*b*c^4*d^2*e^4 + 14*b^2*c^3*d*e^5 + b^3 *c^2*e^6)*f + (12*c^5*d^4*e^2 - 62*b*c^4*d^3*e^3 + 90*b^2*c^3*d^2*e^4 - 27 *b^3*c^2*d*e^5 - 2*b^4*c*e^6)*g)*x^2 - (2*c^5*d^5*e + 607*b*c^4*d^4*e^2...
\[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (417) = 834\).
Time = 0.70 (sec) , antiderivative size = 1067, normalized size of antiderivative = 2.33 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
35/8*(3*c^3*e*f + c^3*d*g - 2*b*c^2*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((32*c^5*d^5*e^2 - 80*b*c^4*d^4*e^3 + 80*b^2*c^ 3*d^3*e^4 - 40*b^3*c^2*d^2*e^5 + 10*b^4*c*d*e^6 - b^5*e^7)*sqrt(-2*c*d + b *e)) + 1/24*(256*c^7*d^4*e*f - 512*b*c^6*d^3*e^2*f + 384*b^2*c^5*d^2*e^3*f - 128*b^3*c^4*d*e^4*f + 16*b^4*c^3*e^5*f + 256*c^7*d^5*g - 768*b*c^6*d^4* e*g + 896*b^2*c^5*d^3*e^2*g - 512*b^3*c^4*d^2*e^3*g + 144*b^4*c^3*d*e^4*g - 16*b^5*c^2*e^5*g - 1152*((e*x + d)*c - 2*c*d + b*e)*c^6*d^3*e*f + 1728*( (e*x + d)*c - 2*c*d + b*e)*b*c^5*d^2*e^2*f - 864*((e*x + d)*c - 2*c*d + b* e)*b^2*c^4*d*e^3*f + 144*((e*x + d)*c - 2*c*d + b*e)*b^3*c^3*e^4*f - 384*( (e*x + d)*c - 2*c*d + b*e)*c^6*d^4*g + 1344*((e*x + d)*c - 2*c*d + b*e)*b* c^5*d^3*e*g - 1440*((e*x + d)*c - 2*c*d + b*e)*b^2*c^4*d^2*e^2*g + 624*((e *x + d)*c - 2*c*d + b*e)*b^3*c^3*d*e^3*g - 96*((e*x + d)*c - 2*c*d + b*e)* b^4*c^2*e^4*g - 2772*((e*x + d)*c - 2*c*d + b*e)^2*c^5*d^2*e*f + 2772*((e* x + d)*c - 2*c*d + b*e)^2*b*c^4*d*e^2*f - 693*((e*x + d)*c - 2*c*d + b*e)^ 2*b^2*c^3*e^3*f - 924*((e*x + d)*c - 2*c*d + b*e)^2*c^5*d^3*g + 2772*((e*x + d)*c - 2*c*d + b*e)^2*b*c^4*d^2*e*g - 2079*((e*x + d)*c - 2*c*d + b*e)^ 2*b^2*c^3*d*e^2*g + 462*((e*x + d)*c - 2*c*d + b*e)^2*b^3*c^2*e^3*g - 1680 *((e*x + d)*c - 2*c*d + b*e)^3*c^4*d*e*f + 840*((e*x + d)*c - 2*c*d + b*e) ^3*b*c^3*e^2*f - 560*((e*x + d)*c - 2*c*d + b*e)^3*c^4*d^2*g + 1400*((e*x + d)*c - 2*c*d + b*e)^3*b*c^3*d*e*g - 560*((e*x + d)*c - 2*c*d + b*e)^3...
Timed out. \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f+g\,x}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d x \]