\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{9/2}} \, dx\) [2257]

Optimal result

Integrand size = 46, antiderivative size = 360 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(2 c d-b e)^{3/2} (5 c e f-9 c d g+2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \]

[Out]

-1/3*(2*b*e*g-9*c*d*g+5*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(e*x+d)^(3/2)-1/5*(2*b*e*g-9*c*d*g+5
*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)^(5/2)-(-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)^(9/2)+(-b*e+2*c*d)^(3/2)*(2*b*e*g-9*c*d*g+5*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)*(2*b*e*g-9*c*d*g+5*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)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {806, 678, 674, 214} \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {(2 c d-b e)^{3/2} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right ) (2 b e g-9 c d g+5 c e f)}{e^2}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (2 b e g-9 c d g+5 c e f)}{5 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-9 c d g+5 c e f)}{3 e^2 (d+e x)^{3/2}}-\frac {(2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-9 c d g+5 c e f)}{e^2 \sqrt {d+e x}} \]

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(9/2),x]

[Out]

-(((2*c*d - b*e)*(5*c*e*f - 9*c*d*g + 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*Sqrt[d + e*x]))
- ((5*c*e*f - 9*c*d*g + 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(d + e*x)^(3/2)) - ((5*c
*e*f - 9*c*d*g + 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)) -
((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)^(9/2)) + ((2*c*d - b*e
)^(3/2)*(5*c*e*f - 9*c*d*g + 2*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqr
t[d + e*x])])/e^2

Rule 214

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]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[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] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 678

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[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] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 806

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] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(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)^{9/2}}-\frac {(5 c e f-9 c d g+2 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx}{2 e (2 c d-b e)} \\ & = -\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(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)^{9/2}}+\frac {((-2 c d+b e) (5 c e f-9 c d g+2 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{2 e (2 c d-b e)} \\ & = -\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(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)^{9/2}}-\frac {((2 c d-b e) (5 c e f-9 c d g+2 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{2 e} \\ & = -\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(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)^{9/2}}-\frac {\left ((2 c d-b e)^2 (5 c e f-9 c d g+2 b e g)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 e} \\ & = -\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(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)^{9/2}}-\left ((2 c d-b e)^2 (5 c e f-9 c d g+2 b e g)\right ) \text {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right ) \\ & = -\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(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)^{9/2}}+\frac {(2 c d-b e)^{3/2} (5 c e f-9 c d g+2 b e g) \tanh ^{-1}\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 )}{e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.71 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {b^2 e^2 (-15 e f+61 d g+46 e g x)+2 b c e \left (-146 d^2 g+5 d e (13 f-21 g x)+e^2 x (35 f+11 g x)\right )+2 c^2 \left (168 d^3 g+e^3 x^2 (5 f+3 g x)-6 d e^2 x (10 f+3 g x)+d^2 e (-95 f+117 g x)\right )}{(d+e x) (-c d+b e+c e x)^2}+\frac {15 (-2 c d+b e)^{3/2} (-5 c e f+9 c d g-2 b e g) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-b e+c (d-e x))^{5/2}}\right )}{15 e^2 (d+e x)^{5/2}} \]

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(9/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((b^2*e^2*(-15*e*f + 61*d*g + 46*e*g*x) + 2*b*c*e*(-146*d^2*g + 5*d*
e*(13*f - 21*g*x) + e^2*x*(35*f + 11*g*x)) + 2*c^2*(168*d^3*g + e^3*x^2*(5*f + 3*g*x) - 6*d*e^2*x*(10*f + 3*g*
x) + d^2*e*(-95*f + 117*g*x)))/((d + e*x)*(-(c*d) + b*e + c*e*x)^2) + (15*(-2*c*d + b*e)^(3/2)*(-5*c*e*f + 9*c
*d*g - 2*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)))/(15*e^2*
(d + e*x)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1127\) vs. \(2(332)=664\).

Time = 0.36 (sec) , antiderivative size = 1128, normalized size of antiderivative = 3.13

method	result	size
default	\(\text {Expression too large to display}\)	\(1128\)

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/15*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-6*c^2*e^3*g*x^3*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+15*b^2*e^3*f
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-336*c^2*d^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+300*arctan((-
c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f-10*c^2*e^3*f*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-
46*b^2*e^3*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-61*b^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2
)+190*c^2*d^2*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-540*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*c^3*d^3*e*g*x+300*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+660*arctan((-c*e*x-b*e+c*
d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g-300*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
+75*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*e^4*f*x-540*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*
d)^(1/2))*c^3*d^4*g-234*c^2*d^2*e*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+120*c^2*d*e^2*f*x*(-c*e*x-b*e+c
*d)^(1/2)*(b*e-2*c*d)^(1/2)+292*b*c*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-130*b*c*d*e^2*f*(-c*e*x-b
*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+660*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-22*b*c*
e^3*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+36*c^2*d*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)
-70*b*c*e^3*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b
^3*d*e^3*g+210*b*c*d*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*
c*d)^(1/2))*b^3*e^4*g*x-255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d*e^3*g*x-300*arctan((-c*e*
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d*e^3*f*x-255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c
*d^2*e^2*g+75*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d*e^3*f)/(e*x+d)^(3/2)/(-c*e*x-b*e+c*d)^(
1/2)/e^2/(b*e-2*c*d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 990, normalized size of antiderivative = 2.75 \[ \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)^{9/2}} \, dx=\left [-\frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\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 (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, 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}}{30 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\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 (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, 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}}{15 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

[-1/30*(15*((5*(2*c^2*d*e^3 - b*c*e^4)*f - (18*c^2*d^2*e^2 - 13*b*c*d*e^3 + 2*b^2*e^4)*g)*x^2 + 5*(2*c^2*d^3*e
- b*c*d^2*e^2)*f - (18*c^2*d^4 - 13*b*c*d^3*e + 2*b^2*d^2*e^2)*g + 2*(5*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (18*c
^2*d^3*e - 13*b*c*d^2*e^2 + 2*b^2*d*e^3)*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*(6*c^2*e^3*g*x^3 + 2*(5*c^2*e^3*f - (18*c^2*d*e^2 - 11*b*c*e^3)*g)*x^2 - 5*(38*c^2*d^2*e - 26*b*
c*d*e^2 + 3*b^2*e^3)*f + (336*c^2*d^3 - 292*b*c*d^2*e + 61*b^2*d*e^2)*g - 2*(5*(12*c^2*d*e^2 - 7*b*c*e^3)*f -
(117*c^2*d^2*e - 105*b*c*d*e^2 + 23*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^4*x^2 + 2*d*e^3*x + d^2*e^2), 1/15*(15*((5*(2*c^2*d*e^3 - b*c*e^4)*f - (18*c^2*d^2*e^2 - 13*b*c*d*e^3 + 2*b^
2*e^4)*g)*x^2 + 5*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (18*c^2*d^4 - 13*b*c*d^3*e + 2*b^2*d^2*e^2)*g + 2*(5*(2*c^2*
d^2*e^2 - b*c*d*e^3)*f - (18*c^2*d^3*e - 13*b*c*d^2*e^2 + 2*b^2*d*e^3)*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)) +
(6*c^2*e^3*g*x^3 + 2*(5*c^2*e^3*f - (18*c^2*d*e^2 - 11*b*c*e^3)*g)*x^2 - 5*(38*c^2*d^2*e - 26*b*c*d*e^2 + 3*b^
2*e^3)*f + (336*c^2*d^3 - 292*b*c*d^2*e + 61*b^2*d*e^2)*g - 2*(5*(12*c^2*d*e^2 - 7*b*c*e^3)*f - (117*c^2*d^2*e
- 105*b*c*d*e^2 + 23*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^4*x^2 + 2*d*
e^3*x + d^2*e^2)]

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]

[In]

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**(9/2), x)

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(9/2), x)

Giac [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.61 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {120 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 240 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g + 180 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g - 30 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g + 10 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f - 30 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 10 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g - 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c g + \frac {15 \, {\left (20 \, c^{4} d^{2} e f - 20 \, b c^{3} d e^{2} f + 5 \, b^{2} c^{2} e^{3} f - 36 \, c^{4} d^{3} g + 44 \, b c^{3} d^{2} e g - 17 \, b^{2} c^{2} d e^{2} g + 2 \, b^{3} c e^{3} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {15 \, {\left (4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{2} e f - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d e^{2} f + \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} e^{3} f - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{3} g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d^{2} e g - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} d e^{2} g\right )}}{{\left (e x + d\right )} c}}{15 \, c e^{2}} \]

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

-1/15*(120*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d*e*f - 60*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*e^2*f - 240*
sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d^2*g + 180*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d*e*g - 30*sqrt(-(e*x
+ d)*c + 2*c*d - b*e)*b^2*c*e^2*g + 10*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*e*f - 30*(-(e*x + d)*c + 2*c*d -
b*e)^(3/2)*c^2*d*g + 10*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*e*g - 6*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e
*x + d)*c + 2*c*d - b*e)*c*g + 15*(20*c^4*d^2*e*f - 20*b*c^3*d*e^2*f + 5*b^2*c^2*e^3*f - 36*c^4*d^3*g + 44*b*c
^3*d^2*e*g - 17*b^2*c^2*d*e^2*g + 2*b^3*c*e^3*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/s
qrt(-2*c*d + b*e) + 15*(4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^2*e*f - 4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*
c^3*d*e^2*f + sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^2*e^3*f - 4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^3*g +
4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^3*d^2*e*g - sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^2*d*e^2*g)/((e*x + d
)*c))/(c*e^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]

[In]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(9/2),x)

[Out]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(9/2), x)