\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\) [222]

Optimal result

Integrand size = 46, antiderivative size = 341 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {(2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (d+e x)^{5/2}}+\frac {(2 c d-b e) (9 c e f-17 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (d+e x)^{3/2}}+\frac {2 c (c e f-5 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 c g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {5 c \sqrt {2 c d-b e} (3 c e f-11 c d g+4 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2} \] Output:

-1/2*(-b*e+2*c*d)^2*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/ 
(e*x+d)^(5/2)+1/4*(-b*e+2*c*d)*(4*b*e*g-17*c*d*g+9*c*e*f)*(d*(-b*e+c*d)-b* 
e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)^(3/2)+2*c*(2*b*e*g-5*c*d*g+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)-2/3*c*g*(d*(-b*e+c*d)- 
b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(e*x+d)^(3/2)-5/4*c*(-b*e+2*c*d)^(1/2)*(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))/e^2
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.74 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {c ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {c^2 \left (-206 d^3 g+d^2 e (54 f-350 g x)+2 d e^2 x (51 f-56 g x)+8 e^3 x^2 (3 f+g x)\right )-6 b^2 e^2 (d g+e (f+2 g x))+b c e \left (107 d^2 g+e^2 x (-27 f+56 g x)+d e (-3 f+187 g x)\right )}{c (d+e x)^2 (-c d+b e+c e x)^2}+\frac {15 \sqrt {-2 c d+b e} (-3 c e f+11 c d g-4 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-b e+c (d-e x))^{5/2}}\right )}{12 e^2 (d+e x)^{5/2}} \] Input:

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x 
)^(11/2),x]
 

Output:

(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[c*d - b*e - c*e*x 
]/Sqrt[-2*c*d + b*e]])/(-(b*e) + c*(d - e*x))^(5/2)))/(12*e^2*(d + e*x)^(5 
/2))
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.96, 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

\(\displaystyle -\frac {\left (-\frac {5}{2} c \left ((2 c d-b e) \left (\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2 c d-b e} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e (d+e x)^{7/2}}\right ) (4 b e g-11 c d g+3 c e f)}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}\)

Input:

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(11/ 
2),x]
 

Output:

-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))
 

Defintions of rubi rules used

rule 221

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 1130

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]
 

rule 1131

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]
 

rule 1136

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]
 

rule 1220

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 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1180\) vs. \(2(309)=618\).

Time = 1.54 (sec) , antiderivative size = 1181, normalized size of antiderivative = 3.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1181\)

Input:

int((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,method 
=_RETURNVERBOSE)
 

Output:

-1/12*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-56*b*c*e^3*g*x^2*(-c*e*x-b*e+c*d) 
^(1/2)*(b*e-2*c*d)^(1/2)+112*c^2*d*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2 
*c*d)^(1/2)+120*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-187*b*c*d*e^2*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+45*arcta 
n((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*f+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+90*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-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^2-570*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-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)-102*c^2*d*e^2*f*x 
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+350*c^2*d^2*e*g*x*(-c*e*x-b*e+c* 
d)^(1/2)*(b*e-2*c*d)^(1/2)+27*b*c*e^3*f*x*(-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^4*g+60 
*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+6*b^2*e^ 
3*f*(-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*e^3*f*x^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)+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+660*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*...
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 961, normalized size of antiderivative = 2.82 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx =\text {Too large to display} \] Input:

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")
 

Output:

[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)/(2*c*d^2 - b 
*d*e + (2*c*d*e - b*e^2)*x)) - (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^...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Timed out} \] Input:

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)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \] Input:

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")
 

Output:

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d) 
^(11/2), x)
 

Giac [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.68 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {24 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} e f - 120 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d g + 48 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e g - 8 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} g + \frac {15 \, {\left (6 \, c^{4} d e f - 3 \, b c^{3} e^{2} f - 22 \, c^{4} d^{2} g + 19 \, b c^{3} d e g - 4 \, b^{2} c^{2} e^{2} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, {\left (28 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{5} d^{2} e f - 28 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{4} d e^{2} f + 7 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{3} e^{3} f - 60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{5} d^{3} g + 76 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{4} d^{2} e g - 31 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{3} d e^{2} g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{3} c^{2} e^{3} g - 18 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{4} d e f + 9 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{3} e^{2} f + 34 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{4} d^{2} g - 25 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{3} d e g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{2} e^{2} g\right )}}{{\left (e x + d\right )}^{2} c^{2}}}{12 \, c e^{2}} \] Input:

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="giac")
 

Output:

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)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(11/ 
2),x)
 

Output:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(11/ 
2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 809, normalized size of antiderivative = 2.37 \[ \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 {-60 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -120 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x -60 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}+165 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -45 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +330 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -90 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x +165 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}-45 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}-6 \sqrt {-c e x -b e +c d}\, b^{2} d \,e^{2} g -6 \sqrt {-c e x -b e +c d}\, b^{2} e^{3} f -12 \sqrt {-c e x -b e +c d}\, b^{2} e^{3} g x +107 \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g -3 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} f +187 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x -27 \sqrt {-c e x -b e +c d}\, b c \,e^{3} f x +56 \sqrt {-c e x -b e +c d}\, b c \,e^{3} g \,x^{2}-206 \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g +54 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f -350 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e g x +102 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} f x -112 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} g \,x^{2}+24 \sqrt {-c e x -b e +c d}\, c^{2} e^{3} f \,x^{2}+8 \sqrt {-c e x -b e +c d}\, c^{2} e^{3} g \,x^{3}}{12 e^{2} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:

int((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)
 

Output:

( - 60*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d) 
)*b*c*d**2*e*g - 120*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqr 
t(b*e - 2*c*d))*b*c*d*e**2*g*x - 60*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c 
*d - c*e*x)/sqrt(b*e - 2*c*d))*b*c*e**3*g*x**2 + 165*sqrt(b*e - 2*c*d)*ata 
n(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**3*g - 45*sqrt(b*e 
- 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**2*e*f 
+ 330*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d)) 
*c**2*d**2*e*g*x - 90*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sq 
rt(b*e - 2*c*d))*c**2*d*e**2*f*x + 165*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e 
+ c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d*e**2*g*x**2 - 45*sqrt(b*e - 2*c*d 
)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*e**3*f*x**2 - 6* 
sqrt( - b*e + c*d - c*e*x)*b**2*d*e**2*g - 6*sqrt( - b*e + c*d - c*e*x)*b* 
*2*e**3*f - 12*sqrt( - b*e + c*d - c*e*x)*b**2*e**3*g*x + 107*sqrt( - b*e 
+ c*d - c*e*x)*b*c*d**2*e*g - 3*sqrt( - b*e + c*d - c*e*x)*b*c*d*e**2*f + 
187*sqrt( - b*e + c*d - c*e*x)*b*c*d*e**2*g*x - 27*sqrt( - b*e + c*d - c*e 
*x)*b*c*e**3*f*x + 56*sqrt( - b*e + c*d - c*e*x)*b*c*e**3*g*x**2 - 206*sqr 
t( - b*e + c*d - c*e*x)*c**2*d**3*g + 54*sqrt( - b*e + c*d - c*e*x)*c**2*d 
**2*e*f - 350*sqrt( - b*e + c*d - c*e*x)*c**2*d**2*e*g*x + 102*sqrt( - b*e 
 + c*d - c*e*x)*c**2*d*e**2*f*x - 112*sqrt( - b*e + c*d - c*e*x)*c**2*d*e* 
*2*g*x**2 + 24*sqrt( - b*e + c*d - c*e*x)*c**2*e**3*f*x**2 + 8*sqrt( - ...