\(\int \frac {(d+e x)^5 (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [187]

Optimal result

Integrand size = 44, antiderivative size = 311 \[ \int \frac {(d+e x)^5 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)^4}{3 c^2 e^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^2}{3 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 (4 c e f+10 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^4 e^2}-\frac {g (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c^3 e^2}+\frac {5 (2 c d-b e) (4 c e f+10 c d g-7 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 c^{9/2} e^2} \] Output:

2/3*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^4/c^2/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2 
)^(3/2)-2/3*(-8*b*e*g+11*c*d*g+5*c*e*f)*(e*x+d)^2/c^3/e^2/(d*(-b*e+c*d)-b* 
e^2*x-c*e^2*x^2)^(1/2)-5/4*(-7*b*e*g+10*c*d*g+4*c*e*f)*(d*(-b*e+c*d)-b*e^2 
*x-c*e^2*x^2)^(1/2)/c^4/e^2-1/2*g*(e*x+d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2) 
^(1/2)/c^3/e^2+5/4*(-b*e+2*c*d)*(-7*b*e*g+10*c*d*g+4*c*e*f)*arctan(c^(1/2) 
*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(9/2)/e^2
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.45 \[ \int \frac {(d+e x)^5 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^5 \left (-7 (c e f+c d g-b e g)+(4 c e f+10 c d g-7 b e g) \left (\frac {-c d+b e+c e x}{-2 c d+b e}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},\frac {c (d+e x)}{2 c d-b e}\right )\right )}{21 c e^2 (-2 c d+b e) ((d+e x) (-b e+c (d-e x)))^{3/2}} \] Input:

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

Output:

(2*(d + e*x)^5*(-7*(c*e*f + c*d*g - b*e*g) + (4*c*e*f + 10*c*d*g - 7*b*e*g 
)*((-(c*d) + b*e + c*e*x)/(-2*c*d + b*e))^(3/2)*Hypergeometric2F1[3/2, 7/2 
, 9/2, (c*(d + e*x))/(2*c*d - b*e)]))/(21*c*e^2*(-2*c*d + b*e)*((d + e*x)* 
(-(b*e) + c*(d - e*x)))^(3/2))
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1218, 1124, 2192, 27, 1160, 1092, 217}

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.

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

\(\Big \downarrow \) 1218

\(\displaystyle \frac {2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-7 b e g+10 c d g+4 c e f) \int \frac {(d+e x)^4}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 1124

\(\displaystyle \frac {2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-7 b e g+10 c d g+4 c e f) \left (\frac {2 (d+e x) (2 c d-b e)^2}{c^3 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\int \frac {c^2 x^2 e^6+c (4 c d-b e) x e^5+\left (7 c^2 d^2-5 b c e d+b^2 e^2\right ) e^4}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c^3 e^4}\right )}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-7 b e g+10 c d g+4 c e f) \left (\frac {2 (d+e x) (2 c d-b e)^2}{c^3 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {-\frac {\int -\frac {c e^6 (2 (5 c d-2 b e) (3 c d-b e)+c e (16 c d-7 b e) x)}{2 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c e^2}-\frac {1}{2} c e^4 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^4}\right )}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-7 b e g+10 c d g+4 c e f) \left (\frac {2 (d+e x) (2 c d-b e)^2}{c^3 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\frac {1}{4} e^4 \int \frac {2 (5 c d-2 b e) (3 c d-b e)+c e (16 c d-7 b e) x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {1}{2} c e^4 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^4}\right )}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-7 b e g+10 c d g+4 c e f) \left (\frac {2 (d+e x) (2 c d-b e)^2}{c^3 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\frac {1}{4} e^4 \left (\frac {15}{2} (2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {(16 c d-7 b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {1}{2} c e^4 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^4}\right )}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-7 b e g+10 c d g+4 c e f) \left (\frac {2 (d+e x) (2 c d-b e)^2}{c^3 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\frac {1}{4} e^4 \left (15 (2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )-\frac {(16 c d-7 b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {1}{2} c e^4 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^4}\right )}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {\left (\frac {2 (d+e x) (2 c d-b e)^2}{c^3 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\frac {1}{4} e^4 \left (\frac {15 (2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt {c} e}-\frac {(16 c d-7 b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {1}{2} c e^4 x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^4}\right ) (-7 b e g+10 c d g+4 c e f)}{3 c e (2 c d-b e)}\)

Input:

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

Output:

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^5)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b 
*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - ((4*c*e*f + 10*c*d*g - 7*b*e*g)*((2*(2 
*c*d - b*e)^2*(d + e*x))/(c^3*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) 
 - (-1/2*(c*e^4*x*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (e^4*(-(((1 
6*c*d - 7*b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/e) + (15*(2*c*d 
- b*e)^2*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - 
c*e^2*x^2])])/(2*Sqrt[c]*e)))/4)/(c^3*e^4)))/(3*c*e*(2*c*d - b*e))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1124

Int[((d_.) + (e_.)*(x_))^(m_.)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[-2*e*(2*c*d - b*e)^(m - 2)*((d + e*x)/(c^(m - 1)*Sqrt[a + 
b*x + c*x^2])), x] + Simp[e^2/c^(m - 1)   Int[(1/Sqrt[a + b*x + c*x^2])*Exp 
andToSum[((2*c*d - b*e)^(m - 1) - c^(m - 1)*(d + e*x)^(m - 1))/(c*d - b*e - 
 c*e*x), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e 
^2, 0] && IGtQ[m, 0]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1218

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( 
c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( 
a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* 
d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e)))   I 
nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d 
, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6818\) vs. \(2(287)=574\).

Time = 5.94 (sec) , antiderivative size = 6819, normalized size of antiderivative = 21.93

method	result	size
default	\(\text {Expression too large to display}\)	\(6819\)

Input:

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

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (287) = 574\).

Time = 3.21 (sec) , antiderivative size = 1233, normalized size of antiderivative = 3.96 \[ \int \frac {(d+e x)^5 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/48*(15*((4*(2*c^4*d*e^3 - b*c^3*e^4)*f + (20*c^4*d^2*e^2 - 24*b*c^3*d* 
e^3 + 7*b^2*c^2*e^4)*g)*x^2 + 4*(2*c^4*d^3*e - 5*b*c^3*d^2*e^2 + 4*b^2*c^2 
*d*e^3 - b^3*c*e^4)*f + (20*c^4*d^4 - 64*b*c^3*d^3*e + 75*b^2*c^2*d^2*e^2 
- 38*b^3*c*d*e^3 + 7*b^4*e^4)*g - 2*(4*(2*c^4*d^2*e^2 - 3*b*c^3*d*e^3 + b^ 
2*c^2*e^4)*f + (20*c^4*d^3*e - 44*b*c^3*d^2*e^2 + 31*b^2*c^2*d*e^3 - 7*b^3 
*c*e^4)*g)*x)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c 
*d*e + b^2*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b 
*e)*sqrt(-c)) + 4*(6*c^4*e^3*g*x^3 + 3*(4*c^4*e^3*f + (16*c^4*d*e^2 - 7*b* 
c^3*e^3)*g)*x^2 + 4*(23*c^4*d^2*e - 40*b*c^3*d*e^2 + 15*b^2*c^2*e^3)*f + ( 
236*c^4*d^3 - 561*b*c^3*d^2*e + 430*b^2*c^2*d*e^2 - 105*b^3*c*e^3)*g - 2*( 
4*(17*c^4*d*e^2 - 10*b*c^3*e^3)*f + (161*c^4*d^2*e - 219*b*c^3*d*e^2 + 70* 
b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^7*e^4*x^ 
2 + c^7*d^2*e^2 - 2*b*c^6*d*e^3 + b^2*c^5*e^4 - 2*(c^7*d*e^3 - b*c^6*e^4)* 
x), -1/24*(15*((4*(2*c^4*d*e^3 - b*c^3*e^4)*f + (20*c^4*d^2*e^2 - 24*b*c^3 
*d*e^3 + 7*b^2*c^2*e^4)*g)*x^2 + 4*(2*c^4*d^3*e - 5*b*c^3*d^2*e^2 + 4*b^2* 
c^2*d*e^3 - b^3*c*e^4)*f + (20*c^4*d^4 - 64*b*c^3*d^3*e + 75*b^2*c^2*d^2*e 
^2 - 38*b^3*c*d*e^3 + 7*b^4*e^4)*g - 2*(4*(2*c^4*d^2*e^2 - 3*b*c^3*d*e^3 + 
 b^2*c^2*e^4)*f + (20*c^4*d^3*e - 44*b*c^3*d^2*e^2 + 31*b^2*c^2*d*e^3 - 7* 
b^3*c*e^4)*g)*x)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b* 
d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*(b*e-2*c*d)>0)', see `assume?` 
 for more
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (287) = 574\).

Time = 2.93 (sec) , antiderivative size = 1051, normalized size of antiderivative = 3.38 \[ \int \frac {(d+e x)^5 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*g*x/(c^3*e) + (4*c^8*e^ 
3*f + 20*c^8*d*e^2*g - 11*b*c^7*e^3*g)/(c^11*e^4)) - 1/8*(8*c^2*d*e*f - 4* 
b*c*e^2*f + 20*c^2*d^2*g - 24*b*c*d*e*g + 7*b^2*e^2*g)*log(abs(-b*c^4*d^4* 
e^2 + 4*b^2*c^3*d^3*e^3 - 6*b^3*c^2*d^2*e^4 + 4*b^4*c*d*e^5 - b^5*e^6 + 2* 
(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*c^4 
*d^4*abs(e) - 12*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d 
*e))*b*sqrt(-c)*c^3*d^3*e*abs(e) + 24*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - 
b*e^2*x + c*d^2 - b*d*e))*b^2*sqrt(-c)*c^2*d^2*e^2*abs(e) - 20*(sqrt(-c*e^ 
2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b^3*sqrt(-c)*c*d*e^3*ab 
s(e) + 6*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b^4 
*sqrt(-c)*e^4*abs(e) - 8*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d 
^2 - b*d*e))^2*c^4*d^3*e + 30*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x 
+ c*d^2 - b*d*e))^2*b*c^3*d^2*e^2 - 36*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - 
 b*e^2*x + c*d^2 - b*d*e))^2*b^2*c^2*d*e^3 + 14*(sqrt(-c*e^2)*x - sqrt(-c* 
e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*b^3*c*e^4 - 12*(sqrt(-c*e^2)*x - sqr 
t(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*sqrt(-c)*c^3*d^2*abs(e) + 28*(s 
qrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*b*sqrt(-c)*c 
^2*d*e*abs(e) - 16*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b 
*d*e))^3*b^2*sqrt(-c)*c*e^2*abs(e) + 8*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - 
 b*e^2*x + c*d^2 - b*d*e))^4*c^3*d*e - 9*(sqrt(-c*e^2)*x - sqrt(-c*e^2*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 2003, normalized size of antiderivative = 6.44 \[ \int \frac {(d+e x)^5 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(840*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - b*e + c*d - c*e* 
x)*i)/sqrt( - b*e + 2*c*d))*b**4*e**4*g - 5400*sqrt(c)*sqrt( - b*e + c*d - 
 c*e*x)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c* 
d*e**3*g - 480*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - b*e + c*d 
 - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c*e**4*f + 840*sqrt(c)*sqrt( - b*e 
 + c*d - c*e*x)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d)) 
*b**3*c*e**4*g*x + 12720*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - 
 b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**2*d**2*e**2*g + 2400* 
sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sq 
rt( - b*e + 2*c*d))*b**2*c**2*d*e**3*f - 4560*sqrt(c)*sqrt( - b*e + c*d - 
c*e*x)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c** 
2*d*e**3*g*x - 480*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - b*e + 
 c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**2*e**4*f*x - 12960*sqrt(c)* 
sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b* 
e + 2*c*d))*b*c**3*d**3*e*g - 3840*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asin 
h((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**3*d**2*e**2*f 
+ 8160*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqrt( - b*e + c*d - c*e*x 
)*i)/sqrt( - b*e + 2*c*d))*b*c**3*d**2*e**2*g*x + 1920*sqrt(c)*sqrt( - b*e 
 + c*d - c*e*x)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d)) 
*b*c**3*d*e**3*f*x + 4800*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*asinh((sqr...