\(\int (d+e x)^2 (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\) [160]

Optimal result

Integrand size = 44, antiderivative size = 425 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {5 (2 c d-b e)^6 (18 c e f+4 c d g-11 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{32768 c^6 e}+\frac {5 (2 c d-b e)^4 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12288 c^5 e}+\frac {(2 c d-b e)^2 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{768 c^4 e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}-\frac {(18 c e f+4 c d g-11 b e g) (32 c d-9 b e+14 c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2016 c^3 e^2}+\frac {5 (2 c d-b e)^8 (18 c e f+4 c d g-11 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{13/2} e^2} \] Output:

5/32768*(-b*e+2*c*d)^6*(-11*b*e*g+4*c*d*g+18*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d 
)-b*e^2*x-c*e^2*x^2)^(1/2)/c^6/e+5/12288*(-b*e+2*c*d)^4*(-11*b*e*g+4*c*d*g 
+18*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^5/e+1/768*(- 
b*e+2*c*d)^2*(-11*b*e*g+4*c*d*g+18*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x- 
c*e^2*x^2)^(5/2)/c^4/e-1/9*g*(e*x+d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7 
/2)/c/e^2-1/2016*(-11*b*e*g+4*c*d*g+18*c*e*f)*(14*c*e*x-9*b*e+32*c*d)*(d*( 
-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^3/e^2+5/32768*(-b*e+2*c*d)^8*(-11*b*e 
*g+4*c*d*g+18*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^(13/2)/e^2
 

Mathematica [A] (verified)

Time = 7.18 (sec) , antiderivative size = 847, normalized size of antiderivative = 1.99 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {(-2 c d+b e)^8 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {\sqrt {c} \left (5670 c^9 e f (d+e x)^8+1260 c^9 d g (d+e x)^8-3465 b c^8 e g (d+e x)^8-49140 c^8 e f (d+e x)^7 (-c d+b e+c e x)+30030 b c^7 e g (d+e x)^7 (-c d+b e+c e x)+188244 c^7 e f (d+e x)^6 (-c d+b e+c e x)^2+41832 c^7 d g (d+e x)^6 (-c d+b e+c e x)^2-115038 b c^6 e g (d+e x)^6 (-c d+b e+c e x)^2+172188 c^6 e f (d+e x)^5 (-c d+b e+c e x)^3-334602 b c^5 e g (d+e x)^5 (-c d+b e+c e x)^3-589824 c^5 e f (d+e x)^4 (-c d+b e+c e x)^4-131072 c^5 d g (d+e x)^4 (-c d+b e+c e x)^4+360448 b c^4 e g (d+e x)^4 (-c d+b e+c e x)^4+417636 c^4 e f (d+e x)^3 (-c d+b e+c e x)^5-255222 b c^3 e g (d+e x)^3 (-c d+b e+c e x)^5-188244 c^3 e f (d+e x)^2 (-c d+b e+c e x)^6-41832 c^3 d g (d+e x)^2 (-c d+b e+c e x)^6+115038 b c^2 e g (d+e x)^2 (-c d+b e+c e x)^6+49140 c^2 e f (d+e x) (-c d+b e+c e x)^7-30030 b c e g (d+e x) (-c d+b e+c e x)^7-5670 c e f (-c d+b e+c e x)^8-1260 c d g (-c d+b e+c e x)^8+3465 b e g (-c d+b e+c e x)^8+10920 c^8 d g (d+e x)^7 (-b e+c (d-e x))-497016 c^6 d g (d+e x)^5 (-b e+c (d-e x))^3-92808 c^4 d g (d+e x)^3 (-b e+c (d-e x))^5-10920 c^2 d g (d+e x) (-b e+c (d-e x))^7\right )}{(2 c d-b e)^9 (d+e x)^2 (-c d+b e+c e x)^2}-\frac {315 (18 c e f+4 c d g-11 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{5/2} (-b e+c (d-e x))^{5/2}}\right )}{2064384 c^{13/2} e^2} \] Input:

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

Output:

((-2*c*d + b*e)^8*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((Sqrt[c]*(5670 
*c^9*e*f*(d + e*x)^8 + 1260*c^9*d*g*(d + e*x)^8 - 3465*b*c^8*e*g*(d + e*x) 
^8 - 49140*c^8*e*f*(d + e*x)^7*(-(c*d) + b*e + c*e*x) + 30030*b*c^7*e*g*(d 
 + e*x)^7*(-(c*d) + b*e + c*e*x) + 188244*c^7*e*f*(d + e*x)^6*(-(c*d) + b* 
e + c*e*x)^2 + 41832*c^7*d*g*(d + e*x)^6*(-(c*d) + b*e + c*e*x)^2 - 115038 
*b*c^6*e*g*(d + e*x)^6*(-(c*d) + b*e + c*e*x)^2 + 172188*c^6*e*f*(d + e*x) 
^5*(-(c*d) + b*e + c*e*x)^3 - 334602*b*c^5*e*g*(d + e*x)^5*(-(c*d) + b*e + 
 c*e*x)^3 - 589824*c^5*e*f*(d + e*x)^4*(-(c*d) + b*e + c*e*x)^4 - 131072*c 
^5*d*g*(d + e*x)^4*(-(c*d) + b*e + c*e*x)^4 + 360448*b*c^4*e*g*(d + e*x)^4 
*(-(c*d) + b*e + c*e*x)^4 + 417636*c^4*e*f*(d + e*x)^3*(-(c*d) + b*e + c*e 
*x)^5 - 255222*b*c^3*e*g*(d + e*x)^3*(-(c*d) + b*e + c*e*x)^5 - 188244*c^3 
*e*f*(d + e*x)^2*(-(c*d) + b*e + c*e*x)^6 - 41832*c^3*d*g*(d + e*x)^2*(-(c 
*d) + b*e + c*e*x)^6 + 115038*b*c^2*e*g*(d + e*x)^2*(-(c*d) + b*e + c*e*x) 
^6 + 49140*c^2*e*f*(d + e*x)*(-(c*d) + b*e + c*e*x)^7 - 30030*b*c*e*g*(d + 
 e*x)*(-(c*d) + b*e + c*e*x)^7 - 5670*c*e*f*(-(c*d) + b*e + c*e*x)^8 - 126 
0*c*d*g*(-(c*d) + b*e + c*e*x)^8 + 3465*b*e*g*(-(c*d) + b*e + c*e*x)^8 + 1 
0920*c^8*d*g*(d + e*x)^7*(-(b*e) + c*(d - e*x)) - 497016*c^6*d*g*(d + e*x) 
^5*(-(b*e) + c*(d - e*x))^3 - 92808*c^4*d*g*(d + e*x)^3*(-(b*e) + c*(d - e 
*x))^5 - 10920*c^2*d*g*(d + e*x)*(-(b*e) + c*(d - e*x))^7))/((2*c*d - b*e) 
^9*(d + e*x)^2*(-(c*d) + b*e + c*e*x)^2) - (315*(18*c*e*f + 4*c*d*g - 1...
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1221, 1134, 1160, 1087, 1087, 1087, 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.

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

\(\Big \downarrow \) 1221

\(\displaystyle \frac {(-11 b e g+4 c d g+18 c e f) \int (d+e x)^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{18 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {(-11 b e g+4 c d g+18 c e f) \left (\frac {9 (2 c d-b e) \int (d+e x) \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{16 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{8 c e}\right )}{18 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {(-11 b e g+4 c d g+18 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {(2 c d-b e) \int \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e}\right )}{16 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{8 c e}\right )}{18 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(-11 b e g+4 c d g+18 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {5 (2 c d-b e)^2 \int \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e}\right )}{16 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{8 c e}\right )}{18 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(-11 b e g+4 c d g+18 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {5 (2 c d-b e)^2 \left (\frac {3 (2 c d-b e)^2 \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e}\right )}{16 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{8 c e}\right )}{18 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(-11 b e g+4 c d g+18 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {5 (2 c d-b e)^2 \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e}\right )}{16 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{8 c e}\right )}{18 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(-11 b e g+4 c d g+18 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {5 (2 c d-b e)^2 \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e}\right )}{16 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{8 c e}\right )}{18 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (\frac {9 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {5 (2 c d-b e)^2 \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e}\right )}{16 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{8 c e}\right ) (-11 b e g+4 c d g+18 c e f)}{18 c e}-\frac {g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}\)

Input:

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

Output:

-1/9*(g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(c*e^2) + 
 ((18*c*e*f + 4*c*d*g - 11*b*e*g)*(-1/8*((d + e*x)*(d*(c*d - b*e) - b*e^2* 
x - c*e^2*x^2)^(7/2))/(c*e) + (9*(2*c*d - b*e)*(-1/7*(d*(c*d - b*e) - b*e^ 
2*x - c*e^2*x^2)^(7/2)/(c*e) + ((2*c*d - b*e)*(((b + 2*c*x)*(d*(c*d - b*e) 
 - b*e^2*x - c*e^2*x^2)^(5/2))/(12*c) + (5*(2*c*d - b*e)^2*(((b + 2*c*x)*( 
d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(8*c) + (3*(2*c*d - b*e)^2*((( 
b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c) + ((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])])/(8*c^(3/2)*e)))/(16*c)))/(24*c)))/(2*c)))/(16*c)))/(18*c*e)
 

Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1134

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

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 1221

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 
)/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   Int[(d + e*x)^m*(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] 
 && NeQ[m + 2*p + 2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2578\) vs. \(2(397)=794\).

Time = 2.86 (sec) , antiderivative size = 2579, normalized size of antiderivative = 6.07

method	result	size
default	\(\text {Expression too large to display}\)	\(2579\)

Input:

int((e*x+d)^2*(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:

d^2*f*(-1/12*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/ 
2)-5/24*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/8*(-2*c*e^2*x-b*e^2)/c 
/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)- 
b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c* 
d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arcta 
n((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))))+e*( 
2*d*g+e*f)*(-1/8*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e^2-9/16*b/c*( 
-1/7*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e^2-1/2*b/c*(-1/12*(-2*c*e^2 
*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-5/24*(-4*c*e^2*(-b* 
d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2 
*x-b*d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*( 
-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e 
^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2 
*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))))+1/8*(-b*d*e+c*d^2)/c/e^2 
*(-1/12*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-5/ 
24*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2* 
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e 
^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^ 
(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c* 
e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))))+d*(d...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1424 vs. \(2 (397) = 794\).

Time = 6.43 (sec) , antiderivative size = 2861, normalized size of antiderivative = 6.73 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2*(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:

[1/8257536*(315*(18*(256*c^9*d^8*e - 1024*b*c^8*d^7*e^2 + 1792*b^2*c^7*d^6 
*e^3 - 1792*b^3*c^6*d^5*e^4 + 1120*b^4*c^5*d^4*e^5 - 448*b^5*c^4*d^3*e^6 + 
 112*b^6*c^3*d^2*e^7 - 16*b^7*c^2*d*e^8 + b^8*c*e^9)*f + (1024*c^9*d^9 - 6 
912*b*c^8*d^8*e + 18432*b^2*c^7*d^7*e^2 - 26880*b^3*c^6*d^6*e^3 + 24192*b^ 
4*c^5*d^5*e^4 - 14112*b^5*c^4*d^4*e^5 + 5376*b^6*c^3*d^3*e^6 - 1296*b^7*c^ 
2*d^2*e^7 + 180*b^8*c*d*e^8 - 11*b^9*e^9)*g)*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*(229376*c^9*e^8*g*x^8 + 1 
4336*(18*c^9*e^8*f + (36*c^9*d*e^7 + 37*b*c^8*e^8)*g)*x^7 + 1024*(18*(32*c 
^9*d*e^7 + 33*b*c^8*e^8)*f - (320*c^9*d^2*e^6 - 1796*b*c^8*d*e^7 - 309*b^2 
*c^7*e^8)*g)*x^6 - 256*(54*(28*c^9*d^2*e^6 - 156*b*c^8*d*e^7 - 27*b^2*c^7* 
e^8)*f + (5712*c^9*d^3*e^5 - 5484*b*c^8*d^2*e^6 - 5928*b^2*c^7*d*e^7 - 5*b 
^3*c^6*e^8)*g)*x^5 - 128*(18*(768*c^9*d^3*e^5 - 732*b*c^8*d^2*e^6 - 804*b^ 
2*c^7*d*e^7 - b^3*c^6*e^8)*f + (3072*c^9*d^4*e^4 + 15504*b*c^8*d^3*e^5 - 2 
2044*b^2*c^7*d^2*e^6 - 120*b^3*c^6*d*e^7 + 11*b^4*c^5*e^8)*g)*x^4 - 16*(18 
*(1680*c^9*d^4*e^4 + 8928*b*c^8*d^3*e^5 - 12552*b^2*c^7*d^2*e^6 - 104*b^3* 
c^6*d*e^7 + 9*b^4*c^5*e^8)*f - (79296*c^9*d^5*e^3 - 232272*b*c^8*d^4*e^4 + 
 148416*b^2*c^7*d^3*e^5 + 5704*b^3*c^6*d^2*e^6 - 1180*b^4*c^5*d*e^7 + 99*b 
^5*c^4*e^8)*g)*x^3 + 8*(54*(4096*c^9*d^5*e^3 - 11920*b*c^8*d^4*e^4 + 7456* 
b^2*c^7*d^3*e^5 + 456*b^3*c^6*d^2*e^6 - 88*b^4*c^5*d*e^7 + 7*b^5*c^4*e^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 18144 vs. \(2 (413) = 826\).

Time = 1.84 (sec) , antiderivative size = 18144, normalized size of antiderivative = 42.69 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)*(c**2*e**6*g*x** 
8/9 - x**7*(-37*b*c**2*e**8*g/18 - 2*c**3*d*e**7*g - c**3*e**8*f)/(8*c*e** 
2) - x**6*(-3*b**2*c*e**8*g - 9*b*c**2*d*e**7*g - 3*b*c**2*e**8*f - 15*b*( 
-37*b*c**2*e**8*g/18 - 2*c**3*d*e**7*g - c**3*e**8*f)/(16*c) + 2*c**3*d**2 
*e**6*g - 2*c**3*d*e**7*f - c**2*e**6*g*(-8*b*d*e + 8*c*d**2)/9)/(7*c*e**2 
) - x**5*(-b**3*e**8*g - 12*b**2*c*d*e**7*g - 3*b**2*c*e**8*f - 3*b*c**2*d 
**2*e**6*g - 9*b*c**2*d*e**7*f - 13*b*(-3*b**2*c*e**8*g - 9*b*c**2*d*e**7* 
g - 3*b*c**2*e**8*f - 15*b*(-37*b*c**2*e**8*g/18 - 2*c**3*d*e**7*g - c**3* 
e**8*f)/(16*c) + 2*c**3*d**2*e**6*g - 2*c**3*d*e**7*f - c**2*e**6*g*(-8*b* 
d*e + 8*c*d**2)/9)/(14*c) + 6*c**3*d**3*e**5*g + 2*c**3*d**2*e**6*f + (-7* 
b*d*e + 7*c*d**2)*(-37*b*c**2*e**8*g/18 - 2*c**3*d*e**7*g - c**3*e**8*f)/( 
8*c*e**2))/(6*c*e**2) - x**4*(-5*b**3*d*e**7*g - b**3*e**8*f - 15*b**2*c*d 
**2*e**6*g - 12*b**2*c*d*e**7*f + 15*b*c**2*d**3*e**5*g - 3*b*c**2*d**2*e* 
*6*f - 11*b*(-b**3*e**8*g - 12*b**2*c*d*e**7*g - 3*b**2*c*e**8*f - 3*b*c** 
2*d**2*e**6*g - 9*b*c**2*d*e**7*f - 13*b*(-3*b**2*c*e**8*g - 9*b*c**2*d*e* 
*7*g - 3*b*c**2*e**8*f - 15*b*(-37*b*c**2*e**8*g/18 - 2*c**3*d*e**7*g - c* 
*3*e**8*f)/(16*c) + 2*c**3*d**2*e**6*g - 2*c**3*d*e**7*f - c**2*e**6*g*(-8 
*b*d*e + 8*c*d**2)/9)/(14*c) + 6*c**3*d**3*e**5*g + 2*c**3*d**2*e**6*f + ( 
-7*b*d*e + 7*c*d**2)*(-37*b*c**2*e**8*g/18 - 2*c**3*d*e**7*g - c**3*e**8*f 
)/(8*c*e**2))/(12*c) + 6*c**3*d**3*e**5*f + (-6*b*d*e + 6*c*d**2)*(-3*b...
 

Maxima [F(-2)]

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

integrate((e*x+d)^2*(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>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1512 vs. \(2 (397) = 794\).

Time = 0.41 (sec) , antiderivative size = 1512, normalized size of antiderivative = 3.56 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2*(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:

1/2064384*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(2*(8*(2*(4*(14 
*(16*c^2*e^6*g*x + (18*c^10*e^20*f + 36*c^10*d*e^19*g + 37*b*c^9*e^20*g)/( 
c^8*e^14))*x + (576*c^10*d*e^19*f + 594*b*c^9*e^20*f - 320*c^10*d^2*e^18*g 
 + 1796*b*c^9*d*e^19*g + 309*b^2*c^8*e^20*g)/(c^8*e^14))*x - (1512*c^10*d^ 
2*e^18*f - 8424*b*c^9*d*e^19*f - 1458*b^2*c^8*e^20*f + 5712*c^10*d^3*e^17* 
g - 5484*b*c^9*d^2*e^18*g - 5928*b^2*c^8*d*e^19*g - 5*b^3*c^7*e^20*g)/(c^8 
*e^14))*x - (13824*c^10*d^3*e^17*f - 13176*b*c^9*d^2*e^18*f - 14472*b^2*c^ 
8*d*e^19*f - 18*b^3*c^7*e^20*f + 3072*c^10*d^4*e^16*g + 15504*b*c^9*d^3*e^ 
17*g - 22044*b^2*c^8*d^2*e^18*g - 120*b^3*c^7*d*e^19*g + 11*b^4*c^6*e^20*g 
)/(c^8*e^14))*x - (30240*c^10*d^4*e^16*f + 160704*b*c^9*d^3*e^17*f - 22593 
6*b^2*c^8*d^2*e^18*f - 1872*b^3*c^7*d*e^19*f + 162*b^4*c^6*e^20*f - 79296* 
c^10*d^5*e^15*g + 232272*b*c^9*d^4*e^16*g - 148416*b^2*c^8*d^3*e^17*g - 57 
04*b^3*c^7*d^2*e^18*g + 1180*b^4*c^6*d*e^19*g - 99*b^5*c^5*e^20*g)/(c^8*e^ 
14))*x + (221184*c^10*d^5*e^15*f - 643680*b*c^9*d^4*e^16*f + 402624*b^2*c^ 
8*d^3*e^17*f + 24624*b^3*c^7*d^2*e^18*f - 4752*b^4*c^6*d*e^19*f + 378*b^5* 
c^5*e^20*f + 106496*c^10*d^6*e^14*g - 192192*b*c^9*d^5*e^15*g + 52752*b^2* 
c^8*d^4*e^16*g + 46144*b^3*c^7*d^3*e^17*g - 16104*b^4*c^6*d^2*e^18*g + 298 
8*b^5*c^5*d*e^19*g - 231*b^6*c^4*e^20*g)/(c^8*e^14))*x + (669312*c^10*d^6* 
e^14*f - 1123200*b*c^9*d^5*e^15*f + 116640*b^2*c^8*d^4*e^16*f + 459072*b^3 
*c^7*d^3*e^17*f - 147528*b^4*c^6*d^2*e^18*f + 25704*b^5*c^5*d*e^19*f - ...
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 20.64 (sec) , antiderivative size = 5315, normalized size of antiderivative = 12.51 \[ \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*( - 3465*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c 
*d))*b**10*e**10*g + 63630*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sq 
rt( - b*e + 2*c*d))*b**9*c*d*e**9*g + 5670*sqrt(c)*asinh((sqrt( - b*e + c* 
d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**9*c*e**10*f - 521640*sqrt(c)*asinh( 
(sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**8*c**2*d**2*e**8*g 
 - 102060*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d 
))*b**8*c**2*d*e**9*f + 2509920*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)* 
i)/sqrt( - b*e + 2*c*d))*b**7*c**3*d**3*e**7*g + 816480*sqrt(c)*asinh((sqr 
t( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**7*c**3*d**2*e**8*f - 7 
832160*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))* 
b**6*c**4*d**4*e**6*g - 3810240*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)* 
i)/sqrt( - b*e + 2*c*d))*b**6*c**4*d**3*e**7*f + 16511040*sqrt(c)*asinh((s 
qrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c**5*d**5*e**5*g + 
 11430720*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d 
))*b**5*c**5*d**4*e**6*f - 23708160*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e 
*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c**6*d**6*e**4*g - 22861440*sqrt(c)*asin 
h((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c**6*d**5*e**5 
*f + 22740480*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2 
*c*d))*b**3*c**7*d**7*e**3*g + 30481920*sqrt(c)*asinh((sqrt( - b*e + c*d - 
 c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**7*d**6*e**4*f - 13789440*sqrt(...