\(\int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [173]

Optimal result

Integrand size = 44, antiderivative size = 276 \[ \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}-\frac {5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (8 c d-3 b e+2 c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{192 c^4 e^2}+\frac {5 (2 c d-b e)^3 (8 c e f+6 c d g-7 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{64 c^{9/2} e^2} \] Output:

-1/24*(-7*b*e*g+6*c*d*g+8*c*e*f)*(e*x+d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2 
)^(1/2)/c^2/e^2-1/4*g*(e*x+d)^3*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c/e 
^2-5/192*(-b*e+2*c*d)*(-7*b*e*g+6*c*d*g+8*c*e*f)*(2*c*e*x-3*b*e+8*c*d)*(d* 
(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^4/e^2+5/64*(-b*e+2*c*d)^3*(-7*b*e*g+ 
6*c*d*g+8*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 [A] (verified)

Time = 0.95 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {(2 c d-b e)^3 \left (-\frac {\sqrt {c} (d+e x) (-b e+c (d-e x)) \left (-105 b^3 e^3 g+10 b^2 c e^2 (12 e f+58 d g+7 e g x)-4 b c^2 e \left (259 d^2 g+2 e^2 x (10 f+7 g x)+2 d e (70 f+39 g x)\right )+8 c^3 \left (72 d^3 g+12 d e^2 x (3 f+2 g x)+2 e^3 x^2 (4 f+3 g x)+d^2 e (88 f+45 g x)\right )\right )}{(2 c d-b e)^3}-15 (8 c e f+6 c d g-7 b e g) \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )\right )}{192 c^{9/2} e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

Integrate[((d + e*x)^3*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2 
],x]
 

Output:

((2*c*d - b*e)^3*(-((Sqrt[c]*(d + e*x)*(-(b*e) + c*(d - e*x))*(-105*b^3*e^ 
3*g + 10*b^2*c*e^2*(12*e*f + 58*d*g + 7*e*g*x) - 4*b*c^2*e*(259*d^2*g + 2* 
e^2*x*(10*f + 7*g*x) + 2*d*e*(70*f + 39*g*x)) + 8*c^3*(72*d^3*g + 12*d*e^2 
*x*(3*f + 2*g*x) + 2*e^3*x^2*(4*f + 3*g*x) + d^2*e*(88*f + 45*g*x))))/(2*c 
*d - b*e)^3) - 15*(8*c*e*f + 6*c*d*g - 7*b*e*g)*Sqrt[d + e*x]*Sqrt[c*d - b 
*e - c*e*x]*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])]))/(192 
*c^(9/2)*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1221, 1134, 1134, 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.

\(\displaystyle \int \frac {(d+e x)^3 (f+g x)}{\sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}} \, dx\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {(-7 b e g+6 c d g+8 c e f) \int \frac {(d+e x)^3}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c e}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {(-7 b e g+6 c d g+8 c e f) \left (\frac {5 (2 c d-b e) \int \frac {(d+e x)^2}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{6 c}-\frac {(d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{8 c e}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {(-7 b e g+6 c d g+8 c e f) \left (\frac {5 (2 c d-b e) \left (\frac {3 (2 c d-b e) \int \frac {d+e x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{4 c}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c e}\right )}{6 c}-\frac {(d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{8 c e}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {(-7 b e g+6 c d g+8 c e f) \left (\frac {5 (2 c d-b e) \left (\frac {3 (2 c d-b e) \left (\frac {(2 c d-b e) \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e}\right )}{4 c}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c e}\right )}{6 c}-\frac {(d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{8 c e}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(-7 b e g+6 c d g+8 c e f) \left (\frac {5 (2 c d-b e) \left (\frac {3 (2 c d-b e) \left (\frac {(2 c d-b e) \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{c}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e}\right )}{4 c}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c e}\right )}{6 c}-\frac {(d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{8 c e}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (\frac {5 (2 c d-b e) \left (\frac {3 (2 c d-b e) \left (\frac {(2 c d-b e) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{3/2} e}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e}\right )}{4 c}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c e}\right )}{6 c}-\frac {(d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right ) (-7 b e g+6 c d g+8 c e f)}{8 c e}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}\)

Input:

Int[((d + e*x)^3*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]
 

Output:

-1/4*(g*(d + e*x)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c*e^2) + ( 
(8*c*e*f + 6*c*d*g - 7*b*e*g)*(-1/3*((d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^ 
2*x - c*e^2*x^2])/(c*e) + (5*(2*c*d - b*e)*(-1/2*((d + e*x)*Sqrt[d*(c*d - 
b*e) - b*e^2*x - c*e^2*x^2])/(c*e) + (3*(2*c*d - b*e)*(-(Sqrt[d*(c*d - b*e 
) - b*e^2*x - c*e^2*x^2]/(c*e)) + ((2*c*d - b*e)*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*c^(3/2)*e)))/(4*c 
)))/(6*c)))/(8*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 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. \(1444\) vs. \(2(256)=512\).

Time = 3.89 (sec) , antiderivative size = 1445, normalized size of antiderivative = 5.24

method	result	size
default	\(\text {Expression too large to display}\)	\(1445\)

Input:

int((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

d^3*f/(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))+e^2*(3*d*g+e*f)*(-1/3*x^2/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d 
*e+c*d^2)^(1/2)-5/6*b/c*(-1/2*x/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/ 
2)-3/4*b/c*(-1/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c/(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)))+1/2*(-b*d*e+c*d^2)/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)))+2/3*(-b*d*e+c*d^2)/c/e^2*(- 
1/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c/(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))))+3*d* 
e*(d*g+e*f)*(-1/2*x/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-3/4*b/c*( 
-1/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c/(c*e^2)^(1/2)*arct 
an((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)))+1/2* 
(-b*d*e+c*d^2)/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^2*(d*g+3*e*f)*(-1/c/e^2*(-c*e^2*x^2-b 
*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c/(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)))+g*e^3*(-1/4*x^3/c/e^2*(-c* 
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-7/8*b/c*(-1/3*x^2/c/e^2*(-c*e^2*x^2-b*e 
^2*x-b*d*e+c*d^2)^(1/2)-5/6*b/c*(-1/2*x/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c* 
d^2)^(1/2)-3/4*b/c*(-1/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/ 
c/(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*...
 

Fricas [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 825, normalized size of antiderivative = 2.99 \[ \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo 
rithm="fricas")
 

Output:

[-1/768*(15*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e 
^4)*f + (48*c^4*d^4 - 128*b*c^3*d^3*e + 120*b^2*c^2*d^2*e^2 - 48*b^3*c*d*e 
^3 + 7*b^4*e^4)*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*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^3*f + (24*c^4*d*e^2 
- 7*b*c^3*e^3)*g)*x^2 + 8*(88*c^4*d^2*e - 70*b*c^3*d*e^2 + 15*b^2*c^2*e^3) 
*f + (576*c^4*d^3 - 1036*b*c^3*d^2*e + 580*b^2*c^2*d*e^2 - 105*b^3*c*e^3)* 
g + 2*(8*(18*c^4*d*e^2 - 5*b*c^3*e^3)*f + (180*c^4*d^2*e - 156*b*c^3*d*e^2 
 + 35*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^5* 
e^2), -1/384*(15*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^ 
3*c*e^4)*f + (48*c^4*d^4 - 128*b*c^3*d^3*e + 120*b^2*c^2*d^2*e^2 - 48*b^3* 
c*d*e^3 + 7*b^4*e^4)*g)*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*e)) + 2*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^3*f + (24*c^4*d*e^2 - 7*b*c^3* 
e^3)*g)*x^2 + 8*(88*c^4*d^2*e - 70*b*c^3*d*e^2 + 15*b^2*c^2*e^3)*f + (576* 
c^4*d^3 - 1036*b*c^3*d^2*e + 580*b^2*c^2*d*e^2 - 105*b^3*c*e^3)*g + 2*(8*( 
18*c^4*d*e^2 - 5*b*c^3*e^3)*f + (180*c^4*d^2*e - 156*b*c^3*d*e^2 + 35*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^5*e^2)]
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1032 vs. \(2 (270) = 540\).

Time = 1.47 (sec) , antiderivative size = 1032, normalized size of antiderivative = 3.74 \[ \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)**3*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x 
)
 

Output:

Piecewise(((-b*(-3*b*(-5*b*(-7*b*e**3*g/(8*c) + 3*d*e**2*g + e**3*f)/(6*c) 
 + 3*d**2*e*g + 3*d*e**2*f + e*g*(-3*b*d*e + 3*c*d**2)/(4*c))/(4*c) + d**3 
*g + 3*d**2*e*f + (-2*b*d*e + 2*c*d**2)*(-7*b*e**3*g/(8*c) + 3*d*e**2*g + 
e**3*f)/(3*c*e**2))/(2*c) + d**3*f + (-b*d*e + c*d**2)*(-5*b*(-7*b*e**3*g/ 
(8*c) + 3*d*e**2*g + e**3*f)/(6*c) + 3*d**2*e*g + 3*d*e**2*f + e*g*(-3*b*d 
*e + 3*c*d**2)/(4*c))/(2*c*e**2))*Piecewise((log(-b*e**2 - 2*c*e**2*x + 2* 
sqrt(-c*e**2)*sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2))/sqrt(-c*e**2 
), Ne(b**2*e**2/(4*c) - b*d*e + c*d**2, 0)), ((b/(2*c) + x)*log(b/(2*c) + 
x)/sqrt(-c*e**2*(b/(2*c) + x)**2), True)) + sqrt(-b*d*e - b*e**2*x + c*d** 
2 - c*e**2*x**2)*(-e*g*x**3/(4*c) - x**2*(-7*b*e**3*g/(8*c) + 3*d*e**2*g + 
 e**3*f)/(3*c*e**2) - x*(-5*b*(-7*b*e**3*g/(8*c) + 3*d*e**2*g + e**3*f)/(6 
*c) + 3*d**2*e*g + 3*d*e**2*f + e*g*(-3*b*d*e + 3*c*d**2)/(4*c))/(2*c*e**2 
) - (-3*b*(-5*b*(-7*b*e**3*g/(8*c) + 3*d*e**2*g + e**3*f)/(6*c) + 3*d**2*e 
*g + 3*d*e**2*f + e*g*(-3*b*d*e + 3*c*d**2)/(4*c))/(4*c) + d**3*g + 3*d**2 
*e*f + (-2*b*d*e + 2*c*d**2)*(-7*b*e**3*g/(8*c) + 3*d*e**2*g + e**3*f)/(3* 
c*e**2))/(c*e**2)), Ne(c*e**2, 0)), (-2*(g*(-b*d*e - b*e**2*x + c*d**2)**( 
9/2)/(9*b**4*e**5) + (-b*d*e - b*e**2*x + c*d**2)**(7/2)*(b*d*e*g - b*e**2 
*f - 4*c*d**2*g)/(7*b**4*e**5) + (-b*d*e - b*e**2*x + c*d**2)**(5/2)*(-3*b 
*c*d**3*e*g + 3*b*c*d**2*e**2*f + 6*c**2*d**4*g)/(5*b**4*e**5) + (-b*d*e - 
 b*e**2*x + c*d**2)**(3/2)*(3*b*c**2*d**5*e*g - 3*b*c**2*d**4*e**2*f - ...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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 [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {1}{192} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, {\left (\frac {6 \, e g x}{c} + \frac {8 \, c^{3} e^{5} f + 24 \, c^{3} d e^{4} g - 7 \, b c^{2} e^{5} g}{c^{4} e^{4}}\right )} x + \frac {144 \, c^{3} d e^{4} f - 40 \, b c^{2} e^{5} f + 180 \, c^{3} d^{2} e^{3} g - 156 \, b c^{2} d e^{4} g + 35 \, b^{2} c e^{5} g}{c^{4} e^{4}}\right )} x + \frac {704 \, c^{3} d^{2} e^{3} f - 560 \, b c^{2} d e^{4} f + 120 \, b^{2} c e^{5} f + 576 \, c^{3} d^{3} e^{2} g - 1036 \, b c^{2} d^{2} e^{3} g + 580 \, b^{2} c d e^{4} g - 105 \, b^{3} e^{5} g}{c^{4} e^{4}}\right )} - \frac {5 \, {\left (64 \, c^{4} d^{3} e f - 96 \, b c^{3} d^{2} e^{2} f + 48 \, b^{2} c^{2} d e^{3} f - 8 \, b^{3} c e^{4} f + 48 \, c^{4} d^{4} g - 128 \, b c^{3} d^{3} e g + 120 \, b^{2} c^{2} d^{2} e^{2} g - 48 \, b^{3} c d e^{3} g + 7 \, b^{4} e^{4} g\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{128 \, \sqrt {-c} c^{4} e {\left | e \right |}} \] Input:

integrate((e*x+d)^3*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo 
rithm="giac")
 

Output:

-1/192*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(6*e*g*x/c + (8*c^ 
3*e^5*f + 24*c^3*d*e^4*g - 7*b*c^2*e^5*g)/(c^4*e^4))*x + (144*c^3*d*e^4*f 
- 40*b*c^2*e^5*f + 180*c^3*d^2*e^3*g - 156*b*c^2*d*e^4*g + 35*b^2*c*e^5*g) 
/(c^4*e^4))*x + (704*c^3*d^2*e^3*f - 560*b*c^2*d*e^4*f + 120*b^2*c*e^5*f + 
 576*c^3*d^3*e^2*g - 1036*b*c^2*d^2*e^3*g + 580*b^2*c*d*e^4*g - 105*b^3*e^ 
5*g)/(c^4*e^4)) - 5/128*(64*c^4*d^3*e*f - 96*b*c^3*d^2*e^2*f + 48*b^2*c^2* 
d*e^3*f - 8*b^3*c*e^4*f + 48*c^4*d^4*g - 128*b*c^3*d^3*e*g + 120*b^2*c^2*d 
^2*e^2*g - 48*b^3*c*d*e^3*g + 7*b^4*e^4*g)*log(abs(-b*e^2 + 2*(sqrt(-c*e^2 
)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*abs(e)))/(sqrt( 
-c)*c^4*e*abs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^3}{\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 1300, normalized size of antiderivative = 4.71 \[ \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(105*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d)) 
*b**5*e**5*g - 930*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b* 
e + 2*c*d))*b**4*c*d*e**4*g - 120*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x 
)*i)/sqrt( - b*e + 2*c*d))*b**4*c*e**5*f + 3240*sqrt(c)*asinh((sqrt( - b*e 
 + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**2*d**2*e**3*g + 960*sqrt( 
c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**2*d* 
e**4*f - 5520*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2 
*c*d))*b**2*c**3*d**3*e**2*g - 2880*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e 
*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**3*d**2*e**3*f + 4560*sqrt(c)*asinh((s 
qrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**4*d**4*e*g + 3840* 
sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**4* 
d**3*e**2*f - 1440*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b* 
e + 2*c*d))*c**5*d**5*g - 1920*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i 
)/sqrt( - b*e + 2*c*d))*c**5*d**4*e*f + 105*sqrt(d + e*x)*sqrt(b*e - 2*c*d 
)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**3*c*e**3*g - 580*sqrt 
(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x 
)*b**2*c**2*d*e**2*g - 120*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2 
*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*e**3*f - 70*sqrt(d + e*x)*sqrt( 
b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*c**2*e** 
3*g*x + 1036*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt(...