\(\int \frac {(d+e x)^2 (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\) [182]

Optimal result

Integrand size = 44, antiderivative size = 168 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 (b e g-c (e f+d g)) (d+e x)}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^2 e^2}-\frac {(2 c e f+4 c d g-3 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c} (d+e x) (-3 b e g+c (2 e f+3 d g-e g x))+(2 c e f+4 c d g-3 b e g) \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{c^{5/2} e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

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

Output:

(Sqrt[c]*(d + e*x)*(-3*b*e*g + c*(2*e*f + 3*d*g - e*g*x)) + (2*c*e*f + 4*c 
*d*g - 3*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])])/(c^(5/2)*e^2*Sqrt[(d + e*x)*(-(b*e) + 
 c*(d - e*x))])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1211, 25, 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.

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

\(\Big \downarrow \) 1211

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

(-2*(b*e*g - c*(e*f + d*g))*(d + e*x))/(c^2*e^2*Sqrt[d*(c*d - b*e) - b*e^2 
*x - c*e^2*x^2]) - (-((g*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/e) + ( 
(2*c*e*f + 4*c*d*g - 3*b*e*g)*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))/(c^2*e)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 1211

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*( 
e*f + d*g) - b*e*g)^n*((d + e*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c* 
x^2])), x] + Simp[1/(c^(m + n - 1)*e^(n - 2))   Int[ExpandToSum[((2*c*d - b 
*e)^(m - 1)*(c*(e*f + d*g) - b*e*g)^n - c^(m + n - 1)*e^n*(d + e*x)^(m - 1) 
*(f + g*x)^n)/(c*d - b*e - c*e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; Free 
Q[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m, 0] 
&& IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(835\) vs. \(2(158)=316\).

Time = 3.01 (sec) , antiderivative size = 836, normalized size of antiderivative = 4.98

method	result	size
default	\(\frac {2 d^{2} f \left (-2 c \,e^{2} x -b \,e^{2}\right )}{\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}+e \left (2 d g +e f \right ) \left (\frac {x}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (\frac {1}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c}-\frac {\arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{c \,e^{2} \sqrt {c \,e^{2}}}\right )+d \left (d g +2 e f \right ) \left (\frac {1}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )+e^{2} g \left (-\frac {x^{2}}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {3 b \left (\frac {x}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (\frac {1}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c}-\frac {\arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2 c}+\frac {2 \left (-b d e +c \,d^{2}\right ) \left (\frac {1}{c \,e^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{c \,e^{2}}\right )\)	\(836\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.99 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (2 \, {\left (c^{2} d e - b c e^{2}\right )} f + {\left (4 \, c^{2} d^{2} - 7 \, b c d e + 3 \, b^{2} e^{2}\right )} g - {\left (2 \, c^{2} e^{2} f + {\left (4 \, c^{2} d e - 3 \, b c e^{2}\right )} g\right )} x\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (c^{2} e g x - 2 \, c^{2} e f - 3 \, {\left (c^{2} d - b c e\right )} g\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{4 \, {\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )}}, -\frac {{\left (2 \, {\left (c^{2} d e - b c e^{2}\right )} f + {\left (4 \, c^{2} d^{2} - 7 \, b c d e + 3 \, b^{2} e^{2}\right )} g - {\left (2 \, c^{2} e^{2} f + {\left (4 \, c^{2} d e - 3 \, b c e^{2}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (c^{2} e g x - 2 \, c^{2} e f - 3 \, {\left (c^{2} d - b c e\right )} g\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{2 \, {\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )}}\right ] \] Input:

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

Output:

[-1/4*((2*(c^2*d*e - b*c*e^2)*f + (4*c^2*d^2 - 7*b*c*d*e + 3*b^2*e^2)*g - 
(2*c^2*e^2*f + (4*c^2*d*e - 3*b*c*e^2)*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*(c^2*e*g*x - 2*c^2*e*f - 
3*(c^2*d - b*c*e)*g)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^3* 
x - c^4*d*e^2 + b*c^3*e^3), -1/2*((2*(c^2*d*e - b*c*e^2)*f + (4*c^2*d^2 - 
7*b*c*d*e + 3*b^2*e^2)*g - (2*c^2*e^2*f + (4*c^2*d*e - 3*b*c*e^2)*g)*x)*sq 
rt(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*(c^2*e*g*x - 
2*c^2*e*f - 3*(c^2*d - b*c*e)*g)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e 
))/(c^4*e^3*x - c^4*d*e^2 + b*c^3*e^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),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. 439 vs. \(2 (158) = 316\).

Time = 0.36 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.61 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} g}{c^{2} e^{2}} + \frac {{\left (2 \, c e f + 4 \, c d g - 3 \, b e g\right )} \log \left ({\left | -b \sqrt {-c} c^{2} d^{2} e^{2} + 2 \, b^{2} \sqrt {-c} c d e^{3} - b^{3} \sqrt {-c} e^{4} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} c^{3} d^{2} {\left | e \right |} + 6 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b c^{2} d e {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b^{2} c e^{2} {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} \sqrt {-c} c^{2} d e + 5 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b \sqrt {-c} c e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} c^{2} {\left | e \right |} \right |}\right )}{6 \, \sqrt {-c} c^{2} e {\left | e \right |}} \] Input:

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

Output:

sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*g/(c^2*e^2) + 1/6*(2*c*e*f + 4* 
c*d*g - 3*b*e*g)*log(abs(-b*sqrt(-c)*c^2*d^2*e^2 + 2*b^2*sqrt(-c)*c*d*e^3 
- b^3*sqrt(-c)*e^4 - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 
 - b*d*e))*c^3*d^2*abs(e) + 6*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x 
+ c*d^2 - b*d*e))*b*c^2*d*e*abs(e) - 4*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - 
 b*e^2*x + c*d^2 - b*d*e))*b^2*c*e^2*abs(e) - 4*(sqrt(-c*e^2)*x - sqrt(-c* 
e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*sqrt(-c)*c^2*d*e + 5*(sqrt(-c*e^2)*x 
 - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*b*sqrt(-c)*c*e^2 + 2*(sqr 
t(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*c^2*abs(e)))/( 
sqrt(-c)*c^2*e*abs(e))
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 551, normalized size of antiderivative = 3.28 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {i \left (12 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b^{2} e^{2} g -40 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b c d e g -8 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b c \,e^{2} f +32 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) c^{2} d^{2} g +16 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) c^{2} d e f -9 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b^{2} e^{2} g +28 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b c d e g +8 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, b c \,e^{2} f -20 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, c^{2} d^{2} g -16 \sqrt {c}\, \sqrt {-c e x -b e +c d}\, c^{2} d e f -12 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, b c e g +12 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{2} d g +8 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{2} e f -4 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, c^{2} e g x \right )}{4 \sqrt {-c e x -b e +c d}\, c^{3} e^{2} \left (b e -2 c d \right )} \] Input:

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

Output:

(i*(12*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*e**2*g - 40*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*d*e*g 
- 8*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*e**2*f + 32*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))*c**2*d**2*g + 
16*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))*c**2*d*e*f - 9*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b 
**2*e**2*g + 28*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c*d*e*g + 8*sqrt(c)*s 
qrt( - b*e + c*d - c*e*x)*b*c*e**2*f - 20*sqrt(c)*sqrt( - b*e + c*d - c*e* 
x)*c**2*d**2*g - 16*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**2*d*e*f - 12*sqr 
t(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*b*c*e*g + 12*sqrt(d + e* 
x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*c**2*d*g + 8*sqrt(d + e*x)*sqrt( 
b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*c**2*e*f - 4*sqrt(d + e*x)*sqrt(b*e - 2* 
c*d)*sqrt( - b*e + 2*c*d)*c**2*e*g*x))/(4*sqrt( - b*e + c*d - c*e*x)*c**3* 
e**2*(b*e - 2*c*d))