\(\int \frac {(d+e x)^3 (f+g x)}{(a+b x+c x^2)^{3/2}} \, dx\) [980]

Optimal result

Integrand size = 27, antiderivative size = 319 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^3 (b f-2 a g+(2 c f-b g) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c f-b g) (d+e x)^2 \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {e \left (32 c^3 d^2 f-15 b^3 e^2 g+4 b c e (3 b e f+9 b d g+13 a e g)-8 c^2 (b d (3 e f+2 d g)+4 a e (e f+3 d g))+2 c e \left (8 c^2 d f+5 b^2 e g-4 c (b e f+b d g+3 a e g)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 e \left (5 b^2 e^2 g+8 c^2 d (e f+d g)-4 c e (b e f+3 b d g+a e g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \] Output:

-2*(e*x+d)^3*(b*f-2*a*g+(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+2 
*e*(-b*g+2*c*f)*(e*x+d)^2*(c*x^2+b*x+a)^(1/2)/c/(-4*a*c+b^2)+1/4*e*(32*c^3 
*d^2*f-15*b^3*e^2*g+4*b*c*e*(13*a*e*g+9*b*d*g+3*b*e*f)-8*c^2*(b*d*(2*d*g+3 
*e*f)+4*a*e*(3*d*g+e*f))+2*c*e*(8*c^2*d*f+5*b^2*e*g-4*c*(3*a*e*g+b*d*g+b*e 
*f))*x)*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)+3/8*e*(5*b^2*e^2*g+8*c^2*d*(d 
*g+e*f)-4*c*e*(a*e*g+3*b*d*g+b*e*f))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+ 
b*x+a)^(1/2))/c^(7/2)
 

Mathematica [A] (verified)

Time = 2.79 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {15 b^4 e^3 g x+b^3 e^2 (15 a e g+c x (-12 e f-36 d g+5 e g x))-2 b^2 c e \left (a e (6 e f+18 d g+31 e g x)+c x \left (-12 d^2 g+6 d e (-2 f+g x)+e^2 x (2 f+g x)\right )\right )+8 c^2 \left (2 c^2 d^3 f x+a^2 e^2 (4 e f+12 d g+3 e g x)+a c \left (-2 d^3 g+6 d e^2 x (-f+g x)-6 d^2 e (f+g x)+e^3 x^2 (2 f+g x)\right )\right )+4 b c \left (-13 a^2 e^3 g+2 c^2 d^2 (-3 e f x+d (f-g x))+a c e \left (6 d^2 g-5 e^2 x (-2 f+g x)+6 d e (f+5 g x)\right )\right )}{4 c^3 \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {3 e \left (-5 b^2 e^2 g-8 c^2 d (e f+d g)+4 c e (b e f+3 b d g+a e g)\right ) \log \left (c^3 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{8 c^{7/2}} \] Input:

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

Output:

(15*b^4*e^3*g*x + b^3*e^2*(15*a*e*g + c*x*(-12*e*f - 36*d*g + 5*e*g*x)) - 
2*b^2*c*e*(a*e*(6*e*f + 18*d*g + 31*e*g*x) + c*x*(-12*d^2*g + 6*d*e*(-2*f 
+ g*x) + e^2*x*(2*f + g*x))) + 8*c^2*(2*c^2*d^3*f*x + a^2*e^2*(4*e*f + 12* 
d*g + 3*e*g*x) + a*c*(-2*d^3*g + 6*d*e^2*x*(-f + g*x) - 6*d^2*e*(f + g*x) 
+ e^3*x^2*(2*f + g*x))) + 4*b*c*(-13*a^2*e^3*g + 2*c^2*d^2*(-3*e*f*x + d*( 
f - g*x)) + a*c*e*(6*d^2*g - 5*e^2*x*(-2*f + g*x) + 6*d*e*(f + 5*g*x))))/( 
4*c^3*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]) + (3*e*(-5*b^2*e^2*g - 8*c^2*d 
*(e*f + d*g) + 4*c*e*(b*e*f + 3*b*d*g + a*e*g))*Log[c^3*(b + 2*c*x - 2*Sqr 
t[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(7/2))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1233, 27, 1225, 1092, 219}

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.

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

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2 \int \frac {e (d+e x) \left (d g b^2+4 c d f b+4 a e g b-8 a c e f-12 a c d g+\left (5 e g b^2+8 c^2 d f-4 c (b e f+b d g+3 a e g)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^2 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \int \frac {(d+e x) \left (d g b^2+4 c d f b+4 a e g b-8 a c e f-12 a c d g+\left (5 e g b^2+8 c^2 d f-4 c (b e f+b d g+3 a e g)\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^2 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {e \left (\frac {3 \left (b^2-4 a c\right ) \left (-4 c e (a e g+3 b d g+b e f)+5 b^2 e^2 g+8 c^2 d (d g+e f)\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c (3 a e g+b d g+b e f)+5 b^2 e g+8 c^2 d f\right )-8 c^2 (4 a e (3 d g+e f)+b d (2 d g+3 e f))+4 b c e (13 a e g+9 b d g+3 b e f)-15 b^3 e^2 g+32 c^3 d^2 f\right )}{4 c^2}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^2 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {e \left (\frac {3 \left (b^2-4 a c\right ) \left (-4 c e (a e g+3 b d g+b e f)+5 b^2 e^2 g+8 c^2 d (d g+e f)\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c (3 a e g+b d g+b e f)+5 b^2 e g+8 c^2 d f\right )-8 c^2 (4 a e (3 d g+e f)+b d (2 d g+3 e f))+4 b c e (13 a e g+9 b d g+3 b e f)-15 b^3 e^2 g+32 c^3 d^2 f\right )}{4 c^2}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^2 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {e \left (\frac {3 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (a e g+3 b d g+b e f)+5 b^2 e^2 g+8 c^2 d (d g+e f)\right )}{8 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c (3 a e g+b d g+b e f)+5 b^2 e g+8 c^2 d f\right )-8 c^2 (4 a e (3 d g+e f)+b d (2 d g+3 e f))+4 b c e (13 a e g+9 b d g+3 b e f)-15 b^3 e^2 g+32 c^3 d^2 f\right )}{4 c^2}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^2 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

Input:

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

Output:

(2*(d + e*x)^2*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e 
*g - c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^ 
2]) + (e*(((32*c^3*d^2*f - 15*b^3*e^2*g + 4*b*c*e*(3*b*e*f + 9*b*d*g + 13* 
a*e*g) - 8*c^2*(b*d*(3*e*f + 2*d*g) + 4*a*e*(e*f + 3*d*g)) + 2*c*e*(8*c^2* 
d*f + 5*b^2*e*g - 4*c*(b*e*f + b*d*g + 3*a*e*g))*x)*Sqrt[a + b*x + c*x^2]) 
/(4*c^2) + (3*(b^2 - 4*a*c)*(5*b^2*e^2*g + 8*c^2*d*(e*f + d*g) - 4*c*e*(b* 
e*f + 3*b*d*g + a*e*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^ 
2])])/(8*c^(5/2))))/(c*(b^2 - 4*a*c))
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[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 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

Maple [A] (verified)

Time = 2.41 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.71

method	result	size
risch	\(-\frac {e^{2} \left (-2 c e g x +7 b e g -12 c d g -4 f c e \right ) \sqrt {c \,x^{2}+b x +a}}{4 c^{3}}-\frac {3 c e \left (4 a c \,e^{2} g -5 b^{2} e^{2} g +12 b c d e g +4 b c \,e^{2} f -8 c^{2} d^{2} g -8 c^{2} d e f \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (-4 a b c \,e^{3} g +24 a \,c^{2} d \,e^{2} g +8 a \,c^{2} e^{3} f -7 b^{3} e^{3} g +12 b^{2} c d \,e^{2} g +4 b^{2} c \,e^{3} f -8 c^{3} d^{3} g -24 d^{2} f \,c^{3} e \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )-\frac {16 c^{3} d^{3} f \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {14 a \,b^{2} e^{3} g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 a^{2} c \,e^{3} g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 a b c \,e^{3} f \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {24 a b c d \,e^{2} g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{8 c^{3}}\)	\(545\)
default	\(\frac {2 d^{3} f \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+e^{2} \left (3 d g +e f \right ) \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+3 d e \left (d g +e f \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+d^{2} \left (d g +3 e f \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+g \,e^{3} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )}{4 c}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )\)	\(769\)

Input:

int((e*x+d)^3*(g*x+f)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*e^2*(-2*c*e*g*x+7*b*e*g-12*c*d*g-4*c*e*f)/c^3*(c*x^2+b*x+a)^(1/2)-1/8 
/c^3*(3*c*e*(4*a*c*e^2*g-5*b^2*e^2*g+12*b*c*d*e*g+4*b*c*e^2*f-8*c^2*d^2*g- 
8*c^2*d*e*f)*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b 
/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^( 
1/2)+(c*x^2+b*x+a)^(1/2)))+(-4*a*b*c*e^3*g+24*a*c^2*d*e^2*g+8*a*c^2*e^3*f- 
7*b^3*e^3*g+12*b^2*c*d*e^2*g+4*b^2*c*e^3*f-8*c^3*d^3*g-24*c^3*d^2*e*f)*(-1 
/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))-16*c 
^3*d^3*f*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-14*a*b^2*e^3*g*(2*c*x+b 
)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+8*a^2*c*e^3*g*(2*c*x+b)/(4*a*c-b^2)/(c*x 
^2+b*x+a)^(1/2)+8*a*b*c*e^3*f*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+24 
*a*b*c*d*e^2*g*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 855 vs. \(2 (301) = 602\).

Time = 1.69 (sec) , antiderivative size = 1713, normalized size of antiderivative = 5.37 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/16*(3*((4*(2*(b^2*c^3 - 4*a*c^4)*d*e^2 - (b^3*c^2 - 4*a*b*c^3)*e^3)*f 
+ (8*(b^2*c^3 - 4*a*c^4)*d^2*e - 12*(b^3*c^2 - 4*a*b*c^3)*d*e^2 + (5*b^4*c 
 - 24*a*b^2*c^2 + 16*a^2*c^3)*e^3)*g)*x^2 + 4*(2*(a*b^2*c^2 - 4*a^2*c^3)*d 
*e^2 - (a*b^3*c - 4*a^2*b*c^2)*e^3)*f + (8*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e - 
 12*(a*b^3*c - 4*a^2*b*c^2)*d*e^2 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)* 
e^3)*g + (4*(2*(b^3*c^2 - 4*a*b*c^3)*d*e^2 - (b^4*c - 4*a*b^2*c^2)*e^3)*f 
+ (8*(b^3*c^2 - 4*a*b*c^3)*d^2*e - 12*(b^4*c - 4*a*b^2*c^2)*d*e^2 + (5*b^5 
 - 24*a*b^3*c + 16*a^2*b*c^2)*e^3)*g)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x 
- b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(2*(b^2*c 
^3 - 4*a*c^4)*e^3*g*x^3 + (4*(b^2*c^3 - 4*a*c^4)*e^3*f + (12*(b^2*c^3 - 4* 
a*c^4)*d*e^2 - 5*(b^3*c^2 - 4*a*b*c^3)*e^3)*g)*x^2 - 4*(2*b*c^4*d^3 - 12*a 
*c^4*d^2*e + 6*a*b*c^3*d*e^2 - (3*a*b^2*c^2 - 8*a^2*c^3)*e^3)*f + (16*a*c^ 
4*d^3 - 24*a*b*c^3*d^2*e + 12*(3*a*b^2*c^2 - 8*a^2*c^3)*d*e^2 - (15*a*b^3* 
c - 52*a^2*b*c^2)*e^3)*g - (4*(4*c^5*d^3 - 6*b*c^4*d^2*e + 6*(b^2*c^3 - 2* 
a*c^4)*d*e^2 - (3*b^3*c^2 - 10*a*b*c^3)*e^3)*f - (8*b*c^4*d^3 - 24*(b^2*c^ 
3 - 2*a*c^4)*d^2*e + 12*(3*b^3*c^2 - 10*a*b*c^3)*d*e^2 - (15*b^4*c - 62*a* 
b^2*c^2 + 24*a^2*c^3)*e^3)*g)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^4 - 4*a^2 
*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4 - 4*a*b*c^5)*x), -1/8*(3*((4*(2* 
(b^2*c^3 - 4*a*c^4)*d*e^2 - (b^3*c^2 - 4*a*b*c^3)*e^3)*f + (8*(b^2*c^3 - 4 
*a*c^4)*d^2*e - 12*(b^3*c^2 - 4*a*b*c^3)*d*e^2 + (5*b^4*c - 24*a*b^2*c^...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^3*(g*x+f)/(c*x^2+b*x+a)^(3/2),x, algorithm="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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.73 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} e^{3} g - 4 \, a c^{3} e^{3} g\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac {4 \, b^{2} c^{2} e^{3} f - 16 \, a c^{3} e^{3} f + 12 \, b^{2} c^{2} d e^{2} g - 48 \, a c^{3} d e^{2} g - 5 \, b^{3} c e^{3} g + 20 \, a b c^{2} e^{3} g}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {16 \, c^{4} d^{3} f - 24 \, b c^{3} d^{2} e f + 24 \, b^{2} c^{2} d e^{2} f - 48 \, a c^{3} d e^{2} f - 12 \, b^{3} c e^{3} f + 40 \, a b c^{2} e^{3} f - 8 \, b c^{3} d^{3} g + 24 \, b^{2} c^{2} d^{2} e g - 48 \, a c^{3} d^{2} e g - 36 \, b^{3} c d e^{2} g + 120 \, a b c^{2} d e^{2} g + 15 \, b^{4} e^{3} g - 62 \, a b^{2} c e^{3} g + 24 \, a^{2} c^{2} e^{3} g}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {8 \, b c^{3} d^{3} f - 48 \, a c^{3} d^{2} e f + 24 \, a b c^{2} d e^{2} f - 12 \, a b^{2} c e^{3} f + 32 \, a^{2} c^{2} e^{3} f - 16 \, a c^{3} d^{3} g + 24 \, a b c^{2} d^{2} e g - 36 \, a b^{2} c d e^{2} g + 96 \, a^{2} c^{2} d e^{2} g + 15 \, a b^{3} e^{3} g - 52 \, a^{2} b c e^{3} g}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (8 \, c^{2} d e^{2} f - 4 \, b c e^{3} f + 8 \, c^{2} d^{2} e g - 12 \, b c d e^{2} g + 5 \, b^{2} e^{3} g - 4 \, a c e^{3} g\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {7}{2}}} \] Input:

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

Output:

1/4*(((2*(b^2*c^2*e^3*g - 4*a*c^3*e^3*g)*x/(b^2*c^3 - 4*a*c^4) + (4*b^2*c^ 
2*e^3*f - 16*a*c^3*e^3*f + 12*b^2*c^2*d*e^2*g - 48*a*c^3*d*e^2*g - 5*b^3*c 
*e^3*g + 20*a*b*c^2*e^3*g)/(b^2*c^3 - 4*a*c^4))*x - (16*c^4*d^3*f - 24*b*c 
^3*d^2*e*f + 24*b^2*c^2*d*e^2*f - 48*a*c^3*d*e^2*f - 12*b^3*c*e^3*f + 40*a 
*b*c^2*e^3*f - 8*b*c^3*d^3*g + 24*b^2*c^2*d^2*e*g - 48*a*c^3*d^2*e*g - 36* 
b^3*c*d*e^2*g + 120*a*b*c^2*d*e^2*g + 15*b^4*e^3*g - 62*a*b^2*c*e^3*g + 24 
*a^2*c^2*e^3*g)/(b^2*c^3 - 4*a*c^4))*x - (8*b*c^3*d^3*f - 48*a*c^3*d^2*e*f 
 + 24*a*b*c^2*d*e^2*f - 12*a*b^2*c*e^3*f + 32*a^2*c^2*e^3*f - 16*a*c^3*d^3 
*g + 24*a*b*c^2*d^2*e*g - 36*a*b^2*c*d*e^2*g + 96*a^2*c^2*d*e^2*g + 15*a*b 
^3*e^3*g - 52*a^2*b*c*e^3*g)/(b^2*c^3 - 4*a*c^4))/sqrt(c*x^2 + b*x + a) - 
3/8*(8*c^2*d*e^2*f - 4*b*c*e^3*f + 8*c^2*d^2*e*g - 12*b*c*d*e^2*g + 5*b^2* 
e^3*g - 4*a*c*e^3*g)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) 
 + b))/c^(7/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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