\(\int \frac {(b+2 c x) (d+e x)^3}{(a+b x+c x^2)^{3/2}} \, dx\) [637]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^3}{\sqrt {a+b x+c x^2}}+\frac {3 e^2 (8 c d-3 b e+2 c e x) \sqrt {a+b x+c x^2}}{2 c^2}+\frac {3 e \left (\left (3 b^2-4 a c\right ) e^2+8 c d (c d-b e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 c^{5/2}} \] Output:

-2*(e*x+d)^3/(c*x^2+b*x+a)^(1/2)+3/2*e^2*(2*c*e*x-3*b*e+8*c*d)*(c*x^2+b*x+ 
a)^(1/2)/c^2+3/4*e*((-4*a*c+3*b^2)*e^2+8*c*d*(-b*e+c*d))*arctanh(1/2*(2*c* 
x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)
 

Mathematica [A] (verified)

Time = 1.82 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.29 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-9 b e^3 (a+b x)-2 c^2 \left (2 d^3+6 d^2 e x-6 d e^2 x^2-e^3 x^3\right )+3 c e^2 (b x (8 d-e x)+2 a (4 d+e x))}{2 c^2 \sqrt {a+x (b+c x)}}+\frac {3 e \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{2 c^{5/2}} \] Input:

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

Output:

(-9*b*e^3*(a + b*x) - 2*c^2*(2*d^3 + 6*d^2*e*x - 6*d*e^2*x^2 - e^3*x^3) + 
3*c*e^2*(b*x*(8*d - e*x) + 2*a*(4*d + e*x)))/(2*c^2*Sqrt[a + x*(b + c*x)]) 
 + (3*e*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(Sqrt[c]*x)/ 
(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(2*c^(5/2))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1222, 1166, 27, 1160, 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.

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

\(\Big \downarrow \) 1222

\(\displaystyle 6 e \int \frac {(d+e x)^2}{\sqrt {c x^2+b x+a}}dx-\frac {2 (d+e x)^3}{\sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

\(\displaystyle 6 e \left (\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{2 c^{3/2}}+\frac {3 e \sqrt {a+b x+c x^2} (2 c d-b e)}{c}}{4 c}+\frac {e (d+e x) \sqrt {a+b x+c x^2}}{2 c}\right )-\frac {2 (d+e x)^3}{\sqrt {a+b x+c x^2}}\)

Input:

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

Output:

(-2*(d + e*x)^3)/Sqrt[a + b*x + c*x^2] + 6*e*((e*(d + e*x)*Sqrt[a + b*x + 
c*x^2])/(2*c) + ((3*e*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/c + ((8*c^2*d^2 
 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a 
+ b*x + c*x^2])])/(2*c^(3/2)))/(4*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 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 1166

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[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1222

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)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1)))   Int[(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 
[2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs. \(2(113)=226\).

Time = 1.95 (sec) , antiderivative size = 442, normalized size of antiderivative = 3.43

method	result	size
risch	\(-\frac {e^{2} \left (-2 c e x +5 b e -12 c d \right ) \sqrt {c \,x^{2}+b x +a}}{2 c^{2}}-\frac {3 c e \left (4 a c \,e^{2}-3 b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (24 d \,e^{2} a \,c^{2}-5 b^{3} e^{3}+12 d \,e^{2} b^{2} c -12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {8 a^{2} c \,e^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {10 b^{2} e^{3} a \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {8 d^{3} b \,c^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {24 b d \,e^{2} a c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{4 c^{2}}\)	\(442\)
default	\(\frac {2 b \,d^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+e^{2} \left (b e +6 c d \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 (b e +2 c d \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 (3 b e +2 c d \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 )+2 e^{3} c \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 )\)	\(772\)

Input:

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

Output:

-1/2*e^2*(-2*c*e*x+5*b*e-12*c*d)/c^2*(c*x^2+b*x+a)^(1/2)-1/4/c^2*(3*c*e*(4 
*a*c*e^2-3*b^2*e^2+8*b*c*d*e-8*c^2*d^2)*(-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)))+(24*a*c^2*d*e^2-5*b^ 
3*e^3+12*b^2*c*d*e^2-12*b*c^2*d^2*e-8*c^3*d^3)*(-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))+8*a^2*c*e^3*(2*c*x+b)/(4*a*c 
-b^2)/(c*x^2+b*x+a)^(1/2)-10*b^2*e^3*a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a) 
^(1/2)-8*d^3*b*c^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+24*b*d*e^2*a* 
c*(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. 306 vs. \(2 (113) = 226\).

Time = 0.27 (sec) , antiderivative size = 615, normalized size of antiderivative = 4.77 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (8 \, a c^{2} d^{2} e - 8 \, a b c d e^{2} + {\left (3 \, a b^{2} - 4 \, a^{2} c\right )} e^{3} + {\left (8 \, c^{3} d^{2} e - 8 \, b c^{2} d e^{2} + {\left (3 \, b^{2} c - 4 \, a c^{2}\right )} e^{3}\right )} x^{2} + {\left (8 \, b c^{2} d^{2} e - 8 \, b^{2} c d e^{2} + {\left (3 \, b^{3} - 4 \, a b c\right )} e^{3}\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (2 \, c^{3} e^{3} x^{3} - 4 \, c^{3} d^{3} + 24 \, a c^{2} d e^{2} - 9 \, a b c e^{3} + 3 \, {\left (4 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (4 \, c^{3} d^{2} e - 8 \, b c^{2} d e^{2} + {\left (3 \, b^{2} c - 2 \, a c^{2}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{8 \, {\left (c^{4} x^{2} + b c^{3} x + a c^{3}\right )}}, -\frac {3 \, {\left (8 \, a c^{2} d^{2} e - 8 \, a b c d e^{2} + {\left (3 \, a b^{2} - 4 \, a^{2} c\right )} e^{3} + {\left (8 \, c^{3} d^{2} e - 8 \, b c^{2} d e^{2} + {\left (3 \, b^{2} c - 4 \, a c^{2}\right )} e^{3}\right )} x^{2} + {\left (8 \, b c^{2} d^{2} e - 8 \, b^{2} c d e^{2} + {\left (3 \, b^{3} - 4 \, a b c\right )} e^{3}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (2 \, c^{3} e^{3} x^{3} - 4 \, c^{3} d^{3} + 24 \, a c^{2} d e^{2} - 9 \, a b c e^{3} + 3 \, {\left (4 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (4 \, c^{3} d^{2} e - 8 \, b c^{2} d e^{2} + {\left (3 \, b^{2} c - 2 \, a c^{2}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (c^{4} x^{2} + b c^{3} x + a c^{3}\right )}}\right ] \] Input:

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

Output:

[-1/8*(3*(8*a*c^2*d^2*e - 8*a*b*c*d*e^2 + (3*a*b^2 - 4*a^2*c)*e^3 + (8*c^3 
*d^2*e - 8*b*c^2*d*e^2 + (3*b^2*c - 4*a*c^2)*e^3)*x^2 + (8*b*c^2*d^2*e - 8 
*b^2*c*d*e^2 + (3*b^3 - 4*a*b*c)*e^3)*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*c^3*e^ 
3*x^3 - 4*c^3*d^3 + 24*a*c^2*d*e^2 - 9*a*b*c*e^3 + 3*(4*c^3*d*e^2 - b*c^2* 
e^3)*x^2 - 3*(4*c^3*d^2*e - 8*b*c^2*d*e^2 + (3*b^2*c - 2*a*c^2)*e^3)*x)*sq 
rt(c*x^2 + b*x + a))/(c^4*x^2 + b*c^3*x + a*c^3), -1/4*(3*(8*a*c^2*d^2*e - 
 8*a*b*c*d*e^2 + (3*a*b^2 - 4*a^2*c)*e^3 + (8*c^3*d^2*e - 8*b*c^2*d*e^2 + 
(3*b^2*c - 4*a*c^2)*e^3)*x^2 + (8*b*c^2*d^2*e - 8*b^2*c*d*e^2 + (3*b^3 - 4 
*a*b*c)*e^3)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt 
(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(2*c^3*e^3*x^3 - 4*c^3*d^3 + 24*a*c^2*d* 
e^2 - 9*a*b*c*e^3 + 3*(4*c^3*d*e^2 - b*c^2*e^3)*x^2 - 3*(4*c^3*d^2*e - 8*b 
*c^2*d*e^2 + (3*b^2*c - 2*a*c^2)*e^3)*x)*sqrt(c*x^2 + b*x + a))/(c^4*x^2 + 
 b*c^3*x + a*c^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((2*c*x+b)*(e*x+d)^3/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (113) = 226\).

Time = 0.26 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.79 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} e^{3} - 4 \, a c^{3} e^{3}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {3 \, {\left (4 \, b^{2} c^{2} d e^{2} - 16 \, a c^{3} d e^{2} - b^{3} c e^{3} + 4 \, a b c^{2} e^{3}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac {3 \, {\left (4 \, b^{2} c^{2} d^{2} e - 16 \, a c^{3} d^{2} e - 8 \, b^{3} c d e^{2} + 32 \, a b c^{2} d e^{2} + 3 \, b^{4} e^{3} - 14 \, a b^{2} c e^{3} + 8 \, a^{2} c^{2} e^{3}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac {4 \, b^{2} c^{2} d^{3} - 16 \, a c^{3} d^{3} - 24 \, a b^{2} c d e^{2} + 96 \, a^{2} c^{2} d e^{2} + 9 \, a b^{3} e^{3} - 36 \, a^{2} b c e^{3}}{b^{2} c^{2} - 4 \, a c^{3}}}{2 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (8 \, c^{2} d^{2} e - 8 \, b c d e^{2} + 3 \, b^{2} e^{3} - 4 \, a c e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{4 \, c^{\frac {5}{2}}} \] Input:

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

Output:

1/2*(((2*(b^2*c^2*e^3 - 4*a*c^3*e^3)*x/(b^2*c^2 - 4*a*c^3) + 3*(4*b^2*c^2* 
d*e^2 - 16*a*c^3*d*e^2 - b^3*c*e^3 + 4*a*b*c^2*e^3)/(b^2*c^2 - 4*a*c^3))*x 
 - 3*(4*b^2*c^2*d^2*e - 16*a*c^3*d^2*e - 8*b^3*c*d*e^2 + 32*a*b*c^2*d*e^2 
+ 3*b^4*e^3 - 14*a*b^2*c*e^3 + 8*a^2*c^2*e^3)/(b^2*c^2 - 4*a*c^3))*x - (4* 
b^2*c^2*d^3 - 16*a*c^3*d^3 - 24*a*b^2*c*d*e^2 + 96*a^2*c^2*d*e^2 + 9*a*b^3 
*e^3 - 36*a^2*b*c*e^3)/(b^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a) - 3/4*(8 
*c^2*d^2*e - 8*b*c*d*e^2 + 3*b^2*e^3 - 4*a*c*e^3)*log(abs(2*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(5/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 962, normalized size of antiderivative = 7.46 \[ \int \frac {(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 72*sqrt(a + b*x + c*x**2)*a*b*c*e**3 + 192*sqrt(a + b*x + c*x**2)*a*c* 
*2*d*e**2 + 48*sqrt(a + b*x + c*x**2)*a*c**2*e**3*x - 72*sqrt(a + b*x + c* 
x**2)*b**2*c*e**3*x + 192*sqrt(a + b*x + c*x**2)*b*c**2*d*e**2*x - 24*sqrt 
(a + b*x + c*x**2)*b*c**2*e**3*x**2 - 32*sqrt(a + b*x + c*x**2)*c**3*d**3 
- 96*sqrt(a + b*x + c*x**2)*c**3*d**2*e*x + 96*sqrt(a + b*x + c*x**2)*c**3 
*d*e**2*x**2 + 16*sqrt(a + b*x + c*x**2)*c**3*e**3*x**3 - 48*sqrt(c)*log(( 
2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c*e 
**3 + 36*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4 
*a*c - b**2))*a*b**2*e**3 - 96*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x** 
2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*d*e**2 - 48*sqrt(c)*log((2*sqrt( 
c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*e**3*x + 
96*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a*c**2*d**2*e - 48*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + 
 b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*e**3*x**2 + 36*sqrt(c)*log((2*sqrt( 
c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*e**3*x - 9 
6*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - 
b**2))*b**2*c*d*e**2*x + 36*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) 
+ b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c*e**3*x**2 + 96*sqrt(c)*log((2*sqrt 
(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**2*d**2*e* 
x - 96*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(...