\(\int \frac {x^4 (A+B x)}{(a+b x+c x^2)^{3/2}} \, dx\) [153]

Optimal result

Integrand size = 23, antiderivative size = 280 \[ \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (7 b^2 B-6 A b c-16 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {\left (105 b^4 B-90 A b^3 c-460 a b^2 B c+312 a A b c^2+256 a^2 B c^2-2 c \left (35 b^3 B-30 A b^2 c-116 a b B c+72 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{24 c^4 \left (b^2-4 a c\right )}-\frac {\left (35 b^3 B-30 A b^2 c-60 a b B c+24 a A c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{9/2}} \] Output:

-2*x^3*(a*(-2*A*c+B*b)+(-A*b*c-2*B*a*c+B*b^2)*x)/c/(-4*a*c+b^2)/(c*x^2+b*x 
+a)^(1/2)+1/3*(-6*A*b*c-16*B*a*c+7*B*b^2)*x^2*(c*x^2+b*x+a)^(1/2)/c^2/(-4* 
a*c+b^2)+1/24*(105*B*b^4-90*A*b^3*c-460*B*a*b^2*c+312*A*a*b*c^2+256*B*a^2* 
c^2-2*c*(72*A*a*c^2-30*A*b^2*c-116*B*a*b*c+35*B*b^3)*x)*(c*x^2+b*x+a)^(1/2 
)/c^4/(-4*a*c+b^2)-1/16*(24*A*a*c^2-30*A*b^2*c-60*B*a*b*c+35*B*b^3)*arctan 
h(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 1.58 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.06 \[ \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {c} \left (-256 a^3 B c^2+b^2 x \left (-105 b^3 B+5 b^2 c (18 A-7 B x)-4 c^3 x^2 (3 A+2 B x)+2 b c^2 x (15 A+7 B x)\right )+a \left (-105 b^4 B+16 c^4 x^3 (3 A+2 B x)-8 b c^3 x^2 (15 A+7 B x)+4 b^2 c^2 x (-93 A+43 B x)+10 b^3 c (9 A+53 B x)\right )+4 a^2 c \left (115 b^2 B+4 c^2 x (9 A-8 B x)-2 b c (39 A+61 B x)\right )\right )-3 \left (b^2-4 a c\right ) \left (35 b^3 B-30 A b^2 c-60 a b B c+24 a A c^2\right ) \sqrt {a+x (b+c x)} \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{48 c^{9/2} \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}} \] Input:

Integrate[(x^4*(A + B*x))/(a + b*x + c*x^2)^(3/2),x]
 

Output:

(2*Sqrt[c]*(-256*a^3*B*c^2 + b^2*x*(-105*b^3*B + 5*b^2*c*(18*A - 7*B*x) - 
4*c^3*x^2*(3*A + 2*B*x) + 2*b*c^2*x*(15*A + 7*B*x)) + a*(-105*b^4*B + 16*c 
^4*x^3*(3*A + 2*B*x) - 8*b*c^3*x^2*(15*A + 7*B*x) + 4*b^2*c^2*x*(-93*A + 4 
3*B*x) + 10*b^3*c*(9*A + 53*B*x)) + 4*a^2*c*(115*b^2*B + 4*c^2*x*(9*A - 8* 
B*x) - 2*b*c*(39*A + 61*B*x))) - 3*(b^2 - 4*a*c)*(35*b^3*B - 30*A*b^2*c - 
60*a*b*B*c + 24*a*A*c^2)*Sqrt[a + x*(b + c*x)]*Log[b + 2*c*x - 2*Sqrt[c]*S 
qrt[a + x*(b + c*x)]])/(48*c^(9/2)*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1233, 27, 1236, 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.

\(\displaystyle \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {2 \int \frac {x^2 \left (6 a (b B-2 A c)+\left (7 B b^2-6 A c b-16 a B c\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {x^2 \left (6 a (b B-2 A c)+\left (7 B b^2-6 A c b-16 a B c\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\frac {\int -\frac {x \left (4 a \left (7 B b^2-6 A c b-16 a B c\right )+\left (35 B b^3-30 A c b^2-116 a B c b+72 a A c^2\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {x^2 \sqrt {a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c}}{c \left (b^2-4 a c\right )}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c}-\frac {\int \frac {x \left (4 a \left (7 B b^2-6 A c b-16 a B c\right )+\left (35 B b^3-30 A c b^2-116 a B c b+72 a A c^2\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}}{c \left (b^2-4 a c\right )}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c}-\frac {\frac {3 \left (b^2-4 a c\right ) \left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}-\frac {\sqrt {a+b x+c x^2} \left (256 a^2 B c^2-2 c x \left (72 a A c^2-116 a b B c-30 A b^2 c+35 b^3 B\right )+312 a A b c^2-460 a b^2 B c-90 A b^3 c+105 b^4 B\right )}{4 c^2}}{6 c}}{c \left (b^2-4 a c\right )}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c}-\frac {\frac {3 \left (b^2-4 a c\right ) \left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}-\frac {\sqrt {a+b x+c x^2} \left (256 a^2 B c^2-2 c x \left (72 a A c^2-116 a b B c-30 A b^2 c+35 b^3 B\right )+312 a A b c^2-460 a b^2 B c-90 A b^3 c+105 b^4 B\right )}{4 c^2}}{6 c}}{c \left (b^2-4 a c\right )}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {x^2 \sqrt {a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c}-\frac {\frac {3 \left (b^2-4 a c\right ) \left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (256 a^2 B c^2-2 c x \left (72 a A c^2-116 a b B c-30 A b^2 c+35 b^3 B\right )+312 a A b c^2-460 a b^2 B c-90 A b^3 c+105 b^4 B\right )}{4 c^2}}{6 c}}{c \left (b^2-4 a c\right )}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\)

Input:

Int[(x^4*(A + B*x))/(a + b*x + c*x^2)^(3/2),x]
 

Output:

(-2*x^3*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)* 
Sqrt[a + b*x + c*x^2]) + (((7*b^2*B - 6*A*b*c - 16*a*B*c)*x^2*Sqrt[a + b*x 
 + c*x^2])/(3*c) - (-1/4*((105*b^4*B - 90*A*b^3*c - 460*a*b^2*B*c + 312*a* 
A*b*c^2 + 256*a^2*B*c^2 - 2*c*(35*b^3*B - 30*A*b^2*c - 116*a*b*B*c + 72*a* 
A*c^2)*x)*Sqrt[a + b*x + c*x^2])/c^2 + (3*(b^2 - 4*a*c)*(35*b^3*B - 30*A*b 
^2*c - 60*a*b*B*c + 24*a*A*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b* 
x + c*x^2])])/(8*c^(5/2)))/(6*c))/(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])
 

rule 1236

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[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.60

method	result	size
risch	\(-\frac {\left (-8 B \,c^{2} x^{2}-12 A \,c^{2} x +22 B b c x +42 A b c +40 a B c -57 B \,b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{24 c^{4}}-\frac {c \left (24 A a \,c^{2}-30 A \,b^{2} c -60 B a b c +35 B \,b^{3}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (-8 A a b \,c^{2}-14 A \,b^{3} c -16 B \,a^{2} c^{2}-12 B a \,b^{2} c +19 B \,b^{4}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {38 B a \,b^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {16 a^{2} A \,c^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {28 A a \,b^{2} c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {56 a^{2} b B c \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{16 c^{4}}\)	\(447\)
default	\(A \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 )+B \left (\frac {x^{4}}{3 c \sqrt {c \,x^{2}+b x +a}}-\frac {7 b \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 )}{6 c}-\frac {4 a \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 c}\right )\)	\(903\)

Input:

int(x^4*(B*x+A)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/24*(-8*B*c^2*x^2-12*A*c^2*x+22*B*b*c*x+42*A*b*c+40*B*a*c-57*B*b^2)/c^4* 
(c*x^2+b*x+a)^(1/2)-1/16/c^4*(c*(24*A*a*c^2-30*A*b^2*c-60*B*a*b*c+35*B*b^3 
)*(-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)))+(-8*A*a*b*c^2-14*A*b^3*c-16*B*a^2*c^2-12*B*a*b^2*c+19*B*b^ 
4)*(-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) 
)+38*B*a*b^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+16*a^2*A*c^2*(2*c*x 
+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-28*A*a*b^2*c*(2*c*x+b)/(4*a*c-b^2)/(c* 
x^2+b*x+a)^(1/2)-56*a^2*b*B*c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1035, normalized size of antiderivative = 3.70 \[ \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(B*x+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

[1/96*(3*(35*B*a*b^5 - 96*A*a^3*c^3 + 48*(5*B*a^3*b + 3*A*a^2*b^2)*c^2 + ( 
35*B*b^5*c - 96*A*a^2*c^4 + 48*(5*B*a^2*b + 3*A*a*b^2)*c^3 - 10*(20*B*a*b^ 
3 + 3*A*b^4)*c^2)*x^2 - 10*(20*B*a^2*b^3 + 3*A*a*b^4)*c + (35*B*b^6 - 96*A 
*a^2*b*c^3 + 48*(5*B*a^2*b^2 + 3*A*a*b^3)*c^2 - 10*(20*B*a*b^4 + 3*A*b^5)* 
c)*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*(105*B*a*b^4*c + 8*(B*b^2*c^4 - 4*B*a*c^5)*x 
^4 + 8*(32*B*a^3 + 39*A*a^2*b)*c^3 - 2*(7*B*b^3*c^3 + 24*A*a*c^5 - 2*(14*B 
*a*b + 3*A*b^2)*c^4)*x^3 - 10*(46*B*a^2*b^2 + 9*A*a*b^3)*c^2 + (35*B*b^4*c 
^2 + 8*(16*B*a^2 + 15*A*a*b)*c^4 - 2*(86*B*a*b^2 + 15*A*b^3)*c^3)*x^2 + (1 
05*B*b^5*c - 144*A*a^2*c^4 + 4*(122*B*a^2*b + 93*A*a*b^2)*c^3 - 10*(53*B*a 
*b^3 + 9*A*b^4)*c^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^5 - 4*a^2*c^6 + (b 
^2*c^6 - 4*a*c^7)*x^2 + (b^3*c^5 - 4*a*b*c^6)*x), 1/48*(3*(35*B*a*b^5 - 96 
*A*a^3*c^3 + 48*(5*B*a^3*b + 3*A*a^2*b^2)*c^2 + (35*B*b^5*c - 96*A*a^2*c^4 
 + 48*(5*B*a^2*b + 3*A*a*b^2)*c^3 - 10*(20*B*a*b^3 + 3*A*b^4)*c^2)*x^2 - 1 
0*(20*B*a^2*b^3 + 3*A*a*b^4)*c + (35*B*b^6 - 96*A*a^2*b*c^3 + 48*(5*B*a^2* 
b^2 + 3*A*a*b^3)*c^2 - 10*(20*B*a*b^4 + 3*A*b^5)*c)*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*( 
105*B*a*b^4*c + 8*(B*b^2*c^4 - 4*B*a*c^5)*x^4 + 8*(32*B*a^3 + 39*A*a^2*b)* 
c^3 - 2*(7*B*b^3*c^3 + 24*A*a*c^5 - 2*(14*B*a*b + 3*A*b^2)*c^4)*x^3 - 10*( 
46*B*a^2*b^2 + 9*A*a*b^3)*c^2 + (35*B*b^4*c^2 + 8*(16*B*a^2 + 15*A*a*b)...
 

Sympy [F]

\[ \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**4*(B*x+A)/(c*x**2+b*x+a)**(3/2),x)
 

Output:

Integral(x**4*(A + B*x)/(a + b*x + c*x**2)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^4*(B*x+A)/(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 = 0.26 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.30 \[ \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (2 \, {\left (\frac {4 \, {\left (B b^{2} c^{3} - 4 \, B a c^{4}\right )} x}{b^{2} c^{4} - 4 \, a c^{5}} - \frac {7 \, B b^{3} c^{2} - 28 \, B a b c^{3} - 6 \, A b^{2} c^{3} + 24 \, A a c^{4}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac {35 \, B b^{4} c - 172 \, B a b^{2} c^{2} - 30 \, A b^{3} c^{2} + 128 \, B a^{2} c^{3} + 120 \, A a b c^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac {105 \, B b^{5} - 530 \, B a b^{3} c - 90 \, A b^{4} c + 488 \, B a^{2} b c^{2} + 372 \, A a b^{2} c^{2} - 144 \, A a^{2} c^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac {105 \, B a b^{4} - 460 \, B a^{2} b^{2} c - 90 \, A a b^{3} c + 256 \, B a^{3} c^{2} + 312 \, A a^{2} b c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}{24 \, \sqrt {c x^{2} + b x + a}} + \frac {{\left (35 \, B b^{3} - 60 \, B a b c - 30 \, A b^{2} c + 24 \, A a c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{16 \, c^{\frac {9}{2}}} \] Input:

integrate(x^4*(B*x+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

1/24*(((2*(4*(B*b^2*c^3 - 4*B*a*c^4)*x/(b^2*c^4 - 4*a*c^5) - (7*B*b^3*c^2 
- 28*B*a*b*c^3 - 6*A*b^2*c^3 + 24*A*a*c^4)/(b^2*c^4 - 4*a*c^5))*x + (35*B* 
b^4*c - 172*B*a*b^2*c^2 - 30*A*b^3*c^2 + 128*B*a^2*c^3 + 120*A*a*b*c^3)/(b 
^2*c^4 - 4*a*c^5))*x + (105*B*b^5 - 530*B*a*b^3*c - 90*A*b^4*c + 488*B*a^2 
*b*c^2 + 372*A*a*b^2*c^2 - 144*A*a^2*c^3)/(b^2*c^4 - 4*a*c^5))*x + (105*B* 
a*b^4 - 460*B*a^2*b^2*c - 90*A*a*b^3*c + 256*B*a^3*c^2 + 312*A*a^2*b*c^2)/ 
(b^2*c^4 - 4*a*c^5))/sqrt(c*x^2 + b*x + a) + 1/16*(35*B*b^3 - 60*B*a*b*c - 
 30*A*b^2*c + 24*A*a*c^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sq 
rt(c) + b))/c^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:

int((x^4*(A + B*x))/(a + b*x + c*x^2)^(3/2),x)
 

Output:

int((x^4*(A + B*x))/(a + b*x + c*x^2)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 1083, normalized size of antiderivative = 3.87 \[ \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x^4*(B*x+A)/(c*x^2+b*x+a)^(3/2),x)
 

Output:

( - 1136*sqrt(a + b*x + c*x**2)*a**3*b*c**3 + 288*sqrt(a + b*x + c*x**2)*a 
**3*c**4*x + 1100*sqrt(a + b*x + c*x**2)*a**2*b**3*c**2 - 1720*sqrt(a + b* 
x + c*x**2)*a**2*b**2*c**3*x - 496*sqrt(a + b*x + c*x**2)*a**2*b*c**4*x**2 
 + 96*sqrt(a + b*x + c*x**2)*a**2*c**5*x**3 - 210*sqrt(a + b*x + c*x**2)*a 
*b**5*c + 1240*sqrt(a + b*x + c*x**2)*a*b**4*c**2*x + 404*sqrt(a + b*x + c 
*x**2)*a*b**3*c**3*x**2 - 136*sqrt(a + b*x + c*x**2)*a*b**2*c**4*x**3 + 64 
*sqrt(a + b*x + c*x**2)*a*b*c**5*x**4 - 210*sqrt(a + b*x + c*x**2)*b**6*c* 
x - 70*sqrt(a + b*x + c*x**2)*b**5*c**2*x**2 + 28*sqrt(a + b*x + c*x**2)*b 
**4*c**3*x**3 - 16*sqrt(a + b*x + c*x**2)*b**3*c**4*x**4 - 288*sqrt(c)*log 
((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*c 
**3 + 1152*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt 
(4*a*c - b**2))*a**3*b**2*c**2 - 288*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + 
 c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**3*x - 288*sqrt(c)*log( 
(2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*c* 
*4*x**2 - 690*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/s 
qrt(4*a*c - b**2))*a**2*b**4*c + 1152*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*b**3*c**2*x + 1152*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*b**2*c**3*x**2 + 105*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**6 - 690*sqrt(c)*log((2*sqrt(c)*sqrt(a...