\(\int \frac {x^2 (A+B x)}{(a+b x+c x^2)^{5/2}} \, dx\) [164]

Optimal result

Integrand size = 23, antiderivative size = 94 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 x^2 (A b-2 a B-(b B-2 A c) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (A b-2 a B) (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \] Output:

-2/3*x^2*(A*b-2*B*a-(-2*A*c+B*b)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)+8/3*( 
A*b-2*B*a)*(b*x+2*a)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)
 

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.17 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-16 a^3 B+b^2 x^2 (3 A b+b B x+2 A c x)+8 a^2 (A b-3 B x (b+c x))+2 a x \left (-3 b B x (b+2 c x)+A \left (6 b^2+6 b c x+4 c^2 x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \] Input:

Integrate[(x^2*(A + B*x))/(a + b*x + c*x^2)^(5/2),x]
 

Output:

(2*(-16*a^3*B + b^2*x^2*(3*A*b + b*B*x + 2*A*c*x) + 8*a^2*(A*b - 3*B*x*(b 
+ c*x)) + 2*a*x*(-3*b*B*x*(b + 2*c*x) + A*(6*b^2 + 6*b*c*x + 4*c^2*x^2)))) 
/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1227, 1158}

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.

\(\displaystyle \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1227

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

\(\Big \downarrow \) 1158

\(\displaystyle \frac {8 (2 a+b x) (A b-2 a B)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 x^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

Input:

Int[(x^2*(A + B*x))/(a + b*x + c*x^2)^(5/2),x]
 

Output:

(-2*x^2*(A*b - 2*a*B - (b*B - 2*A*c)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2 
)^(3/2)) + (8*(A*b - 2*a*B)*(2*a + b*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + 
 c*x^2])
 

Defintions of rubi rules used

rule 1158

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x 
+ c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1227

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[m*((b*( 
e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c)))   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[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.40

method	result	size
trager	\(\frac {\frac {16}{3} A a \,c^{2} x^{3}+\frac {4}{3} A \,b^{2} c \,x^{3}-8 B a b c \,x^{3}+\frac {2}{3} x^{3} B \,b^{3}+8 A a b c \,x^{2}+2 A \,b^{3} x^{2}-16 B \,a^{2} c \,x^{2}-4 B a \,b^{2} x^{2}+8 A a \,b^{2} x -16 B \,a^{2} b x +\frac {16}{3} A \,a^{2} b -\frac {32}{3} B \,a^{3}}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(132\)
gosper	\(\frac {\frac {16}{3} A a \,c^{2} x^{3}+\frac {4}{3} A \,b^{2} c \,x^{3}-8 B a b c \,x^{3}+\frac {2}{3} x^{3} B \,b^{3}+8 A a b c \,x^{2}+2 A \,b^{3} x^{2}-16 B \,a^{2} c \,x^{2}-4 B a \,b^{2} x^{2}+8 A a \,b^{2} x -16 B \,a^{2} b x +\frac {16}{3} A \,a^{2} b -\frac {32}{3} B \,a^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\)	\(141\)
orering	\(\frac {\frac {16}{3} A a \,c^{2} x^{3}+\frac {4}{3} A \,b^{2} c \,x^{3}-8 B a b c \,x^{3}+\frac {2}{3} x^{3} B \,b^{3}+8 A a b c \,x^{2}+2 A \,b^{3} x^{2}-16 B \,a^{2} c \,x^{2}-4 B a \,b^{2} x^{2}+8 A a \,b^{2} x -16 B \,a^{2} b x +\frac {16}{3} A \,a^{2} b -\frac {32}{3} B \,a^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\)	\(141\)
default	\(A \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+B \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{2 c}+\frac {2 a \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{c}\right )\)	\(503\)

Input:

int(x^2*(B*x+A)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/3*(8*A*a*c^2*x^3+2*A*b^2*c*x^3-12*B*a*b*c*x^3+B*b^3*x^3+12*A*a*b*c*x^2+3 
*A*b^3*x^2-24*B*a^2*c*x^2-6*B*a*b^2*x^2+12*A*a*b^2*x-24*B*a^2*b*x+8*A*a^2* 
b-16*B*a^3)/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (87) = 174\).

Time = 0.46 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.64 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (16 \, B a^{3} - 8 \, A a^{2} b - {\left (B b^{3} + 8 \, A a c^{2} - 2 \, {\left (6 \, B a b - A b^{2}\right )} c\right )} x^{3} + 3 \, {\left (2 \, B a b^{2} - A b^{3} + 4 \, {\left (2 \, B a^{2} - A a b\right )} c\right )} x^{2} + 12 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \] Input:

integrate(x^2*(B*x+A)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

-2/3*(16*B*a^3 - 8*A*a^2*b - (B*b^3 + 8*A*a*c^2 - 2*(6*B*a*b - A*b^2)*c)*x 
^3 + 3*(2*B*a*b^2 - A*b^3 + 4*(2*B*a^2 - A*a*b)*c)*x^2 + 12*(2*B*a^2*b - A 
*a*b^2)*x)*sqrt(c*x^2 + b*x + a)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^ 
4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b* 
c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 1 
6*a^3*b*c^2)*x)
 

Sympy [F]

\[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {x^{2} \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**2*(B*x+A)/(c*x**2+b*x+a)**(5/2),x)
                                                                                    
                                                                                    
 

Output:

Integral(x**2*(A + B*x)/(a + b*x + c*x**2)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(B*x+A)/(c*x^2+b*x+a)^(5/2),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. 195 vs. \(2 (87) = 174\).

Time = 0.26 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.07 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {{\left (B b^{3} - 12 \, B a b c + 2 \, A b^{2} c + 8 \, A a c^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} - \frac {3 \, {\left (2 \, B a b^{2} - A b^{3} + 8 \, B a^{2} c - 4 \, A a b c\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {12 \, {\left (2 \, B a^{2} b - A a b^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {8 \, {\left (2 \, B a^{3} - A a^{2} b\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:

integrate(x^2*(B*x+A)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

2/3*((((B*b^3 - 12*B*a*b*c + 2*A*b^2*c + 8*A*a*c^2)*x/(b^4 - 8*a*b^2*c + 1 
6*a^2*c^2) - 3*(2*B*a*b^2 - A*b^3 + 8*B*a^2*c - 4*A*a*b*c)/(b^4 - 8*a*b^2* 
c + 16*a^2*c^2))*x - 12*(2*B*a^2*b - A*a*b^2)/(b^4 - 8*a*b^2*c + 16*a^2*c^ 
2))*x - 8*(2*B*a^3 - A*a^2*b)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b*x 
 + a)^(3/2)
 

Mupad [B] (verification not implemented)

Time = 10.90 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.39 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2\,\left (-16\,B\,a^3-24\,B\,a^2\,b\,x+8\,A\,a^2\,b-24\,B\,a^2\,c\,x^2-6\,B\,a\,b^2\,x^2+12\,A\,a\,b^2\,x-12\,B\,a\,b\,c\,x^3+12\,A\,a\,b\,c\,x^2+8\,A\,a\,c^2\,x^3+B\,b^3\,x^3+3\,A\,b^3\,x^2+2\,A\,b^2\,c\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \] Input:

int((x^2*(A + B*x))/(a + b*x + c*x^2)^(5/2),x)
 

Output:

(2*(3*A*b^3*x^2 - 16*B*a^3 + B*b^3*x^3 + 8*A*a^2*b + 12*A*a*b^2*x - 24*B*a 
^2*b*x - 6*B*a*b^2*x^2 + 8*A*a*c^2*x^3 - 24*B*a^2*c*x^2 + 2*A*b^2*c*x^3 + 
12*A*a*b*c*x^2 - 12*B*a*b*c*x^3))/(3*(4*a*c - b^2)^2*(a + b*x + c*x^2)^(3/ 
2))
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 503, normalized size of antiderivative = 5.35 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-\frac {16 \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{2}}{3}-8 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c^{2} x -8 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{3} x^{2}+\frac {16 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{4} x^{3}}{3}-2 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c^{2} x^{2}-\frac {20 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{3} x^{3}}{3}+\frac {2 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{2} x^{3}}{3}+\frac {16 \sqrt {c}\, a^{4} c^{2}}{3}-12 \sqrt {c}\, a^{3} b^{2} c +\frac {32 \sqrt {c}\, a^{3} b \,c^{2} x}{3}+\frac {32 \sqrt {c}\, a^{3} c^{3} x^{2}}{3}+\frac {10 \sqrt {c}\, a^{2} b^{4}}{3}-24 \sqrt {c}\, a^{2} b^{3} c x -\frac {56 \sqrt {c}\, a^{2} b^{2} c^{2} x^{2}}{3}+\frac {32 \sqrt {c}\, a^{2} b \,c^{3} x^{3}}{3}+\frac {16 \sqrt {c}\, a^{2} c^{4} x^{4}}{3}+\frac {20 \sqrt {c}\, a \,b^{5} x}{3}-\frac {16 \sqrt {c}\, a \,b^{4} c \,x^{2}}{3}-24 \sqrt {c}\, a \,b^{3} c^{2} x^{3}-12 \sqrt {c}\, a \,b^{2} c^{3} x^{4}+\frac {10 \sqrt {c}\, b^{6} x^{2}}{3}+\frac {20 \sqrt {c}\, b^{5} c \,x^{3}}{3}+\frac {10 \sqrt {c}\, b^{4} c^{2} x^{4}}{3}}{c^{2} \left (16 a^{2} c^{4} x^{4}-8 a \,b^{2} c^{3} x^{4}+b^{4} c^{2} x^{4}+32 a^{2} b \,c^{3} x^{3}-16 a \,b^{3} c^{2} x^{3}+2 b^{5} c \,x^{3}+32 a^{3} c^{3} x^{2}-6 a \,b^{4} c \,x^{2}+b^{6} x^{2}+32 a^{3} b \,c^{2} x -16 a^{2} b^{3} c x +2 a \,b^{5} x +16 a^{4} c^{2}-8 a^{3} b^{2} c +a^{2} b^{4}\right )} \] Input:

int(x^2*(B*x+A)/(c*x^2+b*x+a)^(5/2),x)
 

Output:

(2*( - 8*sqrt(a + b*x + c*x**2)*a**3*b*c**2 - 12*sqrt(a + b*x + c*x**2)*a* 
*2*b**2*c**2*x - 12*sqrt(a + b*x + c*x**2)*a**2*b*c**3*x**2 + 8*sqrt(a + b 
*x + c*x**2)*a**2*c**4*x**3 - 3*sqrt(a + b*x + c*x**2)*a*b**3*c**2*x**2 - 
10*sqrt(a + b*x + c*x**2)*a*b**2*c**3*x**3 + sqrt(a + b*x + c*x**2)*b**4*c 
**2*x**3 + 8*sqrt(c)*a**4*c**2 - 18*sqrt(c)*a**3*b**2*c + 16*sqrt(c)*a**3* 
b*c**2*x + 16*sqrt(c)*a**3*c**3*x**2 + 5*sqrt(c)*a**2*b**4 - 36*sqrt(c)*a* 
*2*b**3*c*x - 28*sqrt(c)*a**2*b**2*c**2*x**2 + 16*sqrt(c)*a**2*b*c**3*x**3 
 + 8*sqrt(c)*a**2*c**4*x**4 + 10*sqrt(c)*a*b**5*x - 8*sqrt(c)*a*b**4*c*x** 
2 - 36*sqrt(c)*a*b**3*c**2*x**3 - 18*sqrt(c)*a*b**2*c**3*x**4 + 5*sqrt(c)* 
b**6*x**2 + 10*sqrt(c)*b**5*c*x**3 + 5*sqrt(c)*b**4*c**2*x**4))/(3*c**2*(1 
6*a**4*c**2 - 8*a**3*b**2*c + 32*a**3*b*c**2*x + 32*a**3*c**3*x**2 + a**2* 
b**4 - 16*a**2*b**3*c*x + 32*a**2*b*c**3*x**3 + 16*a**2*c**4*x**4 + 2*a*b* 
*5*x - 6*a*b**4*c*x**2 - 16*a*b**3*c**2*x**3 - 8*a*b**2*c**3*x**4 + b**6*x 
**2 + 2*b**5*c*x**3 + b**4*c**2*x**4))