Integrand size = 23, antiderivative size = 153 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 x \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (3 b^2 B-2 A b c-8 a B c\right ) \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}-\frac {(3 b B-2 A c) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2}} \]
-1/2*(-2*A*c+3*B*b)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^( 5/2)-2*x*(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)+(-2*A*b*c-8*B*a*c+3*B*b^2)*(c*x^2+b*x+a)^(1/2)/c^2/(-4*a*c+b^2 )
Time = 0.63 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {2 \sqrt {c} \left (8 a^2 B c-b^2 x (3 b B-2 A c+B c x)+a \left (-3 b^2 B+4 c^2 x (-A+B x)+2 b c (A+5 B x)\right )\right )}{\sqrt {a+x (b+c x)}}-\left (b^2-4 a c\right ) (3 b B-2 A c) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{2 c^{5/2} \left (-b^2+4 a c\right )} \]
((2*Sqrt[c]*(8*a^2*B*c - b^2*x*(3*b*B - 2*A*c + B*c*x) + a*(-3*b^2*B + 4*c ^2*x*(-A + B*x) + 2*b*c*(A + 5*B*x))))/Sqrt[a + x*(b + c*x)] - (b^2 - 4*a* c)*(3*b*B - 2*A*c)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(2*c^ (5/2)*(-b^2 + 4*a*c))
Time = 0.33 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1233, 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.
| \(\displaystyle \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int \frac {2 a (b B-2 A c)+\left (3 B b^2-2 A c b-8 a B c\right ) x}{2 \sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}-\frac {2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 a (b B-2 A c)+\left (3 B b^2-2 A c b-8 a B c\right ) x}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}-\frac {2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-8 a B c-2 A b c+3 b^2 B\right )}{c}-\frac {\left (b^2-4 a c\right ) (3 b B-2 A c) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c}}{c \left (b^2-4 a c\right )}-\frac {2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-8 a B c-2 A b c+3 b^2 B\right )}{c}-\frac {\left (b^2-4 a c\right ) (3 b B-2 A c) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}}{c \left (b^2-4 a c\right )}-\frac {2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-8 a B c-2 A b c+3 b^2 B\right )}{c}-\frac {\left (b^2-4 a c\right ) (3 b B-2 A c) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2}}}{c \left (b^2-4 a c\right )}-\frac {2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(-2*x*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*Sq rt[a + b*x + c*x^2]) + (((3*b^2*B - 2*A*b*c - 8*a*B*c)*Sqrt[a + b*x + c*x^ 2])/c - ((b^2 - 4*a*c)*(3*b*B - 2*A*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt [a + b*x + c*x^2])])/(2*c^(3/2)))/(c*(b^2 - 4*a*c))
3.10.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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])
Time = 0.51 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(\frac {B \sqrt {c \,x^{2}+b x +a}}{c^{2}}+\frac {-\frac {2 a b B \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (2 A \,c^{2}-3 B b c \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (-2 B a c -B \,b^{2}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c^{2}}\) | \(248\) |
| default | \(B \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+A \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\) | \(308\) |
B/c^2*(c*x^2+b*x+a)^(1/2)+1/2/c^2*(-2*a*b*B*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b *x+a)^(1/2)+(2*A*c^2-3*B*b*c)*(-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)*l n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+(-2*B*a*c-B*b^2)*(-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)))
Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (139) = 278\).
Time = 0.43 (sec) , antiderivative size = 603, normalized size of antiderivative = 3.94 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} + {\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \, {\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} x^{2} - 2 \, {\left (6 \, B a^{2} b + A a b^{2}\right )} c + {\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (3 \, B a b^{2} c - 2 \, {\left (4 \, B a^{2} + A a b\right )} c^{2} + {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} x^{2} + {\left (3 \, B b^{3} c + 4 \, A a c^{3} - 2 \, {\left (5 \, B a b + A b^{2}\right )} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}, \frac {{\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} + {\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \, {\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} x^{2} - 2 \, {\left (6 \, B a^{2} b + A a b^{2}\right )} c + {\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (3 \, B a b^{2} c - 2 \, {\left (4 \, B a^{2} + A a b\right )} c^{2} + {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} x^{2} + {\left (3 \, B b^{3} c + 4 \, A a c^{3} - 2 \, {\left (5 \, B a b + A b^{2}\right )} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}\right ] \]
[-1/4*((3*B*a*b^3 + 8*A*a^2*c^2 + (3*B*b^3*c + 8*A*a*c^3 - 2*(6*B*a*b + A* b^2)*c^2)*x^2 - 2*(6*B*a^2*b + A*a*b^2)*c + (3*B*b^4 + 8*A*a*b*c^2 - 2*(6* B*a*b^2 + A*b^3)*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*(3*B*a*b^2*c - 2*(4*B*a^2 + A*a*b)*c^2 + (B*b^2*c^2 - 4*B*a*c^3)*x^2 + (3*B*b^3*c + 4*A*a*c^3 - 2*(5* B*a*b + A*b^2)*c^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^ 2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x), 1/2*((3*B*a*b^3 + 8*A*a^2 *c^2 + (3*B*b^3*c + 8*A*a*c^3 - 2*(6*B*a*b + A*b^2)*c^2)*x^2 - 2*(6*B*a^2* b + A*a*b^2)*c + (3*B*b^4 + 8*A*a*b*c^2 - 2*(6*B*a*b^2 + A*b^3)*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*(3*B*a*b^2*c - 2*(4*B*a^2 + A*a*b)*c^2 + (B*b^2*c^2 - 4*B*a* c^3)*x^2 + (3*B*b^3*c + 4*A*a*c^3 - 2*(5*B*a*b + A*b^2)*c^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)]
\[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left (\frac {{\left (B b^{2} c - 4 \, B a c^{2}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {3 \, B b^{3} - 10 \, B a b c - 2 \, A b^{2} c + 4 \, A a c^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac {3 \, B a b^{2} - 8 \, B a^{2} c - 2 \, A a b c}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt {c x^{2} + b x + a}} + \frac {{\left (3 \, B b - 2 \, A c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {5}{2}}} \]
(((B*b^2*c - 4*B*a*c^2)*x/(b^2*c^2 - 4*a*c^3) + (3*B*b^3 - 10*B*a*b*c - 2* A*b^2*c + 4*A*a*c^2)/(b^2*c^2 - 4*a*c^3))*x + (3*B*a*b^2 - 8*B*a^2*c - 2*A *a*b*c)/(b^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a) + 1/2*(3*B*b - 2*A*c)*l og(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(5/2)
Timed out. \[ \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]