Integrand size = 21, antiderivative size = 96 \[ \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {B \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}} \] Output:
(-2*a*(-2*A*c+B*b)-2*(-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)+B*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)
Time = 0.65 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {2 \sqrt {c} (a b B+b (b B-A c) x-2 a c (A+B x))}{\sqrt {a+x (b+c x)}}-B \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{3/2} \left (-b^2+4 a c\right )} \] Input:
Integrate[(x*(A + B*x))/(a + b*x + c*x^2)^(3/2),x]
Output:
((2*Sqrt[c]*(a*b*B + b*(b*B - A*c)*x - 2*a*c*(A + B*x)))/Sqrt[a + x*(b + c *x)] - B*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x) ])])/(c^(3/2)*(-b^2 + 4*a*c))
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1224, 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.
\(\displaystyle \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1224 |
\(\displaystyle \frac {B \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{c}-\frac {2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {2 B \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}-\frac {2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {B \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}-\frac {2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
Input:
Int[(x*(A + B*x))/(a + b*x + c*x^2)^(3/2),x]
Output:
(-2*(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]) + (B*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^ 2])])/c^(3/2)
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_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(b^2 - 4*a*c))), 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))/(c*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 1] && !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
Time = 1.23 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.75
method | result | size |
default | \(A \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+B \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\) | \(168\) |
Input:
int(x*(B*x+A)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
A*(-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)) +B*(-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)))
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (86) = 172\).
Time = 0.18 (sec) , antiderivative size = 405, normalized size of antiderivative = 4.22 \[ \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (B a b^{2} - 4 \, B a^{2} c + {\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{2} + {\left (B b^{3} - 4 \, B a b c\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (B a b c - 2 \, A a c^{2} + {\left (B b^{2} c - {\left (2 \, B a + A b\right )} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}}, -\frac {{\left (B a b^{2} - 4 \, B a^{2} c + {\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{2} + {\left (B b^{3} - 4 \, B a b c\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (B a b c - 2 \, A a c^{2} + {\left (B b^{2} c - {\left (2 \, B a + A b\right )} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x}\right ] \] Input:
integrate(x*(B*x+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
[1/2*((B*a*b^2 - 4*B*a^2*c + (B*b^2*c - 4*B*a*c^2)*x^2 + (B*b^3 - 4*B*a*b* 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*(B*a*b*c - 2*A*a*c^2 + (B*b^2*c - (2*B*a + A *b)*c^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a *c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x), -((B*a*b^2 - 4*B*a^2*c + (B*b^2*c - 4*B*a*c^2)*x^2 + (B*b^3 - 4*B*a*b*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*(B*a*b*c - 2*A*a *c^2 + (B*b^2*c - (2*B*a + A*b)*c^2)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)]
\[ \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {x \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x*(B*x+A)/(c*x**2+b*x+a)**(3/2),x)
Output:
Integral(x*(A + B*x)/(a + b*x + c*x**2)**(3/2), x)
Exception generated. \[ \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*(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
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.12 \[ \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {{\left (B b^{2} - 2 \, B a c - A b c\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac {B a b - 2 \, A a c}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {B \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {3}{2}}} \] Input:
integrate(x*(B*x+A)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
-2*((B*b^2 - 2*B*a*c - A*b*c)*x/(b^2*c - 4*a*c^2) + (B*a*b - 2*A*a*c)/(b^2 *c - 4*a*c^2))/sqrt(c*x^2 + b*x + a) - B*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(3/2)
Time = 11.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.16 \[ \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {B\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{c^{3/2}}-\frac {A\,\left (4\,a+2\,b\,x\right )}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {B\,\left (\frac {a\,b}{2}-x\,\left (a\,c-\frac {b^2}{2}\right )\right )}{c\,\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}} \] Input:
int((x*(A + B*x))/(a + b*x + c*x^2)^(3/2),x)
Output:
(B*log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2)))/c^(3/2) - (A*(4*a + 2*b*x))/((4*a*c - b^2)*(a + b*x + c*x^2)^(1/2)) + (B*((a*b)/2 - x*(a*c - b^2/2)))/(c*(a*c - b^2/4)*(a + b*x + c*x^2)^(1/2))
Time = 0.20 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.75 \[ \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-4 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{2}+2 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c -6 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2} x +2 \sqrt {c \,x^{2}+b x +a}\, b^{3} c x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b c -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3}+4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c x +4 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{2} x^{2}-\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{4} x -\sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c \,x^{2}-6 \sqrt {c}\, a^{2} b c +2 \sqrt {c}\, a \,b^{3}-6 \sqrt {c}\, a \,b^{2} c x -6 \sqrt {c}\, a b \,c^{2} x^{2}+2 \sqrt {c}\, b^{4} x +2 \sqrt {c}\, b^{3} c \,x^{2}}{c^{2} \left (4 a \,c^{2} x^{2}-b^{2} c \,x^{2}+4 a b c x -b^{3} x +4 a^{2} c -a \,b^{2}\right )} \] Input:
int(x*(B*x+A)/(c*x^2+b*x+a)^(3/2),x)
Output:
( - 4*sqrt(a + b*x + c*x**2)*a**2*c**2 + 2*sqrt(a + b*x + c*x**2)*a*b**2*c - 6*sqrt(a + b*x + c*x**2)*a*b*c**2*x + 2*sqrt(a + b*x + c*x**2)*b**3*c*x + 4*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*c - 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**3 + 4*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*c*x + 4*sqrt(c)*log((2*sq rt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**2*x** 2 - sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*x - 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*c*x**2 - 6*sqrt(c)*a**2*b*c + 2*sqrt(c)*a*b**3 - 6*sqrt(c)*a*b**2*c*x - 6*sqrt(c)*a*b*c**2*x**2 + 2*sqrt(c)*b**4*x + 2*s qrt(c)*b**3*c*x**2)/(c**2*(4*a**2*c - a*b**2 + 4*a*b*c*x + 4*a*c**2*x**2 - b**3*x - b**2*c*x**2))