Integrand size = 20, antiderivative size = 92 \[ \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx=-\frac {(3 b B-4 A c) \sqrt {b x+c x^2}}{4 c^2}+\frac {B x \sqrt {b x+c x^2}}{2 c}+\frac {b (3 b B-4 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}} \] Output:
-1/4*(-4*A*c+3*B*b)*(c*x^2+b*x)^(1/2)/c^2+1/2*B*x*(c*x^2+b*x)^(1/2)/c+1/4* b*(-4*A*c+3*B*b)*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/c^(5/2)
Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.17 \[ \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx=\frac {\sqrt {x} \left (\sqrt {c} \sqrt {x} (b+c x) (-3 b B+4 A c+2 B c x)+2 b (3 b B-4 A c) \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{4 c^{5/2} \sqrt {x (b+c x)}} \] Input:
Integrate[(x*(A + B*x))/Sqrt[b*x + c*x^2],x]
Output:
(Sqrt[x]*(Sqrt[c]*Sqrt[x]*(b + c*x)*(-3*b*B + 4*A*c + 2*B*c*x) + 2*b*(3*b* B - 4*A*c)*Sqrt[b + c*x]*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c* x])]))/(4*c^(5/2)*Sqrt[x*(b + c*x)])
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1225, 1091, 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.
\(\displaystyle \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {b (3 b B-4 A c) \int \frac {1}{\sqrt {c x^2+b x}}dx}{8 c^2}-\frac {\sqrt {b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle \frac {b (3 b B-4 A c) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{4 c^2}-\frac {\sqrt {b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {b (3 b B-4 A c) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}}-\frac {\sqrt {b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2}\) |
Input:
Int[(x*(A + B*x))/Sqrt[b*x + c*x^2],x]
Output:
-1/4*((3*b*B - 4*A*c - 2*B*c*x)*Sqrt[b*x + c*x^2])/c^2 + (b*(3*b*B - 4*A*c )*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(5/2))
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[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
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]
Time = 1.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {\left (-A b c +\frac {3}{4} B \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+\left (\left (\frac {B x}{2}+A \right ) c^{\frac {3}{2}}-\frac {3 B \sqrt {c}\, b}{4}\right ) \sqrt {x \left (c x +b \right )}}{c^{\frac {5}{2}}}\) | \(64\) |
risch | \(\frac {\left (2 B c x +4 A c -3 B b \right ) x \left (c x +b \right )}{4 c^{2} \sqrt {x \left (c x +b \right )}}-\frac {b \left (4 A c -3 B b \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}\) | \(75\) |
default | \(A \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )+B \left (\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )\) | \(122\) |
Input:
int(x*(B*x+A)/(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/c^(5/2)*((-A*b*c+3/4*B*b^2)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+((1/2*B *x+A)*c^(3/2)-3/4*B*c^(1/2)*b)*(x*(c*x+b))^(1/2))
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.73 \[ \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx=\left [-\frac {{\left (3 \, B b^{2} - 4 \, A b c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (2 \, B c^{2} x - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, c^{3}}, -\frac {{\left (3 \, B b^{2} - 4 \, A b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) - {\left (2 \, B c^{2} x - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, c^{3}}\right ] \] Input:
integrate(x*(B*x+A)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")
Output:
[-1/8*((3*B*b^2 - 4*A*b*c)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqr t(c)) - 2*(2*B*c^2*x - 3*B*b*c + 4*A*c^2)*sqrt(c*x^2 + b*x))/c^3, -1/4*((3 *B*b^2 - 4*A*b*c)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x + b)) - (2*B*c^2*x - 3*B*b*c + 4*A*c^2)*sqrt(c*x^2 + b*x))/c^3]
Time = 0.56 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.64 \[ \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx=\begin {cases} - \frac {b \left (A - \frac {3 B b}{4 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{2 c} + \sqrt {b x + c x^{2}} \left (\frac {B x}{2 c} + \frac {A - \frac {3 B b}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {A \left (b x\right )^{\frac {3}{2}}}{3} + \frac {B \left (b x\right )^{\frac {5}{2}}}{5 b}\right )}{b^{2}} & \text {for}\: b \neq 0 \\\tilde {\infty } \left (\frac {A x^{2}}{2} + \frac {B x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x*(B*x+A)/(c*x**2+b*x)**(1/2),x)
Output:
Piecewise((-b*(A - 3*B*b/(4*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x + c* x**2) + 2*c*x)/sqrt(c), Ne(b**2/c, 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sq rt(c*(b/(2*c) + x)**2), True))/(2*c) + sqrt(b*x + c*x**2)*(B*x/(2*c) + (A - 3*B*b/(4*c))/c), Ne(c, 0)), (2*(A*(b*x)**(3/2)/3 + B*(b*x)**(5/2)/(5*b)) /b**2, Ne(b, 0)), (zoo*(A*x**2/2 + B*x**3/3), True))
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.25 \[ \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx=\frac {\sqrt {c x^{2} + b x} B x}{2 \, c} + \frac {3 \, B b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {A b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} B b}{4 \, c^{2}} + \frac {\sqrt {c x^{2} + b x} A}{c} \] Input:
integrate(x*(B*x+A)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")
Output:
1/2*sqrt(c*x^2 + b*x)*B*x/c + 3/8*B*b^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x )*sqrt(c))/c^(5/2) - 1/2*A*b*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/ c^(3/2) - 3/4*sqrt(c*x^2 + b*x)*B*b/c^2 + sqrt(c*x^2 + b*x)*A/c
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.88 \[ \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx=\frac {1}{4} \, \sqrt {c x^{2} + b x} {\left (\frac {2 \, B x}{c} - \frac {3 \, B b - 4 \, A c}{c^{2}}\right )} - \frac {{\left (3 \, B b^{2} - 4 \, A b c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {5}{2}}} \] Input:
integrate(x*(B*x+A)/(c*x^2+b*x)^(1/2),x, algorithm="giac")
Output:
1/4*sqrt(c*x^2 + b*x)*(2*B*x/c - (3*B*b - 4*A*c)/c^2) - 1/8*(3*B*b^2 - 4*A *b*c)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(5/2)
Timed out. \[ \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx=\int \frac {x\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \] Input:
int((x*(A + B*x))/(b*x + c*x^2)^(1/2),x)
Output:
int((x*(A + B*x))/(b*x + c*x^2)^(1/2), x)
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int \frac {x (A+B x)}{\sqrt {b x+c x^2}} \, dx=\frac {4 \sqrt {x}\, \sqrt {c x +b}\, a \,c^{2}-3 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c +2 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{2} x -4 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) a b c +3 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{3}}{4 c^{3}} \] Input:
int(x*(B*x+A)/(c*x^2+b*x)^(1/2),x)
Output:
(4*sqrt(x)*sqrt(b + c*x)*a*c**2 - 3*sqrt(x)*sqrt(b + c*x)*b**2*c + 2*sqrt( x)*sqrt(b + c*x)*b*c**2*x - 4*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c) )/sqrt(b))*a*b*c + 3*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b) )*b**3)/(4*c**3)