Integrand size = 24, antiderivative size = 68 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b B-A c) \sqrt {x}}{b c \sqrt {b x+c x^2}}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}} \] Output:
-2*(-A*c+B*b)*x^(1/2)/b/c/(c*x^2+b*x)^(1/2)-2*A*arctanh((c*x^2+b*x)^(1/2)/ b^(1/2)/x^(1/2))/b^(3/2)
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {x} \left (\sqrt {b} (b B-A c)+A c \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{b^{3/2} c \sqrt {x (b+c x)}} \] Input:
Integrate[(Sqrt[x]*(A + B*x))/(b*x + c*x^2)^(3/2),x]
Output:
(-2*Sqrt[x]*(Sqrt[b]*(b*B - A*c) + A*c*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x] /Sqrt[b]]))/(b^(3/2)*c*Sqrt[x*(b + c*x)])
Time = 0.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1218, 1136, 220}
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.
\(\displaystyle \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1218 |
\(\displaystyle \frac {A \int \frac {1}{\sqrt {x} \sqrt {c x^2+b x}}dx}{b}-\frac {2 \sqrt {x} (b B-A c)}{b c \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 1136 |
\(\displaystyle \frac {2 A \int \frac {1}{\frac {c x^2+b x}{x}-b}d\frac {\sqrt {c x^2+b x}}{\sqrt {x}}}{b}-\frac {2 \sqrt {x} (b B-A c)}{b c \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -\frac {2 A \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}-\frac {2 \sqrt {x} (b B-A c)}{b c \sqrt {b x+c x^2}}\) |
Input:
Int[(Sqrt[x]*(A + B*x))/(b*x + c*x^2)^(3/2),x]
Output:
(-2*(b*B - A*c)*Sqrt[x])/(b*c*Sqrt[b*x + c*x^2]) - (2*A*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/b^(3/2)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(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[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 0.99 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (A \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c \sqrt {c x +b}-A c \sqrt {b}+B \,b^{\frac {3}{2}}\right )}{b^{\frac {3}{2}} \sqrt {x}\, \left (c x +b \right ) c}\) | \(63\) |
Input:
int(x^(1/2)*(B*x+A)/(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*(x*(c*x+b))^(1/2)/b^(3/2)*(A*arctanh((c*x+b)^(1/2)/b^(1/2))*c*(c*x+b)^( 1/2)-A*c*b^(1/2)+B*b^(3/2))/x^(1/2)/(c*x+b)/c
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.87 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (A c^{2} x^{2} + A b c x\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, {\left (B b^{2} - A b c\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{b^{2} c^{2} x^{2} + b^{3} c x}, \frac {2 \, {\left ({\left (A c^{2} x^{2} + A b c x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (B b^{2} - A b c\right )} \sqrt {c x^{2} + b x} \sqrt {x}\right )}}{b^{2} c^{2} x^{2} + b^{3} c x}\right ] \] Input:
integrate(x^(1/2)*(B*x+A)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")
Output:
[((A*c^2*x^2 + A*b*c*x)*sqrt(b)*log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*x)* sqrt(b)*sqrt(x))/x^2) - 2*(B*b^2 - A*b*c)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^2* c^2*x^2 + b^3*c*x), 2*((A*c^2*x^2 + A*b*c*x)*sqrt(-b)*arctan(sqrt(c*x^2 + b*x)*sqrt(-b)/(b*sqrt(x))) - (B*b^2 - A*b*c)*sqrt(c*x^2 + b*x)*sqrt(x))/(b ^2*c^2*x^2 + b^3*c*x)]
\[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {x} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**(1/2)*(B*x+A)/(c*x**2+b*x)**(3/2),x)
Output:
Integral(sqrt(x)*(A + B*x)/(x*(b + c*x))**(3/2), x)
\[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {x}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^(1/2)*(B*x+A)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*sqrt(x)/(c*x^2 + b*x)^(3/2), x)
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, A \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {2 \, {\left (B b - A c\right )}}{\sqrt {c x + b} b c} \] Input:
integrate(x^(1/2)*(B*x+A)/(c*x^2+b*x)^(3/2),x, algorithm="giac")
Output:
2*A*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b) - 2*(B*b - A*c)/(sqrt(c*x + b)*b*c)
Timed out. \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {x}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \] Input:
int((x^(1/2)*(A + B*x))/(b*x + c*x^2)^(3/2),x)
Output:
int((x^(1/2)*(A + B*x))/(b*x + c*x^2)^(3/2), x)
Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {b}\, \sqrt {c x +b}\, \mathrm {log}\left (\sqrt {c x +b}-\sqrt {b}\right ) a c -\sqrt {b}\, \sqrt {c x +b}\, \mathrm {log}\left (\sqrt {c x +b}+\sqrt {b}\right ) a c +2 a b c -2 b^{3}}{\sqrt {c x +b}\, b^{2} c} \] Input:
int(x^(1/2)*(B*x+A)/(c*x^2+b*x)^(3/2),x)
Output:
(sqrt(b)*sqrt(b + c*x)*log(sqrt(b + c*x) - sqrt(b))*a*c - sqrt(b)*sqrt(b + c*x)*log(sqrt(b + c*x) + sqrt(b))*a*c + 2*a*b*c - 2*b**3)/(sqrt(b + c*x)* b**2*c)