Integrand size = 22, antiderivative size = 110 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx=\frac {3 b B-5 A c}{3 b^2 c x^{3/2}}-\frac {3 b B-5 A c}{b^3 \sqrt {x}}-\frac {b B-A c}{b c x^{3/2} (b+c x)}-\frac {\sqrt {c} (3 b B-5 A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \]
1/3*(-5*A*c+3*B*b)/b^2/c/x^(3/2)+(A*c-B*b)/b/c/x^(3/2)/(c*x+b)-(-5*A*c+3*B *b)*arctan(c^(1/2)*x^(1/2)/b^(1/2))*c^(1/2)/b^(7/2)+(5*A*c-3*B*b)/b^3/x^(1 /2)
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx=\frac {-3 b B x (2 b+3 c x)+A \left (-2 b^2+10 b c x+15 c^2 x^2\right )}{3 b^3 x^{3/2} (b+c x)}+\frac {\sqrt {c} (-3 b B+5 A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \]
(-3*b*B*x*(2*b + 3*c*x) + A*(-2*b^2 + 10*b*c*x + 15*c^2*x^2))/(3*b^3*x^(3/ 2)*(b + c*x)) + (Sqrt[c]*(-3*b*B + 5*A*c)*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b] ])/b^(7/2)
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {9, 87, 61, 61, 73, 218}
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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {A+B x}{x^{5/2} (b+c x)^2}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(3 b B-5 A c) \int \frac {1}{x^{5/2} (b+c x)}dx}{2 b c}-\frac {b B-A c}{b c x^{3/2} (b+c x)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {(3 b B-5 A c) \left (-\frac {c \int \frac {1}{x^{3/2} (b+c x)}dx}{b}-\frac {2}{3 b x^{3/2}}\right )}{2 b c}-\frac {b B-A c}{b c x^{3/2} (b+c x)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {(3 b B-5 A c) \left (-\frac {c \left (-\frac {c \int \frac {1}{\sqrt {x} (b+c x)}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{2 b c}-\frac {b B-A c}{b c x^{3/2} (b+c x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {(3 b B-5 A c) \left (-\frac {c \left (-\frac {2 c \int \frac {1}{b+c x}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{2 b c}-\frac {b B-A c}{b c x^{3/2} (b+c x)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {(3 b B-5 A c) \left (-\frac {c \left (-\frac {2 \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{2 b c}-\frac {b B-A c}{b c x^{3/2} (b+c x)}\) |
-((b*B - A*c)/(b*c*x^(3/2)*(b + c*x))) - ((3*b*B - 5*A*c)*(-2/(3*b*x^(3/2) ) - (c*(-2/(b*Sqrt[x]) - (2*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/b^( 3/2)))/b))/(2*b*c)
3.2.82.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {2 \left (-6 A c x +3 B b x +A b \right )}{3 b^{3} x^{\frac {3}{2}}}+\frac {c \left (\frac {2 \left (\frac {A c}{2}-\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (5 A c -3 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}}\right )}{b^{3}}\) | \(77\) |
| derivativedivides | \(\frac {2 c \left (\frac {\left (\frac {A c}{2}-\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (5 A c -3 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{b^{3}}-\frac {2 A}{3 b^{2} x^{\frac {3}{2}}}-\frac {2 \left (-2 A c +B b \right )}{b^{3} \sqrt {x}}\) | \(81\) |
| default | \(\frac {2 c \left (\frac {\left (\frac {A c}{2}-\frac {B b}{2}\right ) \sqrt {x}}{c x +b}+\frac {\left (5 A c -3 B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{2 \sqrt {b c}}\right )}{b^{3}}-\frac {2 A}{3 b^{2} x^{\frac {3}{2}}}-\frac {2 \left (-2 A c +B b \right )}{b^{3} \sqrt {x}}\) | \(81\) |
-2/3*(-6*A*c*x+3*B*b*x+A*b)/b^3/x^(3/2)+1/b^3*c*(2*(1/2*A*c-1/2*B*b)*x^(1/ 2)/(c*x+b)+(5*A*c-3*B*b)/(b*c)^(1/2)*arctan(c*x^(1/2)/(b*c)^(1/2)))
Time = 0.36 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.38 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx=\left [-\frac {3 \, {\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{3} + {\left (3 \, B b^{2} - 5 \, A b c\right )} x^{2}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x + 2 \, b \sqrt {x} \sqrt {-\frac {c}{b}} - b}{c x + b}\right ) + 2 \, {\left (2 \, A b^{2} + 3 \, {\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt {x}}{6 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, B b c - 5 \, A c^{2}\right )} x^{3} + {\left (3 \, B b^{2} - 5 \, A b c\right )} x^{2}\right )} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) - {\left (2 \, A b^{2} + 3 \, {\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt {x}}{3 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )}}\right ] \]
[-1/6*(3*((3*B*b*c - 5*A*c^2)*x^3 + (3*B*b^2 - 5*A*b*c)*x^2)*sqrt(-c/b)*lo g((c*x + 2*b*sqrt(x)*sqrt(-c/b) - b)/(c*x + b)) + 2*(2*A*b^2 + 3*(3*B*b*c - 5*A*c^2)*x^2 + 2*(3*B*b^2 - 5*A*b*c)*x)*sqrt(x))/(b^3*c*x^3 + b^4*x^2), 1/3*(3*((3*B*b*c - 5*A*c^2)*x^3 + (3*B*b^2 - 5*A*b*c)*x^2)*sqrt(c/b)*arcta n(b*sqrt(c/b)/(c*sqrt(x))) - (2*A*b^2 + 3*(3*B*b*c - 5*A*c^2)*x^2 + 2*(3*B *b^2 - 5*A*b*c)*x)*sqrt(x))/(b^3*c*x^3 + b^4*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (95) = 190\).
Time = 21.15 (sec) , antiderivative size = 882, normalized size of antiderivative = 8.02 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{b^{2}} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{c^{2}} & \text {for}\: b = 0 \\- \frac {4 A b^{2} \sqrt {- \frac {b}{c}}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} + \frac {15 A b c x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} - \frac {15 A b c x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} + \frac {20 A b c x \sqrt {- \frac {b}{c}}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} + \frac {15 A c^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} - \frac {15 A c^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} + \frac {30 A c^{2} x^{2} \sqrt {- \frac {b}{c}}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} - \frac {9 B b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} + \frac {9 B b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} - \frac {12 B b^{2} x \sqrt {- \frac {b}{c}}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} - \frac {9 B b c x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} + \frac {9 B b c x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} - \frac {18 B b c x^{2} \sqrt {- \frac {b}{c}}}{6 b^{4} x^{\frac {3}{2}} \sqrt {- \frac {b}{c}} + 6 b^{3} c x^{\frac {5}{2}} \sqrt {- \frac {b}{c}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2))), Eq(b, 0) & Eq(c, 0) ), ((-2*A/(3*x**(3/2)) - 2*B/sqrt(x))/b**2, Eq(c, 0)), ((-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2)))/c**2, Eq(b, 0)), (-4*A*b**2*sqrt(-b/c)/(6*b**4*x**(3/ 2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)) + 15*A*b*c*x**(3/2)*log(sqrt (x) - sqrt(-b/c))/(6*b**4*x**(3/2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/ c)) - 15*A*b*c*x**(3/2)*log(sqrt(x) + sqrt(-b/c))/(6*b**4*x**(3/2)*sqrt(-b /c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)) + 20*A*b*c*x*sqrt(-b/c)/(6*b**4*x**(3/ 2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)) + 15*A*c**2*x**(5/2)*log(sqr t(x) - sqrt(-b/c))/(6*b**4*x**(3/2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b /c)) - 15*A*c**2*x**(5/2)*log(sqrt(x) + sqrt(-b/c))/(6*b**4*x**(3/2)*sqrt( -b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)) + 30*A*c**2*x**2*sqrt(-b/c)/(6*b**4* x**(3/2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)) - 9*B*b**2*x**(3/2)*lo g(sqrt(x) - sqrt(-b/c))/(6*b**4*x**(3/2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sq rt(-b/c)) + 9*B*b**2*x**(3/2)*log(sqrt(x) + sqrt(-b/c))/(6*b**4*x**(3/2)*s qrt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)) - 12*B*b**2*x*sqrt(-b/c)/(6*b**4 *x**(3/2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)) - 9*B*b*c*x**(5/2)*lo g(sqrt(x) - sqrt(-b/c))/(6*b**4*x**(3/2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sq rt(-b/c)) + 9*B*b*c*x**(5/2)*log(sqrt(x) + sqrt(-b/c))/(6*b**4*x**(3/2)*sq rt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)) - 18*B*b*c*x**2*sqrt(-b/c)/(6*b** 4*x**(3/2)*sqrt(-b/c) + 6*b**3*c*x**(5/2)*sqrt(-b/c)), True))
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx=-\frac {2 \, A b^{2} + 3 \, {\left (3 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} - 5 \, A b c\right )} x}{3 \, {\left (b^{3} c x^{\frac {5}{2}} + b^{4} x^{\frac {3}{2}}\right )}} - \frac {{\left (3 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{3}} \]
-1/3*(2*A*b^2 + 3*(3*B*b*c - 5*A*c^2)*x^2 + 2*(3*B*b^2 - 5*A*b*c)*x)/(b^3* c*x^(5/2) + b^4*x^(3/2)) - (3*B*b*c - 5*A*c^2)*arctan(c*sqrt(x)/sqrt(b*c)) /(sqrt(b*c)*b^3)
Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx=-\frac {{\left (3 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{3}} - \frac {B b c \sqrt {x} - A c^{2} \sqrt {x}}{{\left (c x + b\right )} b^{3}} - \frac {2 \, {\left (3 \, B b x - 6 \, A c x + A b\right )}}{3 \, b^{3} x^{\frac {3}{2}}} \]
-(3*B*b*c - 5*A*c^2)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b^3) - (B*b*c* sqrt(x) - A*c^2*sqrt(x))/((c*x + b)*b^3) - 2/3*(3*B*b*x - 6*A*c*x + A*b)/( b^3*x^(3/2))
Time = 9.98 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{\sqrt {x} \left (b x+c x^2\right )^2} \, dx=\frac {\frac {2\,x\,\left (5\,A\,c-3\,B\,b\right )}{3\,b^2}-\frac {2\,A}{3\,b}+\frac {c\,x^2\,\left (5\,A\,c-3\,B\,b\right )}{b^3}}{b\,x^{3/2}+c\,x^{5/2}}+\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (5\,A\,c-3\,B\,b\right )}{b^{7/2}} \]