Integrand size = 22, antiderivative size = 64 \[ \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=-\frac {(b B-A c) \sqrt {x}}{b c (b+c x)}+\frac {(b B+A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} c^{3/2}} \]
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {(-b B+A c) \sqrt {x}}{b c (b+c x)}+\frac {(b B+A c) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} c^{3/2}} \]
((-(b*B) + A*c)*Sqrt[x])/(b*c*(b + c*x)) + ((b*B + A*c)*ArcTan[(Sqrt[c]*Sq rt[x])/Sqrt[b]])/(b^(3/2)*c^(3/2))
Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {9, 87, 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.
\(\displaystyle \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {A+B x}{\sqrt {x} (b+c x)^2}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(A c+b B) \int \frac {1}{\sqrt {x} (b+c x)}dx}{2 b c}-\frac {\sqrt {x} (b B-A c)}{b c (b+c x)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(A c+b B) \int \frac {1}{b+c x}d\sqrt {x}}{b c}-\frac {\sqrt {x} (b B-A c)}{b c (b+c x)}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(A c+b B) \arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} c^{3/2}}-\frac {\sqrt {x} (b B-A c)}{b c (b+c x)}\) |
-(((b*B - A*c)*Sqrt[x])/(b*c*(b + c*x))) + ((b*B + A*c)*ArcTan[(Sqrt[c]*Sq rt[x])/Sqrt[b]])/(b^(3/2)*c^(3/2))
3.2.80.3.1 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] :> 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 = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\left (A c -B b \right ) \sqrt {x}}{b c \left (c x +b \right )}+\frac {\left (A c +B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{b c \sqrt {b c}}\) | \(57\) |
default | \(\frac {\left (A c -B b \right ) \sqrt {x}}{b c \left (c x +b \right )}+\frac {\left (A c +B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{b c \sqrt {b c}}\) | \(57\) |
Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.77 \[ \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\left [-\frac {{\left (B b^{2} + A b c + {\left (B b c + A c^{2}\right )} x\right )} \sqrt {-b c} \log \left (\frac {c x - b - 2 \, \sqrt {-b c} \sqrt {x}}{c x + b}\right ) + 2 \, {\left (B b^{2} c - A b c^{2}\right )} \sqrt {x}}{2 \, {\left (b^{2} c^{3} x + b^{3} c^{2}\right )}}, -\frac {{\left (B b^{2} + A b c + {\left (B b c + A c^{2}\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right ) + {\left (B b^{2} c - A b c^{2}\right )} \sqrt {x}}{b^{2} c^{3} x + b^{3} c^{2}}\right ] \]
[-1/2*((B*b^2 + A*b*c + (B*b*c + A*c^2)*x)*sqrt(-b*c)*log((c*x - b - 2*sqr t(-b*c)*sqrt(x))/(c*x + b)) + 2*(B*b^2*c - A*b*c^2)*sqrt(x))/(b^2*c^3*x + b^3*c^2), -((B*b^2 + A*b*c + (B*b*c + A*c^2)*x)*sqrt(b*c)*arctan(sqrt(b*c) /(c*sqrt(x))) + (B*b^2*c - A*b*c^2)*sqrt(x))/(b^2*c^3*x + b^3*c^2)]
Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (53) = 106\).
Time = 24.97 (sec) , antiderivative size = 615, normalized size of antiderivative = 9.61 \[ \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{b^{2}} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{c^{2}} & \text {for}\: b = 0 \\\frac {A b c \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} - \frac {A b c \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} + \frac {2 A c^{2} \sqrt {x} \sqrt {- \frac {b}{c}}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} + \frac {A c^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} - \frac {A c^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} + \frac {B b^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} - \frac {B b^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} - \frac {2 B b c \sqrt {x} \sqrt {- \frac {b}{c}}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} + \frac {B b c x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} - \frac {B b c x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{2 b^{2} c^{2} \sqrt {- \frac {b}{c}} + 2 b c^{3} x \sqrt {- \frac {b}{c}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(3*x**(3/2)) - 2*B/sqrt(x)), Eq(b, 0) & Eq(c, 0)), (( 2*A*sqrt(x) + 2*B*x**(3/2)/3)/b**2, Eq(c, 0)), ((-2*A/(3*x**(3/2)) - 2*B/s qrt(x))/c**2, Eq(b, 0)), (A*b*c*log(sqrt(x) - sqrt(-b/c))/(2*b**2*c**2*sqr t(-b/c) + 2*b*c**3*x*sqrt(-b/c)) - A*b*c*log(sqrt(x) + sqrt(-b/c))/(2*b**2 *c**2*sqrt(-b/c) + 2*b*c**3*x*sqrt(-b/c)) + 2*A*c**2*sqrt(x)*sqrt(-b/c)/(2 *b**2*c**2*sqrt(-b/c) + 2*b*c**3*x*sqrt(-b/c)) + A*c**2*x*log(sqrt(x) - sq rt(-b/c))/(2*b**2*c**2*sqrt(-b/c) + 2*b*c**3*x*sqrt(-b/c)) - A*c**2*x*log( sqrt(x) + sqrt(-b/c))/(2*b**2*c**2*sqrt(-b/c) + 2*b*c**3*x*sqrt(-b/c)) + B *b**2*log(sqrt(x) - sqrt(-b/c))/(2*b**2*c**2*sqrt(-b/c) + 2*b*c**3*x*sqrt( -b/c)) - B*b**2*log(sqrt(x) + sqrt(-b/c))/(2*b**2*c**2*sqrt(-b/c) + 2*b*c* *3*x*sqrt(-b/c)) - 2*B*b*c*sqrt(x)*sqrt(-b/c)/(2*b**2*c**2*sqrt(-b/c) + 2* b*c**3*x*sqrt(-b/c)) + B*b*c*x*log(sqrt(x) - sqrt(-b/c))/(2*b**2*c**2*sqrt (-b/c) + 2*b*c**3*x*sqrt(-b/c)) - B*b*c*x*log(sqrt(x) + sqrt(-b/c))/(2*b** 2*c**2*sqrt(-b/c) + 2*b*c**3*x*sqrt(-b/c)), True))
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.91 \[ \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=-\frac {{\left (B b - A c\right )} \sqrt {x}}{b c^{2} x + b^{2} c} + \frac {{\left (B b + A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b c} \]
-(B*b - A*c)*sqrt(x)/(b*c^2*x + b^2*c) + (B*b + A*c)*arctan(c*sqrt(x)/sqrt (b*c))/(sqrt(b*c)*b*c)
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.94 \[ \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (B b + A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b c} - \frac {B b \sqrt {x} - A c \sqrt {x}}{{\left (c x + b\right )} b c} \]
(B*b + A*c)*arctan(c*sqrt(x)/sqrt(b*c))/(sqrt(b*c)*b*c) - (B*b*sqrt(x) - A *c*sqrt(x))/((c*x + b)*b*c)
Time = 10.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80 \[ \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c+B\,b\right )}{b^{3/2}\,c^{3/2}}+\frac {\sqrt {x}\,\left (A\,c-B\,b\right )}{b\,c\,\left (b+c\,x\right )} \]