Integrand size = 18, antiderivative size = 118 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 a^3 (A b-a B)}{3 b^5 (a+b x)^{3/2}}-\frac {2 a^2 (3 A b-4 a B)}{b^5 \sqrt {a+b x}}-\frac {6 a (A b-2 a B) \sqrt {a+b x}}{b^5}+\frac {2 (A b-4 a B) (a+b x)^{3/2}}{3 b^5}+\frac {2 B (a+b x)^{5/2}}{5 b^5} \]
2/3*a^3*(A*b-B*a)/b^5/(b*x+a)^(3/2)+2/3*(A*b-4*B*a)*(b*x+a)^(3/2)/b^5+2/5* B*(b*x+a)^(5/2)/b^5-2*a^2*(3*A*b-4*B*a)/b^5/(b*x+a)^(1/2)-6*a*(A*b-2*B*a)* (b*x+a)^(1/2)/b^5
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \left (128 a^4 B+24 a^2 b^2 x (-5 A+2 B x)+b^4 x^3 (5 A+3 B x)-2 a b^3 x^2 (15 A+4 B x)+a^3 (-80 A b+192 b B x)\right )}{15 b^5 (a+b x)^{3/2}} \]
(2*(128*a^4*B + 24*a^2*b^2*x*(-5*A + 2*B*x) + b^4*x^3*(5*A + 3*B*x) - 2*a* b^3*x^2*(15*A + 4*B*x) + a^3*(-80*A*b + 192*b*B*x)))/(15*b^5*(a + b*x)^(3/ 2))
Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {86, 2009}
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.
\(\displaystyle \int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {a^3 (a B-A b)}{b^4 (a+b x)^{5/2}}-\frac {a^2 (4 a B-3 A b)}{b^4 (a+b x)^{3/2}}+\frac {3 a (2 a B-A b)}{b^4 \sqrt {a+b x}}+\frac {\sqrt {a+b x} (A b-4 a B)}{b^4}+\frac {B (a+b x)^{3/2}}{b^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 a^3 (A b-a B)}{3 b^5 (a+b x)^{3/2}}-\frac {2 a^2 (3 A b-4 a B)}{b^5 \sqrt {a+b x}}-\frac {6 a \sqrt {a+b x} (A b-2 a B)}{b^5}+\frac {2 (a+b x)^{3/2} (A b-4 a B)}{3 b^5}+\frac {2 B (a+b x)^{5/2}}{5 b^5}\) |
(2*a^3*(A*b - a*B))/(3*b^5*(a + b*x)^(3/2)) - (2*a^2*(3*A*b - 4*a*B))/(b^5 *Sqrt[a + b*x]) - (6*a*(A*b - 2*a*B)*Sqrt[a + b*x])/b^5 + (2*(A*b - 4*a*B) *(a + b*x)^(3/2))/(3*b^5) + (2*B*(a + b*x)^(5/2))/(5*b^5)
3.5.45.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 0.53 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(-\frac {32 \left (-\frac {\left (\frac {3 B x}{5}+A \right ) x^{3} b^{4}}{16}+\frac {3 x^{2} \left (\frac {4 B x}{15}+A \right ) a \,b^{3}}{8}+\frac {3 x \left (-\frac {2 B x}{5}+A \right ) a^{2} b^{2}}{2}+a^{3} \left (-\frac {12 B x}{5}+A \right ) b -\frac {8 B \,a^{4}}{5}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{5}}\) | \(75\) |
risch | \(-\frac {2 \left (-3 b^{2} B \,x^{2}-5 A \,b^{2} x +14 B a b x +40 a b A -73 a^{2} B \right ) \sqrt {b x +a}}{15 b^{5}}-\frac {2 a^{2} \left (9 A \,b^{2} x -12 B a b x +8 a b A -11 a^{2} B \right )}{3 b^{5} \left (b x +a \right )^{\frac {3}{2}}}\) | \(88\) |
gosper | \(-\frac {2 \left (-3 B \,x^{4} b^{4}-5 A \,x^{3} b^{4}+8 B \,x^{3} a \,b^{3}+30 A \,x^{2} a \,b^{3}-48 B \,x^{2} a^{2} b^{2}+120 A x \,a^{2} b^{2}-192 B x \,a^{3} b +80 A \,a^{3} b -128 B \,a^{4}\right )}{15 \left (b x +a \right )^{\frac {3}{2}} b^{5}}\) | \(95\) |
trager | \(-\frac {2 \left (-3 B \,x^{4} b^{4}-5 A \,x^{3} b^{4}+8 B \,x^{3} a \,b^{3}+30 A \,x^{2} a \,b^{3}-48 B \,x^{2} a^{2} b^{2}+120 A x \,a^{2} b^{2}-192 B x \,a^{3} b +80 A \,a^{3} b -128 B \,a^{4}\right )}{15 \left (b x +a \right )^{\frac {3}{2}} b^{5}}\) | \(95\) |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 A b \left (b x +a \right )^{\frac {3}{2}}}{3}-\frac {8 B a \left (b x +a \right )^{\frac {3}{2}}}{3}-6 A a b \sqrt {b x +a}+12 B \,a^{2} \sqrt {b x +a}-\frac {2 a^{2} \left (3 A b -4 B a \right )}{\sqrt {b x +a}}+\frac {2 a^{3} \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{5}}\) | \(105\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 A b \left (b x +a \right )^{\frac {3}{2}}}{3}-\frac {8 B a \left (b x +a \right )^{\frac {3}{2}}}{3}-6 A a b \sqrt {b x +a}+12 B \,a^{2} \sqrt {b x +a}-\frac {2 a^{2} \left (3 A b -4 B a \right )}{\sqrt {b x +a}}+\frac {2 a^{3} \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{5}}\) | \(105\) |
-32/3*(-1/16*(3/5*B*x+A)*x^3*b^4+3/8*x^2*(4/15*B*x+A)*a*b^3+3/2*x*(-2/5*B* x+A)*a^2*b^2+a^3*(-12/5*B*x+A)*b-8/5*B*a^4)/(b*x+a)^(3/2)/b^5
Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, B b^{4} x^{4} + 128 \, B a^{4} - 80 \, A a^{3} b - {\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 6 \, {\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 24 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x + a}}{15 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
2/15*(3*B*b^4*x^4 + 128*B*a^4 - 80*A*a^3*b - (8*B*a*b^3 - 5*A*b^4)*x^3 + 6 *(8*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 24*(8*B*a^3*b - 5*A*a^2*b^2)*x)*sqrt(b*x + a)/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)
Time = 1.49 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.13 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {5}{2}}}{5 b} - \frac {a^{3} \left (- A b + B a\right )}{3 b \left (a + b x\right )^{\frac {3}{2}}} + \frac {a^{2} \left (- 3 A b + 4 B a\right )}{b \sqrt {a + b x}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (A b - 4 B a\right )}{3 b} + \frac {\sqrt {a + b x} \left (- 3 A a b + 6 B a^{2}\right )}{b}\right )}{b^{4}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{4}}{4} + \frac {B x^{5}}{5}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(B*(a + b*x)**(5/2)/(5*b) - a**3*(-A*b + B*a)/(3*b*(a + b*x)* *(3/2)) + a**2*(-3*A*b + 4*B*a)/(b*sqrt(a + b*x)) + (a + b*x)**(3/2)*(A*b - 4*B*a)/(3*b) + sqrt(a + b*x)*(-3*A*a*b + 6*B*a**2)/b)/b**4, Ne(b, 0)), ( (A*x**4/4 + B*x**5/5)/a**(5/2), True))
Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} B - 5 \, {\left (4 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 45 \, {\left (2 \, B a^{2} - A a b\right )} \sqrt {b x + a}}{b} - \frac {5 \, {\left (B a^{4} - A a^{3} b - 3 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {3}{2}} b}\right )}}{15 \, b^{4}} \]
2/15*((3*(b*x + a)^(5/2)*B - 5*(4*B*a - A*b)*(b*x + a)^(3/2) + 45*(2*B*a^2 - A*a*b)*sqrt(b*x + a))/b - 5*(B*a^4 - A*a^3*b - 3*(4*B*a^3 - 3*A*a^2*b)* (b*x + a))/((b*x + a)^(3/2)*b))/b^4
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \, {\left (12 \, {\left (b x + a\right )} B a^{3} - B a^{4} - 9 \, {\left (b x + a\right )} A a^{2} b + A a^{3} b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{5}} + \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} B b^{20} - 20 \, {\left (b x + a\right )}^{\frac {3}{2}} B a b^{20} + 90 \, \sqrt {b x + a} B a^{2} b^{20} + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} A b^{21} - 45 \, \sqrt {b x + a} A a b^{21}\right )}}{15 \, b^{25}} \]
2/3*(12*(b*x + a)*B*a^3 - B*a^4 - 9*(b*x + a)*A*a^2*b + A*a^3*b)/((b*x + a )^(3/2)*b^5) + 2/15*(3*(b*x + a)^(5/2)*B*b^20 - 20*(b*x + a)^(3/2)*B*a*b^2 0 + 90*sqrt(b*x + a)*B*a^2*b^20 + 5*(b*x + a)^(3/2)*A*b^21 - 45*sqrt(b*x + a)*A*a*b^21)/b^25
Time = 0.50 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {\left (12\,B\,a^2-6\,A\,a\,b\right )\,\sqrt {a+b\,x}}{b^5}+\frac {2\,B\,{\left (a+b\,x\right )}^{5/2}}{5\,b^5}+\frac {\left (8\,B\,a^3-6\,A\,a^2\,b\right )\,\left (a+b\,x\right )-\frac {2\,B\,a^4}{3}+\frac {2\,A\,a^3\,b}{3}}{b^5\,{\left (a+b\,x\right )}^{3/2}}+\frac {\left (2\,A\,b-8\,B\,a\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^5} \]