Integrand size = 18, antiderivative size = 122 \[ \int x^3 (a+b x)^{5/2} (A+B x) \, dx=-\frac {2 a^3 (A b-a B) (a+b x)^{7/2}}{7 b^5}+\frac {2 a^2 (3 A b-4 a B) (a+b x)^{9/2}}{9 b^5}-\frac {6 a (A b-2 a B) (a+b x)^{11/2}}{11 b^5}+\frac {2 (A b-4 a B) (a+b x)^{13/2}}{13 b^5}+\frac {2 B (a+b x)^{15/2}}{15 b^5} \]
-2/7*a^3*(A*b-B*a)*(b*x+a)^(7/2)/b^5+2/9*a^2*(3*A*b-4*B*a)*(b*x+a)^(9/2)/b ^5-6/11*a*(A*b-2*B*a)*(b*x+a)^(11/2)/b^5+2/13*(A*b-4*B*a)*(b*x+a)^(13/2)/b ^5+2/15*B*(b*x+a)^(15/2)/b^5
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.71 \[ \int x^3 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 (a+b x)^{7/2} \left (128 a^4 B+168 a^2 b^2 x (5 A+6 B x)+231 b^4 x^3 (15 A+13 B x)-16 a^3 b (15 A+28 B x)-42 a b^3 x^2 (45 A+44 B x)\right )}{45045 b^5} \]
(2*(a + b*x)^(7/2)*(128*a^4*B + 168*a^2*b^2*x*(5*A + 6*B*x) + 231*b^4*x^3* (15*A + 13*B*x) - 16*a^3*b*(15*A + 28*B*x) - 42*a*b^3*x^2*(45*A + 44*B*x)) )/(45045*b^5)
Time = 0.24 (sec) , antiderivative size = 122, 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.
| \(\displaystyle \int x^3 (a+b x)^{5/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {a^3 (a+b x)^{5/2} (a B-A b)}{b^4}-\frac {a^2 (a+b x)^{7/2} (4 a B-3 A b)}{b^4}+\frac {(a+b x)^{11/2} (A b-4 a B)}{b^4}+\frac {3 a (a+b x)^{9/2} (2 a B-A b)}{b^4}+\frac {B (a+b x)^{13/2}}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^3 (a+b x)^{7/2} (A b-a B)}{7 b^5}+\frac {2 a^2 (a+b x)^{9/2} (3 A b-4 a B)}{9 b^5}+\frac {2 (a+b x)^{13/2} (A b-4 a B)}{13 b^5}-\frac {6 a (a+b x)^{11/2} (A b-2 a B)}{11 b^5}+\frac {2 B (a+b x)^{15/2}}{15 b^5}\) |
(-2*a^3*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^5) + (2*a^2*(3*A*b - 4*a*B)*(a + b*x)^(9/2))/(9*b^5) - (6*a*(A*b - 2*a*B)*(a + b*x)^(11/2))/(11*b^5) + (2* (A*b - 4*a*B)*(a + b*x)^(13/2))/(13*b^5) + (2*B*(a + b*x)^(15/2))/(15*b^5)
3.5.11.3.1 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 = 0.54 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.61
| method | result | size |
| pseudoelliptic | \(-\frac {32 \left (b x +a \right )^{\frac {7}{2}} \left (-\frac {231 x^{3} \left (\frac {13 B x}{15}+A \right ) b^{4}}{16}+\frac {63 x^{2} \left (\frac {44 B x}{45}+A \right ) a \,b^{3}}{8}-\frac {7 \left (\frac {6 B x}{5}+A \right ) x \,a^{2} b^{2}}{2}+a^{3} \left (\frac {28 B x}{15}+A \right ) b -\frac {8 B \,a^{4}}{15}\right )}{3003 b^{5}}\) | \(75\) |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-3003 B \,x^{4} b^{4}-3465 A \,x^{3} b^{4}+1848 B \,x^{3} a \,b^{3}+1890 A \,x^{2} a \,b^{3}-1008 B \,x^{2} a^{2} b^{2}-840 A x \,a^{2} b^{2}+448 B x \,a^{3} b +240 A \,a^{3} b -128 B \,a^{4}\right )}{45045 b^{5}}\) | \(95\) |
| derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {15}{2}}}{15}+\frac {2 \left (A b -4 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 a^{2} B -3 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-a^{3} B +3 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {2 a^{3} \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{5}}\) | \(110\) |
| default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {15}{2}}}{15}-\frac {2 \left (-A b +4 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{13}-\frac {2 \left (-3 a^{2} B +3 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {11}{2}}}{11}-\frac {2 \left (a^{3} B -3 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {2 a^{3} \left (A b -B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{5}}\) | \(110\) |
| trager | \(-\frac {2 \left (-3003 B \,b^{7} x^{7}-3465 A \,b^{7} x^{6}-7161 B a \,b^{6} x^{6}-8505 A a \,b^{6} x^{5}-4473 B \,a^{2} b^{5} x^{5}-5565 A \,a^{2} b^{5} x^{4}-35 B \,a^{3} b^{4} x^{4}-75 A \,a^{3} b^{4} x^{3}+40 B \,a^{4} b^{3} x^{3}+90 A \,a^{4} b^{3} x^{2}-48 B \,a^{5} b^{2} x^{2}-120 A \,a^{5} b^{2} x +64 B \,a^{6} b x +240 A \,a^{6} b -128 B \,a^{7}\right ) \sqrt {b x +a}}{45045 b^{5}}\) | \(167\) |
| risch | \(-\frac {2 \left (-3003 B \,b^{7} x^{7}-3465 A \,b^{7} x^{6}-7161 B a \,b^{6} x^{6}-8505 A a \,b^{6} x^{5}-4473 B \,a^{2} b^{5} x^{5}-5565 A \,a^{2} b^{5} x^{4}-35 B \,a^{3} b^{4} x^{4}-75 A \,a^{3} b^{4} x^{3}+40 B \,a^{4} b^{3} x^{3}+90 A \,a^{4} b^{3} x^{2}-48 B \,a^{5} b^{2} x^{2}-120 A \,a^{5} b^{2} x +64 B \,a^{6} b x +240 A \,a^{6} b -128 B \,a^{7}\right ) \sqrt {b x +a}}{45045 b^{5}}\) | \(167\) |
-32/3003*(b*x+a)^(7/2)*(-231/16*x^3*(13/15*B*x+A)*b^4+63/8*x^2*(44/45*B*x+ A)*a*b^3-7/2*(6/5*B*x+A)*x*a^2*b^2+a^3*(28/15*B*x+A)*b-8/15*B*a^4)/b^5
Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int x^3 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (3003 \, B b^{7} x^{7} + 128 \, B a^{7} - 240 \, A a^{6} b + 231 \, {\left (31 \, B a b^{6} + 15 \, A b^{7}\right )} x^{6} + 63 \, {\left (71 \, B a^{2} b^{5} + 135 \, A a b^{6}\right )} x^{5} + 35 \, {\left (B a^{3} b^{4} + 159 \, A a^{2} b^{5}\right )} x^{4} - 5 \, {\left (8 \, B a^{4} b^{3} - 15 \, A a^{3} b^{4}\right )} x^{3} + 6 \, {\left (8 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3}\right )} x^{2} - 8 \, {\left (8 \, B a^{6} b - 15 \, A a^{5} b^{2}\right )} x\right )} \sqrt {b x + a}}{45045 \, b^{5}} \]
2/45045*(3003*B*b^7*x^7 + 128*B*a^7 - 240*A*a^6*b + 231*(31*B*a*b^6 + 15*A *b^7)*x^6 + 63*(71*B*a^2*b^5 + 135*A*a*b^6)*x^5 + 35*(B*a^3*b^4 + 159*A*a^ 2*b^5)*x^4 - 5*(8*B*a^4*b^3 - 15*A*a^3*b^4)*x^3 + 6*(8*B*a^5*b^2 - 15*A*a^ 4*b^3)*x^2 - 8*(8*B*a^6*b - 15*A*a^5*b^2)*x)*sqrt(b*x + a)/b^5
Time = 0.82 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.14 \[ \int x^3 (a+b x)^{5/2} (A+B x) \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {15}{2}}}{15 b} + \frac {\left (a + b x\right )^{\frac {13}{2}} \left (A b - 4 B a\right )}{13 b} + \frac {\left (a + b x\right )^{\frac {11}{2}} \left (- 3 A a b + 6 B a^{2}\right )}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \cdot \left (3 A a^{2} b - 4 B a^{3}\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- A a^{3} b + B a^{4}\right )}{7 b}\right )}{b^{4}} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{4}}{4} + \frac {B x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(B*(a + b*x)**(15/2)/(15*b) + (a + b*x)**(13/2)*(A*b - 4*B*a) /(13*b) + (a + b*x)**(11/2)*(-3*A*a*b + 6*B*a**2)/(11*b) + (a + b*x)**(9/2 )*(3*A*a**2*b - 4*B*a**3)/(9*b) + (a + b*x)**(7/2)*(-A*a**3*b + B*a**4)/(7 *b))/b**4, Ne(b, 0)), (a**(5/2)*(A*x**4/4 + B*x**5/5), True))
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.82 \[ \int x^3 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (3003 \, {\left (b x + a\right )}^{\frac {15}{2}} B - 3465 \, {\left (4 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {13}{2}} + 12285 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 5005 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 6435 \, {\left (B a^{4} - A a^{3} b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{45045 \, b^{5}} \]
2/45045*(3003*(b*x + a)^(15/2)*B - 3465*(4*B*a - A*b)*(b*x + a)^(13/2) + 1 2285*(2*B*a^2 - A*a*b)*(b*x + a)^(11/2) - 5005*(4*B*a^3 - 3*A*a^2*b)*(b*x + a)^(9/2) + 6435*(B*a^4 - A*a^3*b)*(b*x + a)^(7/2))/b^5
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (104) = 208\).
Time = 0.29 (sec) , antiderivative size = 612, normalized size of antiderivative = 5.02 \[ \int x^3 (a+b x)^{5/2} (A+B x) \, dx=\frac {2 \, {\left (\frac {1287 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} A a^{3}}{b^{3}} + \frac {143 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} B a^{3}}{b^{4}} + \frac {429 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} A a^{2}}{b^{3}} + \frac {195 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} B a^{2}}{b^{4}} + \frac {195 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} A a}{b^{3}} + \frac {45 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} B a}{b^{4}} + \frac {15 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} A}{b^{3}} + \frac {7 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )} B}{b^{4}}\right )}}{45045 \, b} \]
2/45045*(1287*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/ 2)*a^2 - 35*sqrt(b*x + a)*a^3)*A*a^3/b^3 + 143*(35*(b*x + a)^(9/2) - 180*( b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315 *sqrt(b*x + a)*a^4)*B*a^3/b^4 + 429*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7 /2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*A*a^2/b^3 + 195*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 99 0*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^ 4 - 693*sqrt(b*x + a)*a^5)*B*a^2/b^4 + 195*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155* (b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*A*a/b^3 + 45*(231*(b*x + a)^( 13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a )^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*s qrt(b*x + a)*a^6)*B*a/b^4 + 15*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/ 2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a )^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*A/b^3 + 7 *(429*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/2)* a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32175*(b*x + a)^(7/2)*a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435*sqrt(b*x + a)*a^7)*B/b^4) /b
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.91 \[ \int x^3 (a+b x)^{5/2} (A+B x) \, dx=\frac {\left (12\,B\,a^2-6\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^5}+\frac {2\,B\,{\left (a+b\,x\right )}^{15/2}}{15\,b^5}+\frac {\left (2\,A\,b-8\,B\,a\right )\,{\left (a+b\,x\right )}^{13/2}}{13\,b^5}+\frac {\left (2\,B\,a^4-2\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}-\frac {\left (8\,B\,a^3-6\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5} \]