Integrand size = 18, antiderivative size = 151 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 a^4 (A b-a B) (a+b x)^{5/2}}{5 b^6}-\frac {2 a^3 (4 A b-5 a B) (a+b x)^{7/2}}{7 b^6}+\frac {4 a^2 (3 A b-5 a B) (a+b x)^{9/2}}{9 b^6}-\frac {4 a (2 A b-5 a B) (a+b x)^{11/2}}{11 b^6}+\frac {2 (A b-5 a B) (a+b x)^{13/2}}{13 b^6}+\frac {2 B (a+b x)^{15/2}}{15 b^6} \]
2/5*a^4*(A*b-B*a)*(b*x+a)^(5/2)/b^6-2/7*a^3*(4*A*b-5*B*a)*(b*x+a)^(7/2)/b^ 6+4/9*a^2*(3*A*b-5*B*a)*(b*x+a)^(9/2)/b^6-4/11*a*(2*A*b-5*B*a)*(b*x+a)^(11 /2)/b^6+2/13*(A*b-5*B*a)*(b*x+a)^(13/2)/b^6+2/15*B*(b*x+a)^(15/2)/b^6
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.68 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 (a+b x)^{5/2} \left (-256 a^5 B+1680 a^2 b^3 x^2 (A+B x)+128 a^4 b (3 A+5 B x)-160 a^3 b^2 x (6 A+7 B x)-210 a b^4 x^3 (12 A+11 B x)+231 b^5 x^4 (15 A+13 B x)\right )}{45045 b^6} \]
(2*(a + b*x)^(5/2)*(-256*a^5*B + 1680*a^2*b^3*x^2*(A + B*x) + 128*a^4*b*(3 *A + 5*B*x) - 160*a^3*b^2*x*(6*A + 7*B*x) - 210*a*b^4*x^3*(12*A + 11*B*x) + 231*b^5*x^4*(15*A + 13*B*x)))/(45045*b^6)
Time = 0.27 (sec) , antiderivative size = 151, 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.
| \(\displaystyle \int x^4 (a+b x)^{3/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (-\frac {a^4 (a+b x)^{3/2} (a B-A b)}{b^5}+\frac {a^3 (a+b x)^{5/2} (5 a B-4 A b)}{b^5}-\frac {2 a^2 (a+b x)^{7/2} (5 a B-3 A b)}{b^5}+\frac {(a+b x)^{11/2} (A b-5 a B)}{b^5}+\frac {2 a (a+b x)^{9/2} (5 a B-2 A b)}{b^5}+\frac {B (a+b x)^{13/2}}{b^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^4 (a+b x)^{5/2} (A b-a B)}{5 b^6}-\frac {2 a^3 (a+b x)^{7/2} (4 A b-5 a B)}{7 b^6}+\frac {4 a^2 (a+b x)^{9/2} (3 A b-5 a B)}{9 b^6}+\frac {2 (a+b x)^{13/2} (A b-5 a B)}{13 b^6}-\frac {4 a (a+b x)^{11/2} (2 A b-5 a B)}{11 b^6}+\frac {2 B (a+b x)^{15/2}}{15 b^6}\) |
(2*a^4*(A*b - a*B)*(a + b*x)^(5/2))/(5*b^6) - (2*a^3*(4*A*b - 5*a*B)*(a + b*x)^(7/2))/(7*b^6) + (4*a^2*(3*A*b - 5*a*B)*(a + b*x)^(9/2))/(9*b^6) - (4 *a*(2*A*b - 5*a*B)*(a + b*x)^(11/2))/(11*b^6) + (2*(A*b - 5*a*B)*(a + b*x) ^(13/2))/(13*b^6) + (2*B*(a + b*x)^(15/2))/(15*b^6)
3.4.98.3.1 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 = 91, normalized size of antiderivative = 0.60
| method | result | size |
| pseudoelliptic | \(\frac {256 \left (b x +a \right )^{\frac {5}{2}} \left (\frac {1155 x^{4} \left (\frac {13 B x}{15}+A \right ) b^{5}}{128}-\frac {105 x^{3} \left (\frac {11 B x}{12}+A \right ) a \,b^{4}}{16}+\frac {35 a^{2} x^{2} \left (B x +A \right ) b^{3}}{8}-\frac {5 x \left (\frac {7 B x}{6}+A \right ) a^{3} b^{2}}{2}+a^{4} \left (\frac {5 B x}{3}+A \right ) b -\frac {2 a^{5} B}{3}\right )}{15015 b^{6}}\) | \(91\) |
| gosper | \(\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (3003 b^{5} B \,x^{5}+3465 A \,b^{5} x^{4}-2310 B a \,b^{4} x^{4}-2520 A a \,b^{4} x^{3}+1680 B \,a^{2} b^{3} x^{3}+1680 A \,a^{2} b^{3} x^{2}-1120 B \,a^{3} b^{2} x^{2}-960 a^{3} b^{2} A x +640 a^{4} b B x +384 a^{4} b A -256 a^{5} B \right )}{45045 b^{6}}\) | \(119\) |
| derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {15}{2}}}{15}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (B \,a^{4}-4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 a^{4} \left (A b -B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{6}}\) | \(138\) |
| default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {15}{2}}}{15}+\frac {2 \left (A b -5 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {2 \left (6 a^{2} B -4 a \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-4 a^{3} B +6 a^{2} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (B \,a^{4}-4 a^{3} \left (A b -B a \right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 a^{4} \left (A b -B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}}{b^{6}}\) | \(138\) |
| trager | \(\frac {2 \left (3003 B \,b^{7} x^{7}+3465 A \,b^{7} x^{6}+3696 B a \,b^{6} x^{6}+4410 A a \,b^{6} x^{5}+63 B \,a^{2} b^{5} x^{5}+105 A \,a^{2} b^{5} x^{4}-70 B \,a^{3} b^{4} x^{4}-120 A \,a^{3} b^{4} x^{3}+80 B \,a^{4} b^{3} x^{3}+144 A \,a^{4} b^{3} x^{2}-96 B \,a^{5} b^{2} x^{2}-192 A \,a^{5} b^{2} x +128 B \,a^{6} b x +384 A \,a^{6} b -256 B \,a^{7}\right ) \sqrt {b x +a}}{45045 b^{6}}\) | \(167\) |
| risch | \(\frac {2 \left (3003 B \,b^{7} x^{7}+3465 A \,b^{7} x^{6}+3696 B a \,b^{6} x^{6}+4410 A a \,b^{6} x^{5}+63 B \,a^{2} b^{5} x^{5}+105 A \,a^{2} b^{5} x^{4}-70 B \,a^{3} b^{4} x^{4}-120 A \,a^{3} b^{4} x^{3}+80 B \,a^{4} b^{3} x^{3}+144 A \,a^{4} b^{3} x^{2}-96 B \,a^{5} b^{2} x^{2}-192 A \,a^{5} b^{2} x +128 B \,a^{6} b x +384 A \,a^{6} b -256 B \,a^{7}\right ) \sqrt {b x +a}}{45045 b^{6}}\) | \(167\) |
256/15015*(b*x+a)^(5/2)*(1155/128*x^4*(13/15*B*x+A)*b^5-105/16*x^3*(11/12* B*x+A)*a*b^4+35/8*a^2*x^2*(B*x+A)*b^3-5/2*x*(7/6*B*x+A)*a^3*b^2+a^4*(5/3*B *x+A)*b-2/3*a^5*B)/b^6
Time = 0.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 \, {\left (3003 \, B b^{7} x^{7} - 256 \, B a^{7} + 384 \, A a^{6} b + 231 \, {\left (16 \, B a b^{6} + 15 \, A b^{7}\right )} x^{6} + 63 \, {\left (B a^{2} b^{5} + 70 \, A a b^{6}\right )} x^{5} - 35 \, {\left (2 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 40 \, {\left (2 \, B a^{4} b^{3} - 3 \, A a^{3} b^{4}\right )} x^{3} - 48 \, {\left (2 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3}\right )} x^{2} + 64 \, {\left (2 \, B a^{6} b - 3 \, A a^{5} b^{2}\right )} x\right )} \sqrt {b x + a}}{45045 \, b^{6}} \]
2/45045*(3003*B*b^7*x^7 - 256*B*a^7 + 384*A*a^6*b + 231*(16*B*a*b^6 + 15*A *b^7)*x^6 + 63*(B*a^2*b^5 + 70*A*a*b^6)*x^5 - 35*(2*B*a^3*b^4 - 3*A*a^2*b^ 5)*x^4 + 40*(2*B*a^4*b^3 - 3*A*a^3*b^4)*x^3 - 48*(2*B*a^5*b^2 - 3*A*a^4*b^ 3)*x^2 + 64*(2*B*a^6*b - 3*A*a^5*b^2)*x)*sqrt(b*x + a)/b^6
Time = 0.79 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.11 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\begin {cases} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {15}{2}}}{15 b} + \frac {\left (a + b x\right )^{\frac {13}{2}} \left (A b - 5 B a\right )}{13 b} + \frac {\left (a + b x\right )^{\frac {11}{2}} \left (- 4 A a b + 10 B a^{2}\right )}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \cdot \left (6 A a^{2} b - 10 B a^{3}\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 4 A a^{3} b + 5 B a^{4}\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (A a^{4} b - B a^{5}\right )}{5 b}\right )}{b^{5}} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{5}}{5} + \frac {B x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(B*(a + b*x)**(15/2)/(15*b) + (a + b*x)**(13/2)*(A*b - 5*B*a) /(13*b) + (a + b*x)**(11/2)*(-4*A*a*b + 10*B*a**2)/(11*b) + (a + b*x)**(9/ 2)*(6*A*a**2*b - 10*B*a**3)/(9*b) + (a + b*x)**(7/2)*(-4*A*a**3*b + 5*B*a* *4)/(7*b) + (a + b*x)**(5/2)*(A*a**4*b - B*a**5)/(5*b))/b**5, Ne(b, 0)), ( a**(3/2)*(A*x**5/5 + B*x**6/6), True))
Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 \, {\left (3003 \, {\left (b x + a\right )}^{\frac {15}{2}} B - 3465 \, {\left (5 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {13}{2}} + 8190 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 10010 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 6435 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 9009 \, {\left (B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {5}{2}}\right )}}{45045 \, b^{6}} \]
2/45045*(3003*(b*x + a)^(15/2)*B - 3465*(5*B*a - A*b)*(b*x + a)^(13/2) + 8 190*(5*B*a^2 - 2*A*a*b)*(b*x + a)^(11/2) - 10010*(5*B*a^3 - 3*A*a^2*b)*(b* x + a)^(9/2) + 6435*(5*B*a^4 - 4*A*a^3*b)*(b*x + a)^(7/2) - 9009*(B*a^5 - A*a^4*b)*(b*x + a)^(5/2))/b^6
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (128) = 256\).
Time = 0.28 (sec) , antiderivative size = 494, normalized size of antiderivative = 3.27 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {2 \, {\left (\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 )} A a^{2}}{b^{4}} + \frac {65 \, {\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^{5}} + \frac {130 \, {\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^{4}} + \frac {30 \, {\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^{5}} + \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^{4}} + \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^{5}}\right )}}{45045 \, b} \]
2/45045*(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)*A*a^2/b^4 + 65 *(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)*B*a^2/b^5 + 130*(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 - 69 3*sqrt(b*x + a)*a^5)*A*a/b^4 + 30*(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)*B*a/b^ 5 + 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^4 + 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^5)/b
Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int x^4 (a+b x)^{3/2} (A+B x) \, dx=\frac {\left (20\,B\,a^2-8\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}+\frac {2\,B\,{\left (a+b\,x\right )}^{15/2}}{15\,b^6}+\frac {\left (2\,A\,b-10\,B\,a\right )\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}-\frac {\left (2\,B\,a^5-2\,A\,a^4\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^6}+\frac {\left (10\,B\,a^4-8\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^6}-\frac {\left (20\,B\,a^3-12\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^6} \]