Integrand size = 20, antiderivative size = 183 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{5/2}}{39 a^2 x^{13/2}}-\frac {16 b (2 A b-3 a B) (a+b x)^{5/2}}{429 a^3 x^{11/2}}+\frac {32 b^2 (2 A b-3 a B) (a+b x)^{5/2}}{1287 a^4 x^{9/2}}-\frac {128 b^3 (2 A b-3 a B) (a+b x)^{5/2}}{9009 a^5 x^{7/2}}+\frac {256 b^4 (2 A b-3 a B) (a+b x)^{5/2}}{45045 a^6 x^{5/2}} \]
-2/15*A*(b*x+a)^(5/2)/a/x^(15/2)+2/39*(2*A*b-3*B*a)*(b*x+a)^(5/2)/a^2/x^(1 3/2)-16/429*b*(2*A*b-3*B*a)*(b*x+a)^(5/2)/a^3/x^(11/2)+32/1287*b^2*(2*A*b- 3*B*a)*(b*x+a)^(5/2)/a^4/x^(9/2)-128/9009*b^3*(2*A*b-3*B*a)*(b*x+a)^(5/2)/ a^5/x^(7/2)+256/45045*b^4*(2*A*b-3*B*a)*(b*x+a)^(5/2)/a^6/x^(5/2)
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.61 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 (a+b x)^{5/2} \left (-256 A b^5 x^5+1680 a^3 b^2 x^2 (A+B x)+128 a b^4 x^4 (5 A+3 B x)-160 a^2 b^3 x^3 (7 A+6 B x)-210 a^4 b x (11 A+12 B x)+231 a^5 (13 A+15 B x)\right )}{45045 a^6 x^{15/2}} \]
(-2*(a + b*x)^(5/2)*(-256*A*b^5*x^5 + 1680*a^3*b^2*x^2*(A + B*x) + 128*a*b ^4*x^4*(5*A + 3*B*x) - 160*a^2*b^3*x^3*(7*A + 6*B*x) - 210*a^4*b*x*(11*A + 12*B*x) + 231*a^5*(13*A + 15*B*x)))/(45045*a^6*x^(15/2))
Time = 0.22 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {87, 55, 55, 55, 55, 48}
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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(2 A b-3 a B) \int \frac {(a+b x)^{3/2}}{x^{15/2}}dx}{3 a}-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(2 A b-3 a B) \left (-\frac {8 b \int \frac {(a+b x)^{3/2}}{x^{13/2}}dx}{13 a}-\frac {2 (a+b x)^{5/2}}{13 a x^{13/2}}\right )}{3 a}-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(2 A b-3 a B) \left (-\frac {8 b \left (-\frac {6 b \int \frac {(a+b x)^{3/2}}{x^{11/2}}dx}{11 a}-\frac {2 (a+b x)^{5/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 (a+b x)^{5/2}}{13 a x^{13/2}}\right )}{3 a}-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(2 A b-3 a B) \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {(a+b x)^{3/2}}{x^{9/2}}dx}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{5/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 (a+b x)^{5/2}}{13 a x^{13/2}}\right )}{3 a}-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(2 A b-3 a B) \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {(a+b x)^{3/2}}{x^{7/2}}dx}{7 a}-\frac {2 (a+b x)^{5/2}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{5/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 (a+b x)^{5/2}}{13 a x^{13/2}}\right )}{3 a}-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b (a+b x)^{5/2}}{35 a^2 x^{5/2}}-\frac {2 (a+b x)^{5/2}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{5/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 (a+b x)^{5/2}}{13 a x^{13/2}}\right ) (2 A b-3 a B)}{3 a}-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}\) |
(-2*A*(a + b*x)^(5/2))/(15*a*x^(15/2)) - ((2*A*b - 3*a*B)*((-2*(a + b*x)^( 5/2))/(13*a*x^(13/2)) - (8*b*((-2*(a + b*x)^(5/2))/(11*a*x^(11/2)) - (6*b* ((-2*(a + b*x)^(5/2))/(9*a*x^(9/2)) - (4*b*((-2*(a + b*x)^(5/2))/(7*a*x^(7 /2)) + (4*b*(a + b*x)^(5/2))/(35*a^2*x^(5/2))))/(9*a)))/(11*a)))/(13*a)))/ (3*a)
3.6.1.3.1 Defintions of rubi rules used
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] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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]))))
Time = 0.51 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.68
| method | result | size |
| gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-256 A \,b^{5} x^{5}+384 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-960 B \,a^{2} b^{3} x^{4}-1120 a^{2} A \,b^{3} x^{3}+1680 B \,a^{3} b^{2} x^{3}+1680 a^{3} A \,b^{2} x^{2}-2520 B \,a^{4} b \,x^{2}-2310 a^{4} A b x +3465 a^{5} B x +3003 a^{5} A \right )}{45045 x^{\frac {15}{2}} a^{6}}\) | \(125\) |
| default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-256 A \,b^{6} x^{6}+384 B a \,b^{5} x^{6}+384 A a \,b^{5} x^{5}-576 B \,a^{2} b^{4} x^{5}-480 A \,a^{2} b^{4} x^{4}+720 B \,a^{3} b^{3} x^{4}+560 A \,a^{3} b^{3} x^{3}-840 B \,a^{4} b^{2} x^{3}-630 A \,a^{4} b^{2} x^{2}+945 B \,a^{5} b \,x^{2}+693 A \,a^{5} b x +3465 B \,a^{6} x +3003 A \,a^{6}\right )}{45045 x^{\frac {15}{2}} a^{6}}\) | \(149\) |
| risch | \(-\frac {2 \sqrt {b x +a}\, \left (-256 A \,b^{7} x^{7}+384 B a \,b^{6} x^{7}+128 A a \,b^{6} x^{6}-192 B \,a^{2} b^{5} x^{6}-96 A \,a^{2} b^{5} x^{5}+144 B \,a^{3} b^{4} x^{5}+80 A \,a^{3} b^{4} x^{4}-120 B \,a^{4} b^{3} x^{4}-70 A \,a^{4} b^{3} x^{3}+105 B \,a^{5} b^{2} x^{3}+63 A \,a^{5} b^{2} x^{2}+4410 B \,a^{6} b \,x^{2}+3696 A \,a^{6} b x +3465 B \,a^{7} x +3003 A \,a^{7}\right )}{45045 x^{\frac {15}{2}} a^{6}}\) | \(173\) |
-2/45045*(b*x+a)^(5/2)*(-256*A*b^5*x^5+384*B*a*b^4*x^5+640*A*a*b^4*x^4-960 *B*a^2*b^3*x^4-1120*A*a^2*b^3*x^3+1680*B*a^3*b^2*x^3+1680*A*a^3*b^2*x^2-25 20*B*a^4*b*x^2-2310*A*a^4*b*x+3465*B*a^5*x+3003*A*a^5)/x^(15/2)/a^6
Time = 0.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 \, {\left (3003 \, A a^{7} + 128 \, {\left (3 \, B a b^{6} - 2 \, A b^{7}\right )} x^{7} - 64 \, {\left (3 \, B a^{2} b^{5} - 2 \, A a b^{6}\right )} x^{6} + 48 \, {\left (3 \, B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} x^{5} - 40 \, {\left (3 \, B a^{4} b^{3} - 2 \, A a^{3} b^{4}\right )} x^{4} + 35 \, {\left (3 \, B a^{5} b^{2} - 2 \, A a^{4} b^{3}\right )} x^{3} + 63 \, {\left (70 \, B a^{6} b + A a^{5} b^{2}\right )} x^{2} + 231 \, {\left (15 \, B a^{7} + 16 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{45045 \, a^{6} x^{\frac {15}{2}}} \]
-2/45045*(3003*A*a^7 + 128*(3*B*a*b^6 - 2*A*b^7)*x^7 - 64*(3*B*a^2*b^5 - 2 *A*a*b^6)*x^6 + 48*(3*B*a^3*b^4 - 2*A*a^2*b^5)*x^5 - 40*(3*B*a^4*b^3 - 2*A *a^3*b^4)*x^4 + 35*(3*B*a^5*b^2 - 2*A*a^4*b^3)*x^3 + 63*(70*B*a^6*b + A*a^ 5*b^2)*x^2 + 231*(15*B*a^7 + 16*A*a^6*b)*x)*sqrt(b*x + a)/(a^6*x^(15/2))
Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (147) = 294\).
Time = 0.20 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {256 \, \sqrt {b x^{2} + a x} B b^{6}}{15015 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{7}}{45045 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{5}}{15015 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{6}}{45045 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{4}}{5005 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{5}}{15015 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b^{3}}{3003 \, a^{2} x^{4}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{4}}{9009 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{429 \, a x^{5}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{3}}{1287 \, a^{2} x^{5}} + \frac {3 \, \sqrt {b x^{2} + a x} B b}{715 \, x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{715 \, a x^{6}} + \frac {3 \, \sqrt {b x^{2} + a x} B a}{65 \, x^{7}} + \frac {\sqrt {b x^{2} + a x} A b}{390 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{5 \, x^{8}} + \frac {\sqrt {b x^{2} + a x} A a}{30 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{6 \, x^{9}} \]
-256/15015*sqrt(b*x^2 + a*x)*B*b^6/(a^5*x) + 512/45045*sqrt(b*x^2 + a*x)*A *b^7/(a^6*x) + 128/15015*sqrt(b*x^2 + a*x)*B*b^5/(a^4*x^2) - 256/45045*sqr t(b*x^2 + a*x)*A*b^6/(a^5*x^2) - 32/5005*sqrt(b*x^2 + a*x)*B*b^4/(a^3*x^3) + 64/15015*sqrt(b*x^2 + a*x)*A*b^5/(a^4*x^3) + 16/3003*sqrt(b*x^2 + a*x)* B*b^3/(a^2*x^4) - 32/9009*sqrt(b*x^2 + a*x)*A*b^4/(a^3*x^4) - 2/429*sqrt(b *x^2 + a*x)*B*b^2/(a*x^5) + 4/1287*sqrt(b*x^2 + a*x)*A*b^3/(a^2*x^5) + 3/7 15*sqrt(b*x^2 + a*x)*B*b/x^6 - 2/715*sqrt(b*x^2 + a*x)*A*b^2/(a*x^6) + 3/6 5*sqrt(b*x^2 + a*x)*B*a/x^7 + 1/390*sqrt(b*x^2 + a*x)*A*b/x^7 - 1/5*(b*x^2 + a*x)^(3/2)*B/x^8 + 1/30*sqrt(b*x^2 + a*x)*A*a/x^8 - 1/6*(b*x^2 + a*x)^( 3/2)*A/x^9
Time = 0.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (3 \, B a^{2} b^{14} - 2 \, A a b^{15}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {15 \, {\left (3 \, B a^{3} b^{14} - 2 \, A a^{2} b^{15}\right )}}{a^{7}}\right )} + \frac {195 \, {\left (3 \, B a^{4} b^{14} - 2 \, A a^{3} b^{15}\right )}}{a^{7}}\right )} - \frac {715 \, {\left (3 \, B a^{5} b^{14} - 2 \, A a^{4} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {6435 \, {\left (3 \, B a^{6} b^{14} - 2 \, A a^{5} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )} - \frac {9009 \, {\left (B a^{7} b^{14} - A a^{6} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )}^{\frac {5}{2}} b}{45045 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {15}{2}} {\left | b \right |}} \]
-2/45045*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(3*B*a^2*b^14 - 2*A*a*b^15)*(b*x + a)/a^7 - 15*(3*B*a^3*b^14 - 2*A*a^2*b^15)/a^7) + 195*(3*B*a^4*b^14 - 2* A*a^3*b^15)/a^7) - 715*(3*B*a^5*b^14 - 2*A*a^4*b^15)/a^7)*(b*x + a) + 6435 *(3*B*a^6*b^14 - 2*A*a^5*b^15)/a^7)*(b*x + a) - 9009*(B*a^7*b^14 - A*a^6*b ^15)/a^7)*(b*x + a)^(5/2)*b/(((b*x + a)*b - a*b)^(15/2)*abs(b))
Time = 1.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{15}+x\,\left (\frac {32\,A\,b}{195}+\frac {2\,B\,a}{13}\right )-\frac {x^7\,\left (512\,A\,b^7-768\,B\,a\,b^6\right )}{45045\,a^6}-\frac {2\,b^2\,x^3\,\left (2\,A\,b-3\,B\,a\right )}{1287\,a^2}+\frac {16\,b^3\,x^4\,\left (2\,A\,b-3\,B\,a\right )}{9009\,a^3}-\frac {32\,b^4\,x^5\,\left (2\,A\,b-3\,B\,a\right )}{15015\,a^4}+\frac {128\,b^5\,x^6\,\left (2\,A\,b-3\,B\,a\right )}{45045\,a^5}+\frac {2\,b\,x^2\,\left (A\,b+70\,B\,a\right )}{715\,a}\right )}{x^{15/2}} \]
-((a + b*x)^(1/2)*((2*A*a)/15 + x*((32*A*b)/195 + (2*B*a)/13) - (x^7*(512* A*b^7 - 768*B*a*b^6))/(45045*a^6) - (2*b^2*x^3*(2*A*b - 3*B*a))/(1287*a^2) + (16*b^3*x^4*(2*A*b - 3*B*a))/(9009*a^3) - (32*b^4*x^5*(2*A*b - 3*B*a))/ (15015*a^4) + (128*b^5*x^6*(2*A*b - 3*B*a))/(45045*a^5) + (2*b*x^2*(A*b + 70*B*a))/(715*a)))/x^(15/2)