Integrand size = 20, antiderivative size = 183 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac {2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac {16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}+\frac {32 b^2 (10 A b-13 a B) (a+b x)^{3/2}}{3003 a^4 x^{7/2}}-\frac {128 b^3 (10 A b-13 a B) (a+b x)^{3/2}}{15015 a^5 x^{5/2}}+\frac {256 b^4 (10 A b-13 a B) (a+b x)^{3/2}}{45045 a^6 x^{3/2}} \]
-2/13*A*(b*x+a)^(3/2)/a/x^(13/2)+2/143*(10*A*b-13*B*a)*(b*x+a)^(3/2)/a^2/x ^(11/2)-16/1287*b*(10*A*b-13*B*a)*(b*x+a)^(3/2)/a^3/x^(9/2)+32/3003*b^2*(1 0*A*b-13*B*a)*(b*x+a)^(3/2)/a^4/x^(7/2)-128/15015*b^3*(10*A*b-13*B*a)*(b*x +a)^(3/2)/a^5/x^(5/2)+256/45045*b^4*(10*A*b-13*B*a)*(b*x+a)^(3/2)/a^6/x^(3 /2)
Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {2 (a+b x)^{3/2} \left (-1280 A b^5 x^5+315 a^5 (11 A+13 B x)+128 a b^4 x^4 (15 A+13 B x)-96 a^2 b^3 x^3 (25 A+26 B x)+80 a^3 b^2 x^2 (35 A+39 B x)-70 a^4 b x (45 A+52 B x)\right )}{45045 a^6 x^{13/2}} \]
(-2*(a + b*x)^(3/2)*(-1280*A*b^5*x^5 + 315*a^5*(11*A + 13*B*x) + 128*a*b^4 *x^4*(15*A + 13*B*x) - 96*a^2*b^3*x^3*(25*A + 26*B*x) + 80*a^3*b^2*x^2*(35 *A + 39*B*x) - 70*a^4*b*x*(45*A + 52*B*x)))/(45045*a^6*x^(13/2))
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.
| \(\displaystyle \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(10 A b-13 a B) \int \frac {\sqrt {a+b x}}{x^{13/2}}dx}{13 a}-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(10 A b-13 a B) \left (-\frac {8 b \int \frac {\sqrt {a+b x}}{x^{11/2}}dx}{11 a}-\frac {2 (a+b x)^{3/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(10 A b-13 a B) \left (-\frac {8 b \left (-\frac {2 b \int \frac {\sqrt {a+b x}}{x^{9/2}}dx}{3 a}-\frac {2 (a+b x)^{3/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{3/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(10 A b-13 a B) \left (-\frac {8 b \left (-\frac {2 b \left (-\frac {4 b \int \frac {\sqrt {a+b x}}{x^{7/2}}dx}{7 a}-\frac {2 (a+b x)^{3/2}}{7 a x^{7/2}}\right )}{3 a}-\frac {2 (a+b x)^{3/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{3/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(10 A b-13 a B) \left (-\frac {8 b \left (-\frac {2 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\sqrt {a+b x}}{x^{5/2}}dx}{5 a}-\frac {2 (a+b x)^{3/2}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 (a+b x)^{3/2}}{7 a x^{7/2}}\right )}{3 a}-\frac {2 (a+b x)^{3/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{3/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (-\frac {8 b \left (-\frac {2 b \left (-\frac {4 b \left (\frac {4 b (a+b x)^{3/2}}{15 a^2 x^{3/2}}-\frac {2 (a+b x)^{3/2}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 (a+b x)^{3/2}}{7 a x^{7/2}}\right )}{3 a}-\frac {2 (a+b x)^{3/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{3/2}}{11 a x^{11/2}}\right ) (10 A b-13 a B)}{13 a}-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}\) |
(-2*A*(a + b*x)^(3/2))/(13*a*x^(13/2)) - ((10*A*b - 13*a*B)*((-2*(a + b*x) ^(3/2))/(11*a*x^(11/2)) - (8*b*((-2*(a + b*x)^(3/2))/(9*a*x^(9/2)) - (2*b* ((-2*(a + b*x)^(3/2))/(7*a*x^(7/2)) - (4*b*((-2*(a + b*x)^(3/2))/(5*a*x^(5 /2)) + (4*b*(a + b*x)^(3/2))/(15*a^2*x^(3/2))))/(7*a)))/(3*a)))/(11*a)))/( 13*a)
3.5.89.3.1 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 | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} x^{5}+1664 B a \,b^{4} x^{5}+1920 a A \,b^{4} x^{4}-2496 B \,a^{2} b^{3} x^{4}-2400 a^{2} A \,b^{3} x^{3}+3120 B \,a^{3} b^{2} x^{3}+2800 a^{3} A \,b^{2} x^{2}-3640 B \,a^{4} b \,x^{2}-3150 a^{4} A b x +4095 a^{5} B x +3465 a^{5} A \right )}{45045 x^{\frac {13}{2}} a^{6}}\) | \(125\) |
| default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} x^{5}+1664 B a \,b^{4} x^{5}+1920 a A \,b^{4} x^{4}-2496 B \,a^{2} b^{3} x^{4}-2400 a^{2} A \,b^{3} x^{3}+3120 B \,a^{3} b^{2} x^{3}+2800 a^{3} A \,b^{2} x^{2}-3640 B \,a^{4} b \,x^{2}-3150 a^{4} A b x +4095 a^{5} B x +3465 a^{5} A \right )}{45045 x^{\frac {13}{2}} a^{6}}\) | \(125\) |
| risch | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{6} x^{6}+1664 B a \,b^{5} x^{6}+640 A a \,b^{5} x^{5}-832 B \,a^{2} b^{4} x^{5}-480 A \,a^{2} b^{4} x^{4}+624 B \,a^{3} b^{3} x^{4}+400 A \,a^{3} b^{3} x^{3}-520 B \,a^{4} b^{2} x^{3}-350 A \,a^{4} b^{2} x^{2}+455 B \,a^{5} b \,x^{2}+315 A \,a^{5} b x +4095 B \,a^{6} x +3465 A \,a^{6}\right )}{45045 x^{\frac {13}{2}} a^{6}}\) | \(149\) |
-2/45045*(b*x+a)^(3/2)*(-1280*A*b^5*x^5+1664*B*a*b^4*x^5+1920*A*a*b^4*x^4- 2496*B*a^2*b^3*x^4-2400*A*a^2*b^3*x^3+3120*B*a^3*b^2*x^3+2800*A*a^3*b^2*x^ 2-3640*B*a^4*b*x^2-3150*A*a^4*b*x+4095*B*a^5*x+3465*A*a^5)/x^(13/2)/a^6
Time = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {2 \, {\left (3465 \, A a^{6} + 128 \, {\left (13 \, B a b^{5} - 10 \, A b^{6}\right )} x^{6} - 64 \, {\left (13 \, B a^{2} b^{4} - 10 \, A a b^{5}\right )} x^{5} + 48 \, {\left (13 \, B a^{3} b^{3} - 10 \, A a^{2} b^{4}\right )} x^{4} - 40 \, {\left (13 \, B a^{4} b^{2} - 10 \, A a^{3} b^{3}\right )} x^{3} + 35 \, {\left (13 \, B a^{5} b - 10 \, A a^{4} b^{2}\right )} x^{2} + 315 \, {\left (13 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {b x + a}}{45045 \, a^{6} x^{\frac {13}{2}}} \]
-2/45045*(3465*A*a^6 + 128*(13*B*a*b^5 - 10*A*b^6)*x^6 - 64*(13*B*a^2*b^4 - 10*A*a*b^5)*x^5 + 48*(13*B*a^3*b^3 - 10*A*a^2*b^4)*x^4 - 40*(13*B*a^4*b^ 2 - 10*A*a^3*b^3)*x^3 + 35*(13*B*a^5*b - 10*A*a^4*b^2)*x^2 + 315*(13*B*a^6 + A*a^5*b)*x)*sqrt(b*x + a)/(a^6*x^(13/2))
Timed out. \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {256 \, \sqrt {b x^{2} + a x} B b^{5}}{3465 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{6}}{9009 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{4}}{3465 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{5}}{9009 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{3}}{1155 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{4}}{3003 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b^{2}}{693 \, a^{2} x^{4}} - \frac {160 \, \sqrt {b x^{2} + a x} A b^{3}}{9009 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B b}{99 \, a x^{5}} + \frac {20 \, \sqrt {b x^{2} + a x} A b^{2}}{1287 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{11 \, x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A b}{143 \, a x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{13 \, x^{7}} \]
-256/3465*sqrt(b*x^2 + a*x)*B*b^5/(a^5*x) + 512/9009*sqrt(b*x^2 + a*x)*A*b ^6/(a^6*x) + 128/3465*sqrt(b*x^2 + a*x)*B*b^4/(a^4*x^2) - 256/9009*sqrt(b* x^2 + a*x)*A*b^5/(a^5*x^2) - 32/1155*sqrt(b*x^2 + a*x)*B*b^3/(a^3*x^3) + 6 4/3003*sqrt(b*x^2 + a*x)*A*b^4/(a^4*x^3) + 16/693*sqrt(b*x^2 + a*x)*B*b^2/ (a^2*x^4) - 160/9009*sqrt(b*x^2 + a*x)*A*b^3/(a^3*x^4) - 2/99*sqrt(b*x^2 + a*x)*B*b/(a*x^5) + 20/1287*sqrt(b*x^2 + a*x)*A*b^2/(a^2*x^5) - 2/11*sqrt( b*x^2 + a*x)*B/x^6 - 2/143*sqrt(b*x^2 + a*x)*A*b/(a*x^6) - 2/13*sqrt(b*x^2 + a*x)*A/x^7
Time = 0.31 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a b^{12} - 10 \, A b^{13}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {13 \, {\left (13 \, B a^{2} b^{12} - 10 \, A a b^{13}\right )}}{a^{6}}\right )} + \frac {143 \, {\left (13 \, B a^{3} b^{12} - 10 \, A a^{2} b^{13}\right )}}{a^{6}}\right )} - \frac {429 \, {\left (13 \, B a^{4} b^{12} - 10 \, A a^{3} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )} + \frac {3003 \, {\left (13 \, B a^{5} b^{12} - 10 \, A a^{4} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )} - \frac {15015 \, {\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )}^{\frac {3}{2}} b}{45045 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]
-2/45045*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(13*B*a*b^12 - 10*A*b^13)*(b*x + a)/a^6 - 13*(13*B*a^2*b^12 - 10*A*a*b^13)/a^6) + 143*(13*B*a^3*b^12 - 10* A*a^2*b^13)/a^6) - 429*(13*B*a^4*b^12 - 10*A*a^3*b^13)/a^6)*(b*x + a) + 30 03*(13*B*a^5*b^12 - 10*A*a^4*b^13)/a^6)*(b*x + a) - 15015*(B*a^6*b^12 - A* a^5*b^13)/a^6)*(b*x + a)^(3/2)*b/(((b*x + a)*b - a*b)^(13/2)*abs(b))
Time = 0.88 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{13}+\frac {2\,x\,\left (A\,b+13\,B\,a\right )}{143\,a}+\frac {16\,b^2\,x^3\,\left (10\,A\,b-13\,B\,a\right )}{9009\,a^3}-\frac {32\,b^3\,x^4\,\left (10\,A\,b-13\,B\,a\right )}{15015\,a^4}+\frac {128\,b^4\,x^5\,\left (10\,A\,b-13\,B\,a\right )}{45045\,a^5}-\frac {256\,b^5\,x^6\,\left (10\,A\,b-13\,B\,a\right )}{45045\,a^6}-\frac {2\,b\,x^2\,\left (10\,A\,b-13\,B\,a\right )}{1287\,a^2}\right )}{x^{13/2}} \]
-((a + b*x)^(1/2)*((2*A)/13 + (2*x*(A*b + 13*B*a))/(143*a) + (16*b^2*x^3*( 10*A*b - 13*B*a))/(9009*a^3) - (32*b^3*x^4*(10*A*b - 13*B*a))/(15015*a^4) + (128*b^4*x^5*(10*A*b - 13*B*a))/(45045*a^5) - (256*b^5*x^6*(10*A*b - 13* B*a))/(45045*a^6) - (2*b*x^2*(10*A*b - 13*B*a))/(1287*a^2)))/x^(13/2)