Integrand size = 20, antiderivative size = 150 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac {2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}-\frac {4 b (8 A b-13 a B) (a+b x)^{5/2}}{429 a^3 x^{9/2}}+\frac {16 b^2 (8 A b-13 a B) (a+b x)^{5/2}}{3003 a^4 x^{7/2}}-\frac {32 b^3 (8 A b-13 a B) (a+b x)^{5/2}}{15015 a^5 x^{5/2}} \]
-2/13*A*(b*x+a)^(5/2)/a/x^(13/2)+2/143*(8*A*b-13*B*a)*(b*x+a)^(5/2)/a^2/x^ (11/2)-4/429*b*(8*A*b-13*B*a)*(b*x+a)^(5/2)/a^3/x^(9/2)+16/3003*b^2*(8*A*b -13*B*a)*(b*x+a)^(5/2)/a^4/x^(7/2)-32/15015*b^3*(8*A*b-13*B*a)*(b*x+a)^(5/ 2)/a^5/x^(5/2)
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 (a+b x)^{5/2} \left (128 A b^4 x^4+105 a^4 (11 A+13 B x)-70 a^3 b x (12 A+13 B x)+40 a^2 b^2 x^2 (14 A+13 B x)-16 a b^3 x^3 (20 A+13 B x)\right )}{15015 a^5 x^{13/2}} \]
(-2*(a + b*x)^(5/2)*(128*A*b^4*x^4 + 105*a^4*(11*A + 13*B*x) - 70*a^3*b*x* (12*A + 13*B*x) + 40*a^2*b^2*x^2*(14*A + 13*B*x) - 16*a*b^3*x^3*(20*A + 13 *B*x)))/(15015*a^5*x^(13/2))
Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 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.
\(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(8 A b-13 a B) \int \frac {(a+b x)^{3/2}}{x^{13/2}}dx}{13 a}-\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(8 A b-13 a B) \left (-\frac {6 b \int \frac {(a+b x)^{3/2}}{x^{11/2}}dx}{11 a}-\frac {2 (a+b x)^{5/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(8 A b-13 a B) \left (-\frac {6 b \left (-\frac {4 b \int \frac {(a+b x)^{3/2}}{x^{9/2}}dx}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{5/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(8 A b-13 a B) \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {(a+b x)^{3/2}}{x^{7/2}}dx}{7 a}-\frac {2 (a+b x)^{5/2}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{5/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b (a+b x)^{5/2}}{35 a^2 x^{5/2}}-\frac {2 (a+b x)^{5/2}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{5/2}}{11 a x^{11/2}}\right ) (8 A b-13 a B)}{13 a}-\frac {2 A (a+b x)^{5/2}}{13 a x^{13/2}}\) |
(-2*A*(a + b*x)^(5/2))/(13*a*x^(13/2)) - ((8*A*b - 13*a*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)
3.5.100.3.1 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 = 1.44 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (128 A \,b^{4} x^{4}-208 B a \,b^{3} x^{4}-320 A a \,b^{3} x^{3}+520 B \,a^{2} b^{2} x^{3}+560 A \,a^{2} b^{2} x^{2}-910 B \,a^{3} b \,x^{2}-840 A \,a^{3} b x +1365 B \,a^{4} x +1155 A \,a^{4}\right )}{15015 x^{\frac {13}{2}} a^{5}}\) | \(101\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (128 A \,b^{5} x^{5}-208 B a \,b^{4} x^{5}-192 a A \,b^{4} x^{4}+312 B \,a^{2} b^{3} x^{4}+240 a^{2} A \,b^{3} x^{3}-390 B \,a^{3} b^{2} x^{3}-280 a^{3} A \,b^{2} x^{2}+455 B \,a^{4} b \,x^{2}+315 a^{4} A b x +1365 a^{5} B x +1155 a^{5} A \right )}{15015 x^{\frac {13}{2}} a^{5}}\) | \(125\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{6} x^{6}-208 B a \,b^{5} x^{6}-64 A a \,b^{5} x^{5}+104 B \,a^{2} b^{4} x^{5}+48 A \,a^{2} b^{4} x^{4}-78 B \,a^{3} b^{3} x^{4}-40 A \,a^{3} b^{3} x^{3}+65 B \,a^{4} b^{2} x^{3}+35 A \,a^{4} b^{2} x^{2}+1820 B \,a^{5} b \,x^{2}+1470 A \,a^{5} b x +1365 B \,a^{6} x +1155 A \,a^{6}\right )}{15015 x^{\frac {13}{2}} a^{5}}\) | \(149\) |
-2/15015*(b*x+a)^(5/2)*(128*A*b^4*x^4-208*B*a*b^3*x^4-320*A*a*b^3*x^3+520* B*a^2*b^2*x^3+560*A*a^2*b^2*x^2-910*B*a^3*b*x^2-840*A*a^3*b*x+1365*B*a^4*x +1155*A*a^4)/x^(13/2)/a^5
Time = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 \, {\left (1155 \, A a^{6} - 16 \, {\left (13 \, B a b^{5} - 8 \, A b^{6}\right )} x^{6} + 8 \, {\left (13 \, B a^{2} b^{4} - 8 \, A a b^{5}\right )} x^{5} - 6 \, {\left (13 \, B a^{3} b^{3} - 8 \, A a^{2} b^{4}\right )} x^{4} + 5 \, {\left (13 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3}\right )} x^{3} + 35 \, {\left (52 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 105 \, {\left (13 \, B a^{6} + 14 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{15015 \, a^{5} x^{\frac {13}{2}}} \]
-2/15015*(1155*A*a^6 - 16*(13*B*a*b^5 - 8*A*b^6)*x^6 + 8*(13*B*a^2*b^4 - 8 *A*a*b^5)*x^5 - 6*(13*B*a^3*b^3 - 8*A*a^2*b^4)*x^4 + 5*(13*B*a^4*b^2 - 8*A *a^3*b^3)*x^3 + 35*(52*B*a^5*b + A*a^4*b^2)*x^2 + 105*(13*B*a^6 + 14*A*a^5 *b)*x)*sqrt(b*x + a)/(a^5*x^(13/2))
Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (120) = 240\).
Time = 0.21 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.09 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=\frac {32 \, \sqrt {b x^{2} + a x} B b^{5}}{1155 \, a^{4} x} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{6}}{15015 \, a^{5} x} - \frac {16 \, \sqrt {b x^{2} + a x} B b^{4}}{1155 \, a^{3} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} A b^{5}}{15015 \, a^{4} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} B b^{3}}{385 \, a^{2} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{4}}{5005 \, a^{3} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{231 \, a x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} A b^{3}}{3003 \, a^{2} x^{4}} + \frac {\sqrt {b x^{2} + a x} B b}{132 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{429 \, a x^{5}} + \frac {3 \, \sqrt {b x^{2} + a x} B a}{44 \, x^{6}} + \frac {3 \, \sqrt {b x^{2} + a x} A b}{715 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{4 \, x^{7}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{65 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{5 \, x^{8}} \]
32/1155*sqrt(b*x^2 + a*x)*B*b^5/(a^4*x) - 256/15015*sqrt(b*x^2 + a*x)*A*b^ 6/(a^5*x) - 16/1155*sqrt(b*x^2 + a*x)*B*b^4/(a^3*x^2) + 128/15015*sqrt(b*x ^2 + a*x)*A*b^5/(a^4*x^2) + 4/385*sqrt(b*x^2 + a*x)*B*b^3/(a^2*x^3) - 32/5 005*sqrt(b*x^2 + a*x)*A*b^4/(a^3*x^3) - 2/231*sqrt(b*x^2 + a*x)*B*b^2/(a*x ^4) + 16/3003*sqrt(b*x^2 + a*x)*A*b^3/(a^2*x^4) + 1/132*sqrt(b*x^2 + a*x)* B*b/x^5 - 2/429*sqrt(b*x^2 + a*x)*A*b^2/(a*x^5) + 3/44*sqrt(b*x^2 + a*x)*B *a/x^6 + 3/715*sqrt(b*x^2 + a*x)*A*b/x^6 - 1/4*(b*x^2 + a*x)^(3/2)*B/x^7 + 3/65*sqrt(b*x^2 + a*x)*A*a/x^7 - 1/5*(b*x^2 + a*x)^(3/2)*A/x^8
Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a^{2} b^{12} - 8 \, A a b^{13}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {13 \, {\left (13 \, B a^{3} b^{12} - 8 \, A a^{2} b^{13}\right )}}{a^{6}}\right )} + \frac {143 \, {\left (13 \, B a^{4} b^{12} - 8 \, A a^{3} b^{13}\right )}}{a^{6}}\right )} - \frac {429 \, {\left (13 \, B a^{5} b^{12} - 8 \, A a^{4} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )} + \frac {3003 \, {\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )}^{\frac {5}{2}} b}{15015 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]
2/15015*((2*(b*x + a)*(4*(b*x + a)*(2*(13*B*a^2*b^12 - 8*A*a*b^13)*(b*x + a)/a^6 - 13*(13*B*a^3*b^12 - 8*A*a^2*b^13)/a^6) + 143*(13*B*a^4*b^12 - 8*A *a^3*b^13)/a^6) - 429*(13*B*a^5*b^12 - 8*A*a^4*b^13)/a^6)*(b*x + a) + 3003 *(B*a^6*b^12 - A*a^5*b^13)/a^6)*(b*x + a)^(5/2)*b/(((b*x + a)*b - a*b)^(13 /2)*abs(b))
Time = 0.90 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{13}+x\,\left (\frac {28\,A\,b}{143}+\frac {2\,B\,a}{11}\right )+\frac {x^6\,\left (256\,A\,b^6-416\,B\,a\,b^5\right )}{15015\,a^5}-\frac {2\,b^2\,x^3\,\left (8\,A\,b-13\,B\,a\right )}{3003\,a^2}+\frac {4\,b^3\,x^4\,\left (8\,A\,b-13\,B\,a\right )}{5005\,a^3}-\frac {16\,b^4\,x^5\,\left (8\,A\,b-13\,B\,a\right )}{15015\,a^4}+\frac {2\,b\,x^2\,\left (A\,b+52\,B\,a\right )}{429\,a}\right )}{x^{13/2}} \]