Integrand size = 20, antiderivative size = 150 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}+\frac {2 (8 A b-15 a B) (a+b x)^{7/2}}{195 a^2 x^{13/2}}-\frac {4 b (8 A b-15 a B) (a+b x)^{7/2}}{715 a^3 x^{11/2}}+\frac {16 b^2 (8 A b-15 a B) (a+b x)^{7/2}}{6435 a^4 x^{9/2}}-\frac {32 b^3 (8 A b-15 a B) (a+b x)^{7/2}}{45045 a^5 x^{7/2}} \]
-2/15*A*(b*x+a)^(7/2)/a/x^(15/2)+2/195*(8*A*b-15*B*a)*(b*x+a)^(7/2)/a^2/x^ (13/2)-4/715*b*(8*A*b-15*B*a)*(b*x+a)^(7/2)/a^3/x^(11/2)+16/6435*b^2*(8*A* b-15*B*a)*(b*x+a)^(7/2)/a^4/x^(9/2)-32/45045*b^3*(8*A*b-15*B*a)*(b*x+a)^(7 /2)/a^5/x^(7/2)
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 (a+b x)^{7/2} \left (128 A b^4 x^4+168 a^2 b^2 x^2 (6 A+5 B x)+231 a^4 (13 A+15 B x)-16 a b^3 x^3 (28 A+15 B x)-42 a^3 b x (44 A+45 B x)\right )}{45045 a^5 x^{15/2}} \]
(-2*(a + b*x)^(7/2)*(128*A*b^4*x^4 + 168*a^2*b^2*x^2*(6*A + 5*B*x) + 231*a ^4*(13*A + 15*B*x) - 16*a*b^3*x^3*(28*A + 15*B*x) - 42*a^3*b*x*(44*A + 45* B*x)))/(45045*a^5*x^(15/2))
Time = 0.21 (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.
\(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(8 A b-15 a B) \int \frac {(a+b x)^{5/2}}{x^{15/2}}dx}{15 a}-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(8 A b-15 a B) \left (-\frac {6 b \int \frac {(a+b x)^{5/2}}{x^{13/2}}dx}{13 a}-\frac {2 (a+b x)^{7/2}}{13 a x^{13/2}}\right )}{15 a}-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(8 A b-15 a B) \left (-\frac {6 b \left (-\frac {4 b \int \frac {(a+b x)^{5/2}}{x^{11/2}}dx}{11 a}-\frac {2 (a+b x)^{7/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 (a+b x)^{7/2}}{13 a x^{13/2}}\right )}{15 a}-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(8 A b-15 a B) \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {(a+b x)^{5/2}}{x^{9/2}}dx}{9 a}-\frac {2 (a+b x)^{7/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{7/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 (a+b x)^{7/2}}{13 a x^{13/2}}\right )}{15 a}-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b (a+b x)^{7/2}}{63 a^2 x^{7/2}}-\frac {2 (a+b x)^{7/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 (a+b x)^{7/2}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 (a+b x)^{7/2}}{13 a x^{13/2}}\right ) (8 A b-15 a B)}{15 a}-\frac {2 A (a+b x)^{7/2}}{15 a x^{15/2}}\) |
(-2*A*(a + b*x)^(7/2))/(15*a*x^(15/2)) - ((8*A*b - 15*a*B)*((-2*(a + b*x)^ (7/2))/(13*a*x^(13/2)) - (6*b*((-2*(a + b*x)^(7/2))/(11*a*x^(11/2)) - (4*b *((-2*(a + b*x)^(7/2))/(9*a*x^(9/2)) + (4*b*(a + b*x)^(7/2))/(63*a^2*x^(7/ 2))))/(11*a)))/(13*a)))/(15*a)
3.6.12.3.1 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 = 1.46 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (128 A \,b^{4} x^{4}-240 B a \,b^{3} x^{4}-448 A a \,b^{3} x^{3}+840 B \,a^{2} b^{2} x^{3}+1008 A \,a^{2} b^{2} x^{2}-1890 B \,a^{3} b \,x^{2}-1848 A \,a^{3} b x +3465 B \,a^{4} x +3003 A \,a^{4}\right )}{45045 x^{\frac {15}{2}} a^{5}}\) | \(101\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (128 A \,b^{6} x^{6}-240 B a \,b^{5} x^{6}-192 A a \,b^{5} x^{5}+360 B \,a^{2} b^{4} x^{5}+240 A \,a^{2} b^{4} x^{4}-450 B \,a^{3} b^{3} x^{4}-280 A \,a^{3} b^{3} x^{3}+525 B \,a^{4} b^{2} x^{3}+315 A \,a^{4} b^{2} x^{2}+5040 B \,a^{5} b \,x^{2}+4158 A \,a^{5} b x +3465 B \,a^{6} x +3003 A \,a^{6}\right )}{45045 x^{\frac {15}{2}} a^{5}}\) | \(149\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{7} x^{7}-240 B a \,b^{6} x^{7}-64 A a \,b^{6} x^{6}+120 B \,a^{2} b^{5} x^{6}+48 A \,a^{2} b^{5} x^{5}-90 B \,a^{3} b^{4} x^{5}-40 A \,a^{3} b^{4} x^{4}+75 B \,a^{4} b^{3} x^{4}+35 A \,a^{4} b^{3} x^{3}+5565 B \,a^{5} b^{2} x^{3}+4473 A \,a^{5} b^{2} x^{2}+8505 B \,a^{6} b \,x^{2}+7161 A \,a^{6} b x +3465 B \,a^{7} x +3003 A \,a^{7}\right )}{45045 x^{\frac {15}{2}} a^{5}}\) | \(173\) |
-2/45045*(b*x+a)^(7/2)*(128*A*b^4*x^4-240*B*a*b^3*x^4-448*A*a*b^3*x^3+840* B*a^2*b^2*x^3+1008*A*a^2*b^2*x^2-1890*B*a^3*b*x^2-1848*A*a^3*b*x+3465*B*a^ 4*x+3003*A*a^4)/x^(15/2)/a^5
Time = 0.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 \, {\left (3003 \, A a^{7} - 16 \, {\left (15 \, B a b^{6} - 8 \, A b^{7}\right )} x^{7} + 8 \, {\left (15 \, B a^{2} b^{5} - 8 \, A a b^{6}\right )} x^{6} - 6 \, {\left (15 \, B a^{3} b^{4} - 8 \, A a^{2} b^{5}\right )} x^{5} + 5 \, {\left (15 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{4} + 35 \, {\left (159 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 63 \, {\left (135 \, B a^{6} b + 71 \, A a^{5} b^{2}\right )} x^{2} + 231 \, {\left (15 \, B a^{7} + 31 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{45045 \, a^{5} x^{\frac {15}{2}}} \]
-2/45045*(3003*A*a^7 - 16*(15*B*a*b^6 - 8*A*b^7)*x^7 + 8*(15*B*a^2*b^5 - 8 *A*a*b^6)*x^6 - 6*(15*B*a^3*b^4 - 8*A*a^2*b^5)*x^5 + 5*(15*B*a^4*b^3 - 8*A *a^3*b^4)*x^4 + 35*(159*B*a^5*b^2 + A*a^4*b^3)*x^3 + 63*(135*B*a^6*b + 71* A*a^5*b^2)*x^2 + 231*(15*B*a^7 + 31*A*a^6*b)*x)*sqrt(b*x + a)/(a^5*x^(15/2 ))
Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (120) = 240\).
Time = 0.20 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.64 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=\frac {32 \, \sqrt {b x^{2} + a x} B b^{6}}{3003 \, a^{4} x} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{7}}{45045 \, a^{5} x} - \frac {16 \, \sqrt {b x^{2} + a x} B b^{5}}{3003 \, a^{3} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} A b^{6}}{45045 \, a^{4} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} B b^{4}}{1001 \, a^{2} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{5}}{15015 \, a^{3} x^{3}} - \frac {10 \, \sqrt {b x^{2} + a x} B b^{3}}{3003 \, a x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} A b^{4}}{9009 \, a^{2} x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} B b^{2}}{1716 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{3}}{1287 \, a x^{5}} - \frac {3 \, \sqrt {b x^{2} + a x} B a b}{1144 \, x^{6}} + \frac {\sqrt {b x^{2} + a x} A b^{2}}{715 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} B a^{2}}{104 \, x^{7}} - \frac {\sqrt {b x^{2} + a x} A a b}{780 \, x^{7}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{8 \, x^{8}} - \frac {\sqrt {b x^{2} + a x} A a^{2}}{60 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{4 \, x^{9}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{12 \, x^{9}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{5 \, x^{10}} \]
32/3003*sqrt(b*x^2 + a*x)*B*b^6/(a^4*x) - 256/45045*sqrt(b*x^2 + a*x)*A*b^ 7/(a^5*x) - 16/3003*sqrt(b*x^2 + a*x)*B*b^5/(a^3*x^2) + 128/45045*sqrt(b*x ^2 + a*x)*A*b^6/(a^4*x^2) + 4/1001*sqrt(b*x^2 + a*x)*B*b^4/(a^2*x^3) - 32/ 15015*sqrt(b*x^2 + a*x)*A*b^5/(a^3*x^3) - 10/3003*sqrt(b*x^2 + a*x)*B*b^3/ (a*x^4) + 16/9009*sqrt(b*x^2 + a*x)*A*b^4/(a^2*x^4) + 5/1716*sqrt(b*x^2 + a*x)*B*b^2/x^5 - 2/1287*sqrt(b*x^2 + a*x)*A*b^3/(a*x^5) - 3/1144*sqrt(b*x^ 2 + a*x)*B*a*b/x^6 + 1/715*sqrt(b*x^2 + a*x)*A*b^2/x^6 - 3/104*sqrt(b*x^2 + a*x)*B*a^2/x^7 - 1/780*sqrt(b*x^2 + a*x)*A*a*b/x^7 + 1/8*(b*x^2 + a*x)^( 3/2)*B*a/x^8 - 1/60*sqrt(b*x^2 + a*x)*A*a^2/x^8 - 1/4*(b*x^2 + a*x)^(5/2)* B/x^9 + 1/12*(b*x^2 + a*x)^(3/2)*A*a/x^9 - 1/5*(b*x^2 + a*x)^(5/2)*A/x^10
Time = 0.37 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (15 \, B a^{3} b^{14} - 8 \, A a^{2} b^{15}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {15 \, {\left (15 \, B a^{4} b^{14} - 8 \, A a^{3} b^{15}\right )}}{a^{7}}\right )} + \frac {195 \, {\left (15 \, B a^{5} b^{14} - 8 \, A a^{4} b^{15}\right )}}{a^{7}}\right )} - \frac {715 \, {\left (15 \, B a^{6} b^{14} - 8 \, A a^{5} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {6435 \, {\left (B a^{7} b^{14} - A a^{6} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{45045 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {15}{2}} {\left | b \right |}} \]
2/45045*((2*(b*x + a)*(4*(b*x + a)*(2*(15*B*a^3*b^14 - 8*A*a^2*b^15)*(b*x + a)/a^7 - 15*(15*B*a^4*b^14 - 8*A*a^3*b^15)/a^7) + 195*(15*B*a^5*b^14 - 8 *A*a^4*b^15)/a^7) - 715*(15*B*a^6*b^14 - 8*A*a^5*b^15)/a^7)*(b*x + a) + 64 35*(B*a^7*b^14 - A*a^6*b^15)/a^7)*(b*x + a)^(7/2)*b/(((b*x + a)*b - a*b)^( 15/2)*abs(b))
Time = 0.91 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{17/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{15}+\frac {x^7\,\left (256\,A\,b^7-480\,B\,a\,b^6\right )}{45045\,a^5}+\frac {2\,a\,x\,\left (31\,A\,b+15\,B\,a\right )}{195}+\frac {2\,b\,x^2\,\left (71\,A\,b+135\,B\,a\right )}{715}-\frac {2\,b^3\,x^4\,\left (8\,A\,b-15\,B\,a\right )}{9009\,a^2}+\frac {4\,b^4\,x^5\,\left (8\,A\,b-15\,B\,a\right )}{15015\,a^3}-\frac {16\,b^5\,x^6\,\left (8\,A\,b-15\,B\,a\right )}{45045\,a^4}+\frac {2\,b^2\,x^3\,\left (A\,b+159\,B\,a\right )}{1287\,a}\right )}{x^{15/2}} \]
-((a + b*x)^(1/2)*((2*A*a^2)/15 + (x^7*(256*A*b^7 - 480*B*a*b^6))/(45045*a ^5) + (2*a*x*(31*A*b + 15*B*a))/195 + (2*b*x^2*(71*A*b + 135*B*a))/715 - ( 2*b^3*x^4*(8*A*b - 15*B*a))/(9009*a^2) + (4*b^4*x^5*(8*A*b - 15*B*a))/(150 15*a^3) - (16*b^5*x^6*(8*A*b - 15*B*a))/(45045*a^4) + (2*b^2*x^3*(A*b + 15 9*B*a))/(1287*a)))/x^(15/2)