Integrand size = 20, antiderivative size = 117 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}-\frac {8 b (6 A b-11 a B) (a+b x)^{5/2}}{693 a^3 x^{7/2}}+\frac {16 b^2 (6 A b-11 a B) (a+b x)^{5/2}}{3465 a^4 x^{5/2}} \]
-2/11*A*(b*x+a)^(5/2)/a/x^(11/2)+2/99*(6*A*b-11*B*a)*(b*x+a)^(5/2)/a^2/x^( 9/2)-8/693*b*(6*A*b-11*B*a)*(b*x+a)^(5/2)/a^3/x^(7/2)+16/3465*b^2*(6*A*b-1 1*B*a)*(b*x+a)^(5/2)/a^4/x^(5/2)
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.65 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx=-\frac {2 (a+b x)^{5/2} \left (-48 A b^3 x^3+35 a^3 (9 A+11 B x)+8 a b^2 x^2 (15 A+11 B x)-10 a^2 b x (21 A+22 B x)\right )}{3465 a^4 x^{11/2}} \]
(-2*(a + b*x)^(5/2)*(-48*A*b^3*x^3 + 35*a^3*(9*A + 11*B*x) + 8*a*b^2*x^2*( 15*A + 11*B*x) - 10*a^2*b*x*(21*A + 22*B*x)))/(3465*a^4*x^(11/2))
Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {87, 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.
\(\displaystyle \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(6 A b-11 a B) \int \frac {(a+b x)^{3/2}}{x^{11/2}}dx}{11 a}-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(6 A b-11 a B) \left (-\frac {4 b \int \frac {(a+b x)^{3/2}}{x^{9/2}}dx}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(6 A b-11 a B) \left (-\frac {4 b \left (-\frac {2 b \int \frac {(a+b x)^{3/2}}{x^{7/2}}dx}{7 a}-\frac {2 (a+b x)^{5/2}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (-\frac {4 b \left (\frac {4 b (a+b x)^{5/2}}{35 a^2 x^{5/2}}-\frac {2 (a+b x)^{5/2}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 (a+b x)^{5/2}}{9 a x^{9/2}}\right ) (6 A b-11 a B)}{11 a}-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}\) |
(-2*A*(a + b*x)^(5/2))/(11*a*x^(11/2)) - ((6*A*b - 11*a*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)
3.5.99.3.1 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 = 77, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-48 A \,b^{3} x^{3}+88 B a \,b^{2} x^{3}+120 a A \,b^{2} x^{2}-220 B \,a^{2} b \,x^{2}-210 a^{2} A b x +385 a^{3} B x +315 a^{3} A \right )}{3465 x^{\frac {11}{2}} a^{4}}\) | \(77\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-48 A \,b^{4} x^{4}+88 B a \,b^{3} x^{4}+72 A a \,b^{3} x^{3}-132 B \,a^{2} b^{2} x^{3}-90 A \,a^{2} b^{2} x^{2}+165 B \,a^{3} b \,x^{2}+105 A \,a^{3} b x +385 B \,a^{4} x +315 A \,a^{4}\right )}{3465 x^{\frac {11}{2}} a^{4}}\) | \(101\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{5} x^{5}+88 B a \,b^{4} x^{5}+24 a A \,b^{4} x^{4}-44 B \,a^{2} b^{3} x^{4}-18 a^{2} A \,b^{3} x^{3}+33 B \,a^{3} b^{2} x^{3}+15 a^{3} A \,b^{2} x^{2}+550 B \,a^{4} b \,x^{2}+420 a^{4} A b x +385 a^{5} B x +315 a^{5} A \right )}{3465 x^{\frac {11}{2}} a^{4}}\) | \(125\) |
-2/3465*(b*x+a)^(5/2)*(-48*A*b^3*x^3+88*B*a*b^2*x^3+120*A*a*b^2*x^2-220*B* a^2*b*x^2-210*A*a^2*b*x+385*B*a^3*x+315*A*a^3)/x^(11/2)/a^4
Time = 0.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx=-\frac {2 \, {\left (315 \, A a^{5} + 8 \, {\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 4 \, {\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} + 3 \, {\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} + 5 \, {\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3465 \, a^{4} x^{\frac {11}{2}}} \]
-2/3465*(315*A*a^5 + 8*(11*B*a*b^4 - 6*A*b^5)*x^5 - 4*(11*B*a^2*b^3 - 6*A* a*b^4)*x^4 + 3*(11*B*a^3*b^2 - 6*A*a^2*b^3)*x^3 + 5*(110*B*a^4*b + 3*A*a^3 *b^2)*x^2 + 35*(11*B*a^5 + 12*A*a^4*b)*x)*sqrt(b*x + a)/(a^4*x^(11/2))
Leaf count of result is larger than twice the leaf count of optimal. 2351 vs. \(2 (116) = 232\).
Time = 113.67 (sec) , antiderivative size = 2351, normalized size of antiderivative = 20.09 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx=\text {Too large to display} \]
-630*A*a**10*b**(33/2)*sqrt(a/(b*x) + 1)/(3465*a**9*b**16*x**5 + 13860*a** 8*b**17*x**6 + 20790*a**7*b**18*x**7 + 13860*a**6*b**19*x**8 + 3465*a**5*b **20*x**9) - 2590*A*a**9*b**(35/2)*x*sqrt(a/(b*x) + 1)/(3465*a**9*b**16*x* *5 + 13860*a**8*b**17*x**6 + 20790*a**7*b**18*x**7 + 13860*a**6*b**19*x**8 + 3465*a**5*b**20*x**9) - 3980*A*a**8*b**(37/2)*x**2*sqrt(a/(b*x) + 1)/(3 465*a**9*b**16*x**5 + 13860*a**8*b**17*x**6 + 20790*a**7*b**18*x**7 + 1386 0*a**6*b**19*x**8 + 3465*a**5*b**20*x**9) - 2716*A*a**7*b**(39/2)*x**3*sqr t(a/(b*x) + 1)/(3465*a**9*b**16*x**5 + 13860*a**8*b**17*x**6 + 20790*a**7* b**18*x**7 + 13860*a**6*b**19*x**8 + 3465*a**5*b**20*x**9) - 70*A*a**7*b** (21/2)*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a **5*b**11*x**6 + 315*a**4*b**12*x**7) - 686*A*a**6*b**(41/2)*x**4*sqrt(a/( b*x) + 1)/(3465*a**9*b**16*x**5 + 13860*a**8*b**17*x**6 + 20790*a**7*b**18 *x**7 + 13860*a**6*b**19*x**8 + 3465*a**5*b**20*x**9) - 220*A*a**6*b**(23/ 2)*x*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a** 5*b**11*x**6 + 315*a**4*b**12*x**7) - 70*A*a**5*b**(43/2)*x**5*sqrt(a/(b*x ) + 1)/(3465*a**9*b**16*x**5 + 13860*a**8*b**17*x**6 + 20790*a**7*b**18*x* *7 + 13860*a**6*b**19*x**8 + 3465*a**5*b**20*x**9) - 228*A*a**5*b**(25/2)* x**2*sqrt(a/(b*x) + 1)/(315*a**7*b**9*x**4 + 945*a**6*b**10*x**5 + 945*a** 5*b**11*x**6 + 315*a**4*b**12*x**7) - 560*A*a**4*b**(45/2)*x**6*sqrt(a/(b* x) + 1)/(3465*a**9*b**16*x**5 + 13860*a**8*b**17*x**6 + 20790*a**7*b**1...
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (93) = 186\).
Time = 0.19 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.29 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx=-\frac {16 \, \sqrt {b x^{2} + a x} B b^{4}}{315 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{5}}{1155 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b^{3}}{315 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{4}}{1155 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{3}}{385 \, a^{2} x^{3}} + \frac {\sqrt {b x^{2} + a x} B b}{63 \, x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{231 \, a x^{4}} + \frac {\sqrt {b x^{2} + a x} B a}{9 \, x^{5}} + \frac {\sqrt {b x^{2} + a x} A b}{132 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{3 \, x^{6}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{44 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{4 \, x^{7}} \]
-16/315*sqrt(b*x^2 + a*x)*B*b^4/(a^3*x) + 32/1155*sqrt(b*x^2 + a*x)*A*b^5/ (a^4*x) + 8/315*sqrt(b*x^2 + a*x)*B*b^3/(a^2*x^2) - 16/1155*sqrt(b*x^2 + a *x)*A*b^4/(a^3*x^2) - 2/105*sqrt(b*x^2 + a*x)*B*b^2/(a*x^3) + 4/385*sqrt(b *x^2 + a*x)*A*b^3/(a^2*x^3) + 1/63*sqrt(b*x^2 + a*x)*B*b/x^4 - 2/231*sqrt( b*x^2 + a*x)*A*b^2/(a*x^4) + 1/9*sqrt(b*x^2 + a*x)*B*a/x^5 + 1/132*sqrt(b* x^2 + a*x)*A*b/x^5 - 1/3*(b*x^2 + a*x)^(3/2)*B/x^6 + 3/44*sqrt(b*x^2 + a*x )*A*a/x^6 - 1/4*(b*x^2 + a*x)^(3/2)*A/x^7
Time = 0.34 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx=-\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a^{2} b^{10} - 6 \, A a b^{11}\right )} {\left (b x + a\right )}}{a^{5}} - \frac {11 \, {\left (11 \, B a^{3} b^{10} - 6 \, A a^{2} b^{11}\right )}}{a^{5}}\right )} + \frac {99 \, {\left (11 \, B a^{4} b^{10} - 6 \, A a^{3} b^{11}\right )}}{a^{5}}\right )} - \frac {693 \, {\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{5}}\right )} {\left (b x + a\right )}^{\frac {5}{2}} b}{3465 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \]
-2/3465*((b*x + a)*(4*(b*x + a)*(2*(11*B*a^2*b^10 - 6*A*a*b^11)*(b*x + a)/ a^5 - 11*(11*B*a^3*b^10 - 6*A*a^2*b^11)/a^5) + 99*(11*B*a^4*b^10 - 6*A*a^3 *b^11)/a^5) - 693*(B*a^5*b^10 - A*a^4*b^11)/a^5)*(b*x + a)^(5/2)*b/(((b*x + a)*b - a*b)^(11/2)*abs(b))
Time = 0.80 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{11}+\frac {x\,\left (770\,B\,a^5+840\,A\,b\,a^4\right )}{3465\,a^4}-\frac {x^5\,\left (96\,A\,b^5-176\,B\,a\,b^4\right )}{3465\,a^4}-\frac {2\,b^2\,x^3\,\left (6\,A\,b-11\,B\,a\right )}{1155\,a^2}+\frac {8\,b^3\,x^4\,\left (6\,A\,b-11\,B\,a\right )}{3465\,a^3}+\frac {2\,b\,x^2\,\left (3\,A\,b+110\,B\,a\right )}{693\,a}\right )}{x^{11/2}} \]