Integrand size = 20, antiderivative size = 84 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx=-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}}+\frac {2 (4 A b-11 a B) (a+b x)^{7/2}}{99 a^2 x^{9/2}}-\frac {4 b (4 A b-11 a B) (a+b x)^{7/2}}{693 a^3 x^{7/2}} \]
-2/11*A*(b*x+a)^(7/2)/a/x^(11/2)+2/99*(4*A*b-11*B*a)*(b*x+a)^(7/2)/a^2/x^( 9/2)-4/693*b*(4*A*b-11*B*a)*(b*x+a)^(7/2)/a^3/x^(7/2)
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx=-\frac {2 (a+b x)^{7/2} \left (63 a^2 A-28 a A b x+77 a^2 B x+8 A b^2 x^2-22 a b B x^2\right )}{693 a^3 x^{11/2}} \]
(-2*(a + b*x)^(7/2)*(63*a^2*A - 28*a*A*b*x + 77*a^2*B*x + 8*A*b^2*x^2 - 22 *a*b*B*x^2))/(693*a^3*x^(11/2))
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {87, 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.
\(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {(4 A b-11 a B) \int \frac {(a+b x)^{5/2}}{x^{11/2}}dx}{11 a}-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {(4 A b-11 a B) \left (-\frac {2 b \int \frac {(a+b x)^{5/2}}{x^{9/2}}dx}{9 a}-\frac {2 (a+b x)^{7/2}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {4 b (a+b x)^{7/2}}{63 a^2 x^{7/2}}-\frac {2 (a+b x)^{7/2}}{9 a x^{9/2}}\right ) (4 A b-11 a B)}{11 a}-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}}\) |
(-2*A*(a + b*x)^(7/2))/(11*a*x^(11/2)) - ((4*A*b - 11*a*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)
3.6.10.3.1 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 = 1.45 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (8 A \,b^{2} x^{2}-22 B a b \,x^{2}-28 a A b x +77 a^{2} B x +63 a^{2} A \right )}{693 x^{\frac {11}{2}} a^{3}}\) | \(53\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (8 A \,b^{4} x^{4}-22 B a \,b^{3} x^{4}-12 A a \,b^{3} x^{3}+33 B \,a^{2} b^{2} x^{3}+15 A \,a^{2} b^{2} x^{2}+132 B \,a^{3} b \,x^{2}+98 A \,a^{3} b x +77 B \,a^{4} x +63 A \,a^{4}\right )}{693 x^{\frac {11}{2}} a^{3}}\) | \(101\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (8 A \,b^{5} x^{5}-22 B a \,b^{4} x^{5}-4 a A \,b^{4} x^{4}+11 B \,a^{2} b^{3} x^{4}+3 a^{2} A \,b^{3} x^{3}+165 B \,a^{3} b^{2} x^{3}+113 a^{3} A \,b^{2} x^{2}+209 B \,a^{4} b \,x^{2}+161 a^{4} A b x +77 a^{5} B x +63 a^{5} A \right )}{693 x^{\frac {11}{2}} a^{3}}\) | \(125\) |
Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx=-\frac {2 \, {\left (63 \, A a^{5} - 2 \, {\left (11 \, B a b^{4} - 4 \, A b^{5}\right )} x^{5} + {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{4} + 3 \, {\left (55 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + {\left (209 \, B a^{4} b + 113 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (11 \, B a^{5} + 23 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{693 \, a^{3} x^{\frac {11}{2}}} \]
-2/693*(63*A*a^5 - 2*(11*B*a*b^4 - 4*A*b^5)*x^5 + (11*B*a^2*b^3 - 4*A*a*b^ 4)*x^4 + 3*(55*B*a^3*b^2 + A*a^2*b^3)*x^3 + (209*B*a^4*b + 113*A*a^3*b^2)* x^2 + 7*(11*B*a^5 + 23*A*a^4*b)*x)*sqrt(b*x + a)/(a^3*x^(11/2))
Leaf count of result is larger than twice the leaf count of optimal. 2790 vs. \(2 (82) = 164\).
Time = 119.93 (sec) , antiderivative size = 2790, normalized size of antiderivative = 33.21 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx=\text {Too large to display} \]
-630*A*a**11*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**10*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**9*b**(37/2)*x**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 + 138 60*a**6*b**19*x**8 + 3465*a**5*b**20*x**9) - 2716*A*a**8*b**(39/2)*x**3*sq rt(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) - 140*A*a**8*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**7*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) - 440*A*a**7*b**(2 3/2)*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**6*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) - 456*A*a**6*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**5*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*...
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (66) = 132\).
Time = 0.20 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.62 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx=\frac {4 \, \sqrt {b x^{2} + a x} B b^{4}}{63 \, a^{2} x} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{5}}{693 \, a^{3} x} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{3}}{63 \, a x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} A b^{4}}{693 \, a^{2} x^{2}} + \frac {\sqrt {b x^{2} + a x} B b^{2}}{42 \, x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{3}}{231 \, a x^{3}} - \frac {5 \, \sqrt {b x^{2} + a x} B a b}{252 \, x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} A b^{2}}{693 \, x^{4}} - \frac {5 \, \sqrt {b x^{2} + a x} B a^{2}}{36 \, x^{5}} - \frac {5 \, \sqrt {b x^{2} + a x} A a b}{792 \, x^{5}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{12 \, x^{6}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{88 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{2 \, x^{7}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{24 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{3 \, x^{8}} \]
4/63*sqrt(b*x^2 + a*x)*B*b^4/(a^2*x) - 16/693*sqrt(b*x^2 + a*x)*A*b^5/(a^3 *x) - 2/63*sqrt(b*x^2 + a*x)*B*b^3/(a*x^2) + 8/693*sqrt(b*x^2 + a*x)*A*b^4 /(a^2*x^2) + 1/42*sqrt(b*x^2 + a*x)*B*b^2/x^3 - 2/231*sqrt(b*x^2 + a*x)*A* b^3/(a*x^3) - 5/252*sqrt(b*x^2 + a*x)*B*a*b/x^4 + 5/693*sqrt(b*x^2 + a*x)* A*b^2/x^4 - 5/36*sqrt(b*x^2 + a*x)*B*a^2/x^5 - 5/792*sqrt(b*x^2 + a*x)*A*a *b/x^5 + 5/12*(b*x^2 + a*x)^(3/2)*B*a/x^6 - 5/88*sqrt(b*x^2 + a*x)*A*a^2/x ^6 - 1/2*(b*x^2 + a*x)^(5/2)*B/x^7 + 5/24*(b*x^2 + a*x)^(3/2)*A*a/x^7 - 1/ 3*(b*x^2 + a*x)^(5/2)*A/x^8
Time = 0.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a^{3} b^{10} - 4 \, A a^{2} b^{11}\right )} {\left (b x + a\right )}}{a^{5}} - \frac {11 \, {\left (11 \, B a^{4} b^{10} - 4 \, A a^{3} b^{11}\right )}}{a^{5}}\right )} + \frac {99 \, {\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{5}}\right )} b}{693 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \]
2/693*(b*x + a)^(7/2)*((b*x + a)*(2*(11*B*a^3*b^10 - 4*A*a^2*b^11)*(b*x + a)/a^5 - 11*(11*B*a^4*b^10 - 4*A*a^3*b^11)/a^5) + 99*(B*a^5*b^10 - A*a^4*b ^11)/a^5)*b/(((b*x + a)*b - a*b)^(11/2)*abs(b))
Time = 0.85 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{11}+\frac {x\,\left (154\,B\,a^5+322\,A\,b\,a^4\right )}{693\,a^3}+\frac {x^5\,\left (16\,A\,b^5-44\,B\,a\,b^4\right )}{693\,a^3}+\frac {2\,b\,x^2\,\left (113\,A\,b+209\,B\,a\right )}{693}-\frac {2\,b^3\,x^4\,\left (4\,A\,b-11\,B\,a\right )}{693\,a^2}+\frac {2\,b^2\,x^3\,\left (A\,b+55\,B\,a\right )}{231\,a}\right )}{x^{11/2}} \]