Integrand size = 20, antiderivative size = 183 \[ \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx=-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}+\frac {2 (10 A b-11 a B) \sqrt {a+b x}}{99 a^2 x^{9/2}}-\frac {16 b (10 A b-11 a B) \sqrt {a+b x}}{693 a^3 x^{7/2}}+\frac {32 b^2 (10 A b-11 a B) \sqrt {a+b x}}{1155 a^4 x^{5/2}}-\frac {128 b^3 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^5 x^{3/2}}+\frac {256 b^4 (10 A b-11 a B) \sqrt {a+b x}}{3465 a^6 \sqrt {x}} \]
-2/11*A*(b*x+a)^(1/2)/a/x^(11/2)+2/99*(10*A*b-11*B*a)*(b*x+a)^(1/2)/a^2/x^ (9/2)-16/693*b*(10*A*b-11*B*a)*(b*x+a)^(1/2)/a^3/x^(7/2)+32/1155*b^2*(10*A *b-11*B*a)*(b*x+a)^(1/2)/a^4/x^(5/2)-128/3465*b^3*(10*A*b-11*B*a)*(b*x+a)^ (1/2)/a^5/x^(3/2)+256/3465*b^4*(10*A*b-11*B*a)*(b*x+a)^(1/2)/a^6/x^(1/2)
Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x} \left (-1280 A b^5 x^5+128 a b^4 x^4 (5 A+11 B x)+35 a^5 (9 A+11 B x)-32 a^2 b^3 x^3 (15 A+22 B x)+16 a^3 b^2 x^2 (25 A+33 B x)-10 a^4 b x (35 A+44 B x)\right )}{3465 a^6 x^{11/2}} \]
(-2*Sqrt[a + b*x]*(-1280*A*b^5*x^5 + 128*a*b^4*x^4*(5*A + 11*B*x) + 35*a^5 *(9*A + 11*B*x) - 32*a^2*b^3*x^3*(15*A + 22*B*x) + 16*a^3*b^2*x^2*(25*A + 33*B*x) - 10*a^4*b*x*(35*A + 44*B*x)))/(3465*a^6*x^(11/2))
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.
| \(\displaystyle \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(10 A b-11 a B) \int \frac {1}{x^{11/2} \sqrt {a+b x}}dx}{11 a}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(10 A b-11 a B) \left (-\frac {8 b \int \frac {1}{x^{9/2} \sqrt {a+b x}}dx}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(10 A b-11 a B) \left (-\frac {8 b \left (-\frac {6 b \int \frac {1}{x^{7/2} \sqrt {a+b x}}dx}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(10 A b-11 a B) \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {1}{x^{5/2} \sqrt {a+b x}}dx}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(10 A b-11 a B) \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {1}{x^{3/2} \sqrt {a+b x}}dx}{3 a}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}\right )}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}\right )}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right ) (10 A b-11 a B)}{11 a}-\frac {2 A \sqrt {a+b x}}{11 a x^{11/2}}\) |
(-2*A*Sqrt[a + b*x])/(11*a*x^(11/2)) - ((10*A*b - 11*a*B)*((-2*Sqrt[a + b* x])/(9*a*x^(9/2)) - (8*b*((-2*Sqrt[a + b*x])/(7*a*x^(7/2)) - (6*b*((-2*Sqr t[a + b*x])/(5*a*x^(5/2)) - (4*b*((-2*Sqrt[a + b*x])/(3*a*x^(3/2)) + (4*b* Sqrt[a + b*x])/(3*a^2*Sqrt[x])))/(5*a)))/(7*a)))/(9*a)))/(11*a)
3.6.24.3.1 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 = 125, normalized size of antiderivative = 0.68
| method | result | size |
| gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{5} x^{5}+1408 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-704 B \,a^{2} b^{3} x^{4}-480 a^{2} A \,b^{3} x^{3}+528 B \,a^{3} b^{2} x^{3}+400 a^{3} A \,b^{2} x^{2}-440 B \,a^{4} b \,x^{2}-350 a^{4} A b x +385 a^{5} B x +315 a^{5} A \right )}{3465 x^{\frac {11}{2}} a^{6}}\) | \(125\) |
| default | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{5} x^{5}+1408 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-704 B \,a^{2} b^{3} x^{4}-480 a^{2} A \,b^{3} x^{3}+528 B \,a^{3} b^{2} x^{3}+400 a^{3} A \,b^{2} x^{2}-440 B \,a^{4} b \,x^{2}-350 a^{4} A b x +385 a^{5} B x +315 a^{5} A \right )}{3465 x^{\frac {11}{2}} a^{6}}\) | \(125\) |
| risch | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{5} x^{5}+1408 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-704 B \,a^{2} b^{3} x^{4}-480 a^{2} A \,b^{3} x^{3}+528 B \,a^{3} b^{2} x^{3}+400 a^{3} A \,b^{2} x^{2}-440 B \,a^{4} b \,x^{2}-350 a^{4} A b x +385 a^{5} B x +315 a^{5} A \right )}{3465 x^{\frac {11}{2}} a^{6}}\) | \(125\) |
-2/3465*(b*x+a)^(1/2)*(-1280*A*b^5*x^5+1408*B*a*b^4*x^5+640*A*a*b^4*x^4-70 4*B*a^2*b^3*x^4-480*A*a^2*b^3*x^3+528*B*a^3*b^2*x^3+400*A*a^3*b^2*x^2-440* B*a^4*b*x^2-350*A*a^4*b*x+385*B*a^5*x+315*A*a^5)/x^(11/2)/a^6
Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (315 \, A a^{5} + 128 \, {\left (11 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} - 64 \, {\left (11 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 48 \, {\left (11 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 40 \, {\left (11 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3465 \, a^{6} x^{\frac {11}{2}}} \]
-2/3465*(315*A*a^5 + 128*(11*B*a*b^4 - 10*A*b^5)*x^5 - 64*(11*B*a^2*b^3 - 10*A*a*b^4)*x^4 + 48*(11*B*a^3*b^2 - 10*A*a^2*b^3)*x^3 - 40*(11*B*a^4*b - 10*A*a^3*b^2)*x^2 + 35*(11*B*a^5 - 10*A*a^4*b)*x)*sqrt(b*x + a)/(a^6*x^(11 /2))
Leaf count of result is larger than twice the leaf count of optimal. 1809 vs. \(2 (184) = 368\).
Time = 107.81 (sec) , antiderivative size = 1809, normalized size of antiderivative = 9.89 \[ \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx=\text {Too large to display} \]
-126*A*a**10*b**(51/2)*sqrt(a/(b*x) + 1)/(693*a**11*b**25*x**5 + 3465*a**1 0*b**26*x**6 + 6930*a**9*b**27*x**7 + 6930*a**8*b**28*x**8 + 3465*a**7*b** 29*x**9 + 693*a**6*b**30*x**10) - 490*A*a**9*b**(53/2)*x*sqrt(a/(b*x) + 1) /(693*a**11*b**25*x**5 + 3465*a**10*b**26*x**6 + 6930*a**9*b**27*x**7 + 69 30*a**8*b**28*x**8 + 3465*a**7*b**29*x**9 + 693*a**6*b**30*x**10) - 720*A* a**8*b**(55/2)*x**2*sqrt(a/(b*x) + 1)/(693*a**11*b**25*x**5 + 3465*a**10*b **26*x**6 + 6930*a**9*b**27*x**7 + 6930*a**8*b**28*x**8 + 3465*a**7*b**29* x**9 + 693*a**6*b**30*x**10) - 468*A*a**7*b**(57/2)*x**3*sqrt(a/(b*x) + 1) /(693*a**11*b**25*x**5 + 3465*a**10*b**26*x**6 + 6930*a**9*b**27*x**7 + 69 30*a**8*b**28*x**8 + 3465*a**7*b**29*x**9 + 693*a**6*b**30*x**10) - 126*A* a**6*b**(59/2)*x**4*sqrt(a/(b*x) + 1)/(693*a**11*b**25*x**5 + 3465*a**10*b **26*x**6 + 6930*a**9*b**27*x**7 + 6930*a**8*b**28*x**8 + 3465*a**7*b**29* x**9 + 693*a**6*b**30*x**10) + 126*A*a**5*b**(61/2)*x**5*sqrt(a/(b*x) + 1) /(693*a**11*b**25*x**5 + 3465*a**10*b**26*x**6 + 6930*a**9*b**27*x**7 + 69 30*a**8*b**28*x**8 + 3465*a**7*b**29*x**9 + 693*a**6*b**30*x**10) + 1260*A *a**4*b**(63/2)*x**6*sqrt(a/(b*x) + 1)/(693*a**11*b**25*x**5 + 3465*a**10* b**26*x**6 + 6930*a**9*b**27*x**7 + 6930*a**8*b**28*x**8 + 3465*a**7*b**29 *x**9 + 693*a**6*b**30*x**10) + 3360*A*a**3*b**(65/2)*x**7*sqrt(a/(b*x) + 1)/(693*a**11*b**25*x**5 + 3465*a**10*b**26*x**6 + 6930*a**9*b**27*x**7 + 6930*a**8*b**28*x**8 + 3465*a**7*b**29*x**9 + 693*a**6*b**30*x**10) + 4...
Time = 0.19 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.33 \[ \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx=-\frac {256 \, \sqrt {b x^{2} + a x} B b^{4}}{315 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{5}}{693 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{3}}{315 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{4}}{693 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{3}}{231 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b}{63 \, a^{2} x^{4}} - \frac {160 \, \sqrt {b x^{2} + a x} A b^{2}}{693 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{9 \, a x^{5}} + \frac {20 \, \sqrt {b x^{2} + a x} A b}{99 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{11 \, a x^{6}} \]
-256/315*sqrt(b*x^2 + a*x)*B*b^4/(a^5*x) + 512/693*sqrt(b*x^2 + a*x)*A*b^5 /(a^6*x) + 128/315*sqrt(b*x^2 + a*x)*B*b^3/(a^4*x^2) - 256/693*sqrt(b*x^2 + a*x)*A*b^4/(a^5*x^2) - 32/105*sqrt(b*x^2 + a*x)*B*b^2/(a^3*x^3) + 64/231 *sqrt(b*x^2 + a*x)*A*b^3/(a^4*x^3) + 16/63*sqrt(b*x^2 + a*x)*B*b/(a^2*x^4) - 160/693*sqrt(b*x^2 + a*x)*A*b^2/(a^3*x^4) - 2/9*sqrt(b*x^2 + a*x)*B/(a* x^5) + 20/99*sqrt(b*x^2 + a*x)*A*b/(a^2*x^5) - 2/11*sqrt(b*x^2 + a*x)*A/(a *x^6)
Time = 0.31 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a b^{10} - 10 \, A b^{11}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {11 \, {\left (11 \, B a^{2} b^{10} - 10 \, A a b^{11}\right )}}{a^{6}}\right )} + \frac {99 \, {\left (11 \, B a^{3} b^{10} - 10 \, A a^{2} b^{11}\right )}}{a^{6}}\right )} - \frac {231 \, {\left (11 \, B a^{4} b^{10} - 10 \, A a^{3} b^{11}\right )}}{a^{6}}\right )} {\left (b x + a\right )} + \frac {1155 \, {\left (11 \, B a^{5} b^{10} - 10 \, A a^{4} b^{11}\right )}}{a^{6}}\right )} {\left (b x + a\right )} - \frac {3465 \, {\left (B a^{6} b^{10} - A a^{5} b^{11}\right )}}{a^{6}}\right )} \sqrt {b x + a} b}{3465 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \]
-2/3465*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(11*B*a*b^10 - 10*A*b^11)*(b*x + a)/a^6 - 11*(11*B*a^2*b^10 - 10*A*a*b^11)/a^6) + 99*(11*B*a^3*b^10 - 10*A* a^2*b^11)/a^6) - 231*(11*B*a^4*b^10 - 10*A*a^3*b^11)/a^6)*(b*x + a) + 1155 *(11*B*a^5*b^10 - 10*A*a^4*b^11)/a^6)*(b*x + a) - 3465*(B*a^6*b^10 - A*a^5 *b^11)/a^6)*sqrt(b*x + a)*b/(((b*x + a)*b - a*b)^(11/2)*abs(b))
Time = 1.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x}{x^{13/2} \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{11\,a}+\frac {x\,\left (770\,B\,a^5-700\,A\,a^4\,b\right )}{3465\,a^6}-\frac {32\,b^2\,x^3\,\left (10\,A\,b-11\,B\,a\right )}{1155\,a^4}+\frac {128\,b^3\,x^4\,\left (10\,A\,b-11\,B\,a\right )}{3465\,a^5}-\frac {256\,b^4\,x^5\,\left (10\,A\,b-11\,B\,a\right )}{3465\,a^6}+\frac {16\,b\,x^2\,\left (10\,A\,b-11\,B\,a\right )}{693\,a^3}\right )}{x^{11/2}} \]