Integrand size = 20, antiderivative size = 216 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}-\frac {64 b^3 (12 A b-13 a B) \sqrt {a+b x}}{3003 a^5 x^{5/2}}+\frac {256 b^4 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^6 x^{3/2}}-\frac {512 b^5 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^7 \sqrt {x}} \]
-2/13*A*(b*x+a)^(1/2)/a/x^(13/2)+2/143*(12*A*b-13*B*a)*(b*x+a)^(1/2)/a^2/x ^(11/2)-20/1287*b*(12*A*b-13*B*a)*(b*x+a)^(1/2)/a^3/x^(9/2)+160/9009*b^2*( 12*A*b-13*B*a)*(b*x+a)^(1/2)/a^4/x^(7/2)-64/3003*b^3*(12*A*b-13*B*a)*(b*x+ a)^(1/2)/a^5/x^(5/2)+256/9009*b^4*(12*A*b-13*B*a)*(b*x+a)^(1/2)/a^6/x^(3/2 )-512/9009*b^5*(12*A*b-13*B*a)*(b*x+a)^(1/2)/a^7/x^(1/2)
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x} \left (3072 A b^6 x^6-256 a b^5 x^5 (6 A+13 B x)+128 a^2 b^4 x^4 (9 A+13 B x)-96 a^3 b^3 x^3 (10 A+13 B x)+63 a^6 (11 A+13 B x)+40 a^4 b^2 x^2 (21 A+26 B x)-14 a^5 b x (54 A+65 B x)\right )}{9009 a^7 x^{13/2}} \]
(-2*Sqrt[a + b*x]*(3072*A*b^6*x^6 - 256*a*b^5*x^5*(6*A + 13*B*x) + 128*a^2 *b^4*x^4*(9*A + 13*B*x) - 96*a^3*b^3*x^3*(10*A + 13*B*x) + 63*a^6*(11*A + 13*B*x) + 40*a^4*b^2*x^2*(21*A + 26*B*x) - 14*a^5*b*x*(54*A + 65*B*x)))/(9 009*a^7*x^(13/2))
Time = 0.23 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {87, 55, 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.
| \(\displaystyle \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(12 A b-13 a B) \int \frac {1}{x^{13/2} \sqrt {a+b x}}dx}{13 a}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(12 A b-13 a B) \left (-\frac {10 b \int \frac {1}{x^{11/2} \sqrt {a+b x}}dx}{11 a}-\frac {2 \sqrt {a+b x}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(12 A b-13 a B) \left (-\frac {10 b \left (-\frac {8 b \int \frac {1}{x^{9/2} \sqrt {a+b x}}dx}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 \sqrt {a+b x}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(12 A b-13 a B) \left (-\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \int \frac {1}{x^{7/2} \sqrt {a+b x}}dx}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 \sqrt {a+b x}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(12 A b-13 a B) \left (-\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \int \frac {1}{x^{5/2} \sqrt {a+b x}}dx}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 \sqrt {a+b x}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(12 A b-13 a B) \left (-\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {1}{x^{3/2} \sqrt {a+b x}}dx}{3 a}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}\right )}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 \sqrt {a+b x}}{11 a x^{11/2}}\right )}{13 a}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (-\frac {10 b \left (-\frac {8 b \left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}\right )}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{9 a}-\frac {2 \sqrt {a+b x}}{9 a x^{9/2}}\right )}{11 a}-\frac {2 \sqrt {a+b x}}{11 a x^{11/2}}\right ) (12 A b-13 a B)}{13 a}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}\) |
(-2*A*Sqrt[a + b*x])/(13*a*x^(13/2)) - ((12*A*b - 13*a*B)*((-2*Sqrt[a + b* x])/(11*a*x^(11/2)) - (10*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*Sqrt[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)))/(13*a)
3.6.25.3.1 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 = 149, normalized size of antiderivative = 0.69
| method | result | size |
| gosper | \(-\frac {2 \sqrt {b x +a}\, \left (3072 A \,b^{6} x^{6}-3328 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1664 B \,a^{2} b^{4} x^{5}+1152 A \,a^{2} b^{4} x^{4}-1248 B \,a^{3} b^{3} x^{4}-960 A \,a^{3} b^{3} x^{3}+1040 B \,a^{4} b^{2} x^{3}+840 A \,a^{4} b^{2} x^{2}-910 B \,a^{5} b \,x^{2}-756 A \,a^{5} b x +819 B \,a^{6} x +693 A \,a^{6}\right )}{9009 x^{\frac {13}{2}} a^{7}}\) | \(149\) |
| default | \(-\frac {2 \sqrt {b x +a}\, \left (3072 A \,b^{6} x^{6}-3328 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1664 B \,a^{2} b^{4} x^{5}+1152 A \,a^{2} b^{4} x^{4}-1248 B \,a^{3} b^{3} x^{4}-960 A \,a^{3} b^{3} x^{3}+1040 B \,a^{4} b^{2} x^{3}+840 A \,a^{4} b^{2} x^{2}-910 B \,a^{5} b \,x^{2}-756 A \,a^{5} b x +819 B \,a^{6} x +693 A \,a^{6}\right )}{9009 x^{\frac {13}{2}} a^{7}}\) | \(149\) |
| risch | \(-\frac {2 \sqrt {b x +a}\, \left (3072 A \,b^{6} x^{6}-3328 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1664 B \,a^{2} b^{4} x^{5}+1152 A \,a^{2} b^{4} x^{4}-1248 B \,a^{3} b^{3} x^{4}-960 A \,a^{3} b^{3} x^{3}+1040 B \,a^{4} b^{2} x^{3}+840 A \,a^{4} b^{2} x^{2}-910 B \,a^{5} b \,x^{2}-756 A \,a^{5} b x +819 B \,a^{6} x +693 A \,a^{6}\right )}{9009 x^{\frac {13}{2}} a^{7}}\) | \(149\) |
-2/9009*(b*x+a)^(1/2)*(3072*A*b^6*x^6-3328*B*a*b^5*x^6-1536*A*a*b^5*x^5+16 64*B*a^2*b^4*x^5+1152*A*a^2*b^4*x^4-1248*B*a^3*b^3*x^4-960*A*a^3*b^3*x^3+1 040*B*a^4*b^2*x^3+840*A*a^4*b^2*x^2-910*B*a^5*b*x^2-756*A*a^5*b*x+819*B*a^ 6*x+693*A*a^6)/x^(13/2)/a^7
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (693 \, A a^{6} - 256 \, {\left (13 \, B a b^{5} - 12 \, A b^{6}\right )} x^{6} + 128 \, {\left (13 \, B a^{2} b^{4} - 12 \, A a b^{5}\right )} x^{5} - 96 \, {\left (13 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{4} + 80 \, {\left (13 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} - 70 \, {\left (13 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (13 \, B a^{6} - 12 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{9009 \, a^{7} x^{\frac {13}{2}}} \]
-2/9009*(693*A*a^6 - 256*(13*B*a*b^5 - 12*A*b^6)*x^6 + 128*(13*B*a^2*b^4 - 12*A*a*b^5)*x^5 - 96*(13*B*a^3*b^3 - 12*A*a^2*b^4)*x^4 + 80*(13*B*a^4*b^2 - 12*A*a^3*b^3)*x^3 - 70*(13*B*a^5*b - 12*A*a^4*b^2)*x^2 + 63*(13*B*a^6 - 12*A*a^5*b)*x)*sqrt(b*x + a)/(a^7*x^(13/2))
Timed out. \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=\frac {512 \, \sqrt {b x^{2} + a x} B b^{5}}{693 \, a^{6} x} - \frac {2048 \, \sqrt {b x^{2} + a x} A b^{6}}{3003 \, a^{7} x} - \frac {256 \, \sqrt {b x^{2} + a x} B b^{4}}{693 \, a^{5} x^{2}} + \frac {1024 \, \sqrt {b x^{2} + a x} A b^{5}}{3003 \, a^{6} x^{2}} + \frac {64 \, \sqrt {b x^{2} + a x} B b^{3}}{231 \, a^{4} x^{3}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{4}}{1001 \, a^{5} x^{3}} - \frac {160 \, \sqrt {b x^{2} + a x} B b^{2}}{693 \, a^{3} x^{4}} + \frac {640 \, \sqrt {b x^{2} + a x} A b^{3}}{3003 \, a^{4} x^{4}} + \frac {20 \, \sqrt {b x^{2} + a x} B b}{99 \, a^{2} x^{5}} - \frac {80 \, \sqrt {b x^{2} + a x} A b^{2}}{429 \, a^{3} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{11 \, a x^{6}} + \frac {24 \, \sqrt {b x^{2} + a x} A b}{143 \, a^{2} x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{13 \, a x^{7}} \]
512/693*sqrt(b*x^2 + a*x)*B*b^5/(a^6*x) - 2048/3003*sqrt(b*x^2 + a*x)*A*b^ 6/(a^7*x) - 256/693*sqrt(b*x^2 + a*x)*B*b^4/(a^5*x^2) + 1024/3003*sqrt(b*x ^2 + a*x)*A*b^5/(a^6*x^2) + 64/231*sqrt(b*x^2 + a*x)*B*b^3/(a^4*x^3) - 256 /1001*sqrt(b*x^2 + a*x)*A*b^4/(a^5*x^3) - 160/693*sqrt(b*x^2 + a*x)*B*b^2/ (a^3*x^4) + 640/3003*sqrt(b*x^2 + a*x)*A*b^3/(a^4*x^4) + 20/99*sqrt(b*x^2 + a*x)*B*b/(a^2*x^5) - 80/429*sqrt(b*x^2 + a*x)*A*b^2/(a^3*x^5) - 2/11*sqr t(b*x^2 + a*x)*B/(a*x^6) + 24/143*sqrt(b*x^2 + a*x)*A*b/(a^2*x^6) - 2/13*s qrt(b*x^2 + a*x)*A/(a*x^7)
Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a b^{12} - 12 \, A b^{13}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {13 \, {\left (13 \, B a^{2} b^{12} - 12 \, A a b^{13}\right )}}{a^{7}}\right )} + \frac {143 \, {\left (13 \, B a^{3} b^{12} - 12 \, A a^{2} b^{13}\right )}}{a^{7}}\right )} - \frac {429 \, {\left (13 \, B a^{4} b^{12} - 12 \, A a^{3} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {3003 \, {\left (13 \, B a^{5} b^{12} - 12 \, A a^{4} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} - \frac {3003 \, {\left (13 \, B a^{6} b^{12} - 12 \, A a^{5} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {9009 \, {\left (B a^{7} b^{12} - A a^{6} b^{13}\right )}}{a^{7}}\right )} \sqrt {b x + a} b}{9009 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]
2/9009*((2*(8*(2*(b*x + a)*(4*(b*x + a)*(2*(13*B*a*b^12 - 12*A*b^13)*(b*x + a)/a^7 - 13*(13*B*a^2*b^12 - 12*A*a*b^13)/a^7) + 143*(13*B*a^3*b^12 - 12 *A*a^2*b^13)/a^7) - 429*(13*B*a^4*b^12 - 12*A*a^3*b^13)/a^7)*(b*x + a) + 3 003*(13*B*a^5*b^12 - 12*A*a^4*b^13)/a^7)*(b*x + a) - 3003*(13*B*a^6*b^12 - 12*A*a^5*b^13)/a^7)*(b*x + a) + 9009*(B*a^7*b^12 - A*a^6*b^13)/a^7)*sqrt( b*x + a)*b/(((b*x + a)*b - a*b)^(13/2)*abs(b))
Time = 0.99 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{13\,a}+\frac {x\,\left (1638\,B\,a^6-1512\,A\,a^5\,b\right )}{9009\,a^7}-\frac {160\,b^2\,x^3\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^4}+\frac {64\,b^3\,x^4\,\left (12\,A\,b-13\,B\,a\right )}{3003\,a^5}-\frac {256\,b^4\,x^5\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^6}+\frac {512\,b^5\,x^6\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^7}+\frac {20\,b\,x^2\,\left (12\,A\,b-13\,B\,a\right )}{1287\,a^3}\right )}{x^{13/2}} \]
-((a + b*x)^(1/2)*((2*A)/(13*a) + (x*(1638*B*a^6 - 1512*A*a^5*b))/(9009*a^ 7) - (160*b^2*x^3*(12*A*b - 13*B*a))/(9009*a^4) + (64*b^3*x^4*(12*A*b - 13 *B*a))/(3003*a^5) - (256*b^4*x^5*(12*A*b - 13*B*a))/(9009*a^6) + (512*b^5* x^6*(12*A*b - 13*B*a))/(9009*a^7) + (20*b*x^2*(12*A*b - 13*B*a))/(1287*a^3 )))/x^(13/2)