Integrand size = 20, antiderivative size = 210 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx=-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}-\frac {2 (4 A b-3 a B)}{9 a^2 x^{7/2} (a+b x)^{3/2}}-\frac {20 (4 A b-3 a B)}{9 a^3 x^{7/2} \sqrt {a+b x}}+\frac {160 (4 A b-3 a B) \sqrt {a+b x}}{63 a^4 x^{7/2}}-\frac {64 b (4 A b-3 a B) \sqrt {a+b x}}{21 a^5 x^{5/2}}+\frac {256 b^2 (4 A b-3 a B) \sqrt {a+b x}}{63 a^6 x^{3/2}}-\frac {512 b^3 (4 A b-3 a B) \sqrt {a+b x}}{63 a^7 \sqrt {x}} \]
-2/9*A/a/x^(9/2)/(b*x+a)^(3/2)-2/9*(4*A*b-3*B*a)/a^2/x^(7/2)/(b*x+a)^(3/2) -20/9*(4*A*b-3*B*a)/a^3/x^(7/2)/(b*x+a)^(1/2)+160/63*(4*A*b-3*B*a)*(b*x+a) ^(1/2)/a^4/x^(7/2)-64/21*b*(4*A*b-3*B*a)*(b*x+a)^(1/2)/a^5/x^(5/2)+256/63* b^2*(4*A*b-3*B*a)*(b*x+a)^(1/2)/a^6/x^(3/2)-512/63*b^3*(4*A*b-3*B*a)*(b*x+ a)^(1/2)/a^7/x^(1/2)
Time = 0.39 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.60 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx=-\frac {2 \left (1024 A b^6 x^6+384 a^2 b^4 x^4 (A-3 B x)-768 a b^5 x^5 (-2 A+B x)+24 a^4 b^2 x^2 (A+2 B x)-6 a^5 b x (2 A+3 B x)-32 a^3 b^3 x^3 (2 A+9 B x)+a^6 (7 A+9 B x)\right )}{63 a^7 x^{9/2} (a+b x)^{3/2}} \]
(-2*(1024*A*b^6*x^6 + 384*a^2*b^4*x^4*(A - 3*B*x) - 768*a*b^5*x^5*(-2*A + B*x) + 24*a^4*b^2*x^2*(A + 2*B*x) - 6*a^5*b*x*(2*A + 3*B*x) - 32*a^3*b^3*x ^3*(2*A + 9*B*x) + a^6*(7*A + 9*B*x)))/(63*a^7*x^(9/2)*(a + b*x)^(3/2))
Time = 0.23 (sec) , antiderivative size = 197, 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.
| \(\displaystyle \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(4 A b-3 a B) \int \frac {1}{x^{9/2} (a+b x)^{5/2}}dx}{3 a}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(4 A b-3 a B) \left (\frac {10 \int \frac {1}{x^{9/2} (a+b x)^{3/2}}dx}{3 a}+\frac {2}{3 a x^{7/2} (a+b x)^{3/2}}\right )}{3 a}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(4 A b-3 a B) \left (\frac {10 \left (\frac {8 \int \frac {1}{x^{9/2} \sqrt {a+b x}}dx}{a}+\frac {2}{a x^{7/2} \sqrt {a+b x}}\right )}{3 a}+\frac {2}{3 a x^{7/2} (a+b x)^{3/2}}\right )}{3 a}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(4 A b-3 a B) \left (\frac {10 \left (\frac {8 \left (-\frac {6 b \int \frac {1}{x^{7/2} \sqrt {a+b x}}dx}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{a}+\frac {2}{a x^{7/2} \sqrt {a+b x}}\right )}{3 a}+\frac {2}{3 a x^{7/2} (a+b x)^{3/2}}\right )}{3 a}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(4 A b-3 a B) \left (\frac {10 \left (\frac {8 \left (-\frac {6 b \left (-\frac {4 b \int \frac {1}{x^{5/2} \sqrt {a+b x}}dx}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{a}+\frac {2}{a x^{7/2} \sqrt {a+b x}}\right )}{3 a}+\frac {2}{3 a x^{7/2} (a+b x)^{3/2}}\right )}{3 a}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {(4 A b-3 a B) \left (\frac {10 \left (\frac {8 \left (-\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {1}{x^{3/2} \sqrt {a+b x}}dx}{3 a}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}\right )}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{a}+\frac {2}{a x^{7/2} \sqrt {a+b x}}\right )}{3 a}+\frac {2}{3 a x^{7/2} (a+b x)^{3/2}}\right )}{3 a}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (\frac {10 \left (\frac {8 \left (-\frac {6 b \left (-\frac {4 b \left (\frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}\right )}{5 a}-\frac {2 \sqrt {a+b x}}{5 a x^{5/2}}\right )}{7 a}-\frac {2 \sqrt {a+b x}}{7 a x^{7/2}}\right )}{a}+\frac {2}{a x^{7/2} \sqrt {a+b x}}\right )}{3 a}+\frac {2}{3 a x^{7/2} (a+b x)^{3/2}}\right ) (4 A b-3 a B)}{3 a}-\frac {2 A}{9 a x^{9/2} (a+b x)^{3/2}}\) |
(-2*A)/(9*a*x^(9/2)*(a + b*x)^(3/2)) - ((4*A*b - 3*a*B)*(2/(3*a*x^(7/2)*(a + b*x)^(3/2)) + (10*(2/(a*x^(7/2)*Sqrt[a + b*x]) + (8*((-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)))/a))/(3*a)))/(3*a)
3.6.46.3.1 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 = 0.54 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(-\frac {2 \sqrt {b x +a}\, \left (667 A \,b^{4} x^{4}-474 B a \,b^{3} x^{4}-176 A a \,b^{3} x^{3}+111 B \,a^{2} b^{2} x^{3}+69 A \,a^{2} b^{2} x^{2}-36 B \,a^{3} b \,x^{2}-26 A \,a^{3} b x +9 B \,a^{4} x +7 A \,a^{4}\right )}{63 a^{7} x^{\frac {9}{2}}}-\frac {2 b^{4} \left (17 A \,b^{2} x -14 B a b x +18 a b A -15 a^{2} B \right ) \sqrt {x}}{3 \left (b x +a \right )^{\frac {3}{2}} a^{7}}\) | \(145\) |
| gosper | \(-\frac {2 \left (1024 A \,b^{6} x^{6}-768 B a \,b^{5} x^{6}+1536 A a \,b^{5} x^{5}-1152 B \,a^{2} b^{4} x^{5}+384 A \,a^{2} b^{4} x^{4}-288 B \,a^{3} b^{3} x^{4}-64 A \,a^{3} b^{3} x^{3}+48 B \,a^{4} b^{2} x^{3}+24 A \,a^{4} b^{2} x^{2}-18 B \,a^{5} b \,x^{2}-12 A \,a^{5} b x +9 B \,a^{6} x +7 A \,a^{6}\right )}{63 x^{\frac {9}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{7}}\) | \(149\) |
| default | \(-\frac {2 \left (1024 A \,b^{6} x^{6}-768 B a \,b^{5} x^{6}+1536 A a \,b^{5} x^{5}-1152 B \,a^{2} b^{4} x^{5}+384 A \,a^{2} b^{4} x^{4}-288 B \,a^{3} b^{3} x^{4}-64 A \,a^{3} b^{3} x^{3}+48 B \,a^{4} b^{2} x^{3}+24 A \,a^{4} b^{2} x^{2}-18 B \,a^{5} b \,x^{2}-12 A \,a^{5} b x +9 B \,a^{6} x +7 A \,a^{6}\right )}{63 x^{\frac {9}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{7}}\) | \(149\) |
-2/63*(b*x+a)^(1/2)*(667*A*b^4*x^4-474*B*a*b^3*x^4-176*A*a*b^3*x^3+111*B*a ^2*b^2*x^3+69*A*a^2*b^2*x^2-36*B*a^3*b*x^2-26*A*a^3*b*x+9*B*a^4*x+7*A*a^4) /a^7/x^(9/2)-2/3*b^4*(17*A*b^2*x-14*B*a*b*x+18*A*a*b-15*B*a^2)*x^(1/2)/(b* x+a)^(3/2)/a^7
Time = 0.24 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx=-\frac {2 \, {\left (7 \, A a^{6} - 256 \, {\left (3 \, B a b^{5} - 4 \, A b^{6}\right )} x^{6} - 384 \, {\left (3 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )} x^{5} - 96 \, {\left (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4}\right )} x^{4} + 16 \, {\left (3 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3}\right )} x^{3} - 6 \, {\left (3 \, B a^{5} b - 4 \, A a^{4} b^{2}\right )} x^{2} + 3 \, {\left (3 \, B a^{6} - 4 \, A a^{5} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{63 \, {\left (a^{7} b^{2} x^{7} + 2 \, a^{8} b x^{6} + a^{9} x^{5}\right )}} \]
-2/63*(7*A*a^6 - 256*(3*B*a*b^5 - 4*A*b^6)*x^6 - 384*(3*B*a^2*b^4 - 4*A*a* b^5)*x^5 - 96*(3*B*a^3*b^3 - 4*A*a^2*b^4)*x^4 + 16*(3*B*a^4*b^2 - 4*A*a^3* b^3)*x^3 - 6*(3*B*a^5*b - 4*A*a^4*b^2)*x^2 + 3*(3*B*a^6 - 4*A*a^5*b)*x)*sq rt(b*x + a)*sqrt(x)/(a^7*b^2*x^7 + 2*a^8*b*x^6 + a^9*x^5)
Timed out. \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx=-\frac {64 \, B b^{3} x}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} + \frac {512 \, B b^{4} x}{21 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {256 \, A b^{4} x}{63 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{5}} - \frac {2048 \, A b^{5} x}{63 \, \sqrt {b x^{2} + a x} a^{7}} - \frac {32 \, B b^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} + \frac {256 \, B b^{3}}{21 \, \sqrt {b x^{2} + a x} a^{5}} + \frac {128 \, A b^{3}}{63 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} - \frac {1024 \, A b^{4}}{63 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {4 \, B b}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x} - \frac {16 \, A b^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} x} - \frac {2 \, B}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{2}} + \frac {8 \, A b}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x^{2}} - \frac {2 \, A}{9 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{3}} \]
-64/21*B*b^3*x/((b*x^2 + a*x)^(3/2)*a^4) + 512/21*B*b^4*x/(sqrt(b*x^2 + a* x)*a^6) + 256/63*A*b^4*x/((b*x^2 + a*x)^(3/2)*a^5) - 2048/63*A*b^5*x/(sqrt (b*x^2 + a*x)*a^7) - 32/21*B*b^2/((b*x^2 + a*x)^(3/2)*a^3) + 256/21*B*b^3/ (sqrt(b*x^2 + a*x)*a^5) + 128/63*A*b^3/((b*x^2 + a*x)^(3/2)*a^4) - 1024/63 *A*b^4/(sqrt(b*x^2 + a*x)*a^6) + 4/7*B*b/((b*x^2 + a*x)^(3/2)*a^2*x) - 16/ 21*A*b^2/((b*x^2 + a*x)^(3/2)*a^3*x) - 2/7*B/((b*x^2 + a*x)^(3/2)*a*x^2) + 8/21*A*b/((b*x^2 + a*x)^(3/2)*a^2*x^2) - 2/9*A/((b*x^2 + a*x)^(3/2)*a*x^3 )
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (168) = 336\).
Time = 0.44 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.99 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (474 \, B a^{19} b^{13} - 667 \, A a^{18} b^{14}\right )} {\left (b x + a\right )}}{a^{25} b^{4} {\left | b \right |}} - \frac {9 \, {\left (223 \, B a^{20} b^{13} - 316 \, A a^{19} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} + \frac {63 \, {\left (51 \, B a^{21} b^{13} - 73 \, A a^{20} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} - \frac {210 \, {\left (11 \, B a^{22} b^{13} - 16 \, A a^{21} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {315 \, {\left (2 \, B a^{23} b^{13} - 3 \, A a^{22} b^{14}\right )}}{a^{25} b^{4} {\left | b \right |}}\right )} \sqrt {b x + a}}{63 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}}} + \frac {4 \, {\left (12 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {11}{2}} + 30 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {13}{2}} - 15 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {13}{2}} + 14 \, B a^{3} b^{\frac {15}{2}} - 36 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {15}{2}} - 17 \, A a^{2} b^{\frac {17}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{6} {\left | b \right |}} \]
2/63*(((b*x + a)*((b*x + a)*((474*B*a^19*b^13 - 667*A*a^18*b^14)*(b*x + a) /(a^25*b^4*abs(b)) - 9*(223*B*a^20*b^13 - 316*A*a^19*b^14)/(a^25*b^4*abs(b ))) + 63*(51*B*a^21*b^13 - 73*A*a^20*b^14)/(a^25*b^4*abs(b))) - 210*(11*B* a^22*b^13 - 16*A*a^21*b^14)/(a^25*b^4*abs(b)))*(b*x + a) + 315*(2*B*a^23*b ^13 - 3*A*a^22*b^14)/(a^25*b^4*abs(b)))*sqrt(b*x + a)/((b*x + a)*b - a*b)^ (9/2) + 4/3*(12*B*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^ (11/2) + 30*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(1 3/2) - 15*A*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(13/2) + 14*B*a^3*b^(15/2) - 36*A*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a* b))^2*b^(15/2) - 17*A*a^2*b^(17/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^6*abs(b))
Time = 1.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{5/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{9\,a\,b^2}-\frac {32\,x^3\,\left (4\,A\,b-3\,B\,a\right )}{63\,a^4}+\frac {4\,x^2\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^3\,b}+\frac {256\,b^2\,x^5\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^6}+\frac {512\,b^3\,x^6\,\left (4\,A\,b-3\,B\,a\right )}{63\,a^7}+\frac {64\,b\,x^4\,\left (4\,A\,b-3\,B\,a\right )}{21\,a^5}+\frac {x\,\left (18\,B\,a^6-24\,A\,a^5\,b\right )}{63\,a^7\,b^2}\right )}{x^{13/2}+\frac {2\,a\,x^{11/2}}{b}+\frac {a^2\,x^{9/2}}{b^2}} \]
-((a + b*x)^(1/2)*((2*A)/(9*a*b^2) - (32*x^3*(4*A*b - 3*B*a))/(63*a^4) + ( 4*x^2*(4*A*b - 3*B*a))/(21*a^3*b) + (256*b^2*x^5*(4*A*b - 3*B*a))/(21*a^6) + (512*b^3*x^6*(4*A*b - 3*B*a))/(63*a^7) + (64*b*x^4*(4*A*b - 3*B*a))/(21 *a^5) + (x*(18*B*a^6 - 24*A*a^5*b))/(63*a^7*b^2)))/(x^(13/2) + (2*a*x^(11/ 2))/b + (a^2*x^(9/2))/b^2)