Integrand size = 16, antiderivative size = 175 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=-\frac {5 a^5 \sqrt {x} \sqrt {a-b x}}{512 b^3}-\frac {5 a^4 x^{3/2} \sqrt {a-b x}}{768 b^2}-\frac {a^3 x^{5/2} \sqrt {a-b x}}{192 b}+\frac {9}{32} a^2 x^{7/2} \sqrt {a-b x}-\frac {5}{12} a b x^{9/2} \sqrt {a-b x}+\frac {1}{6} b^2 x^{11/2} \sqrt {a-b x}+\frac {5 a^6 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{512 b^{7/2}} \] Output:
-5/512*a^5*x^(1/2)*(-b*x+a)^(1/2)/b^3-5/768*a^4*x^(3/2)*(-b*x+a)^(1/2)/b^2 -1/192*a^3*x^(5/2)*(-b*x+a)^(1/2)/b+9/32*a^2*x^(7/2)*(-b*x+a)^(1/2)-5/12*a *b*x^(9/2)*(-b*x+a)^(1/2)+1/6*b^2*x^(11/2)*(-b*x+a)^(1/2)+5/512*a^6*arctan (b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(7/2)
Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a-b x} \left (-15 a^5-10 a^4 b x-8 a^3 b^2 x^2+432 a^2 b^3 x^3-640 a b^4 x^4+256 b^5 x^5\right )+30 a^6 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{1536 b^{7/2}} \] Input:
Integrate[x^(5/2)*(a - b*x)^(5/2),x]
Output:
(Sqrt[b]*Sqrt[x]*Sqrt[a - b*x]*(-15*a^5 - 10*a^4*b*x - 8*a^3*b^2*x^2 + 432 *a^2*b^3*x^3 - 640*a*b^4*x^4 + 256*b^5*x^5) + 30*a^6*ArcTan[(Sqrt[b]*Sqrt[ x])/(-Sqrt[a] + Sqrt[a - b*x])])/(1536*b^(7/2))
Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {60, 60, 60, 60, 60, 60, 65, 216}
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.
| \(\displaystyle \int x^{5/2} (a-b x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \int x^{5/2} (a-b x)^{3/2}dx+\frac {1}{6} x^{7/2} (a-b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \int x^{5/2} \sqrt {a-b x}dx+\frac {1}{5} x^{7/2} (a-b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a-b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \int \frac {x^{5/2}}{\sqrt {a-b x}}dx+\frac {1}{4} x^{7/2} \sqrt {a-b x}\right )+\frac {1}{5} x^{7/2} (a-b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a-b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {5 a \int \frac {x^{3/2}}{\sqrt {a-b x}}dx}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a-b x}\right )+\frac {1}{5} x^{7/2} (a-b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a-b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {5 a \left (\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a-b x}}dx}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a-b x}\right )+\frac {1}{5} x^{7/2} (a-b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a-b x)^{5/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {5 a \left (\frac {3 a \left (\frac {a \int \frac {1}{\sqrt {x} \sqrt {a-b x}}dx}{2 b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a-b x}\right )+\frac {1}{5} x^{7/2} (a-b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a-b x)^{5/2}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {5 a \left (\frac {3 a \left (\frac {a \int \frac {1}{\frac {b x}{a-b x}+1}d\frac {\sqrt {x}}{\sqrt {a-b x}}}{b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a-b x}\right )+\frac {1}{5} x^{7/2} (a-b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a-b x)^{5/2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {5}{12} a \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {5 a \left (\frac {3 a \left (\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{4 b}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}\right )}{6 b}-\frac {x^{5/2} \sqrt {a-b x}}{3 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a-b x}\right )+\frac {1}{5} x^{7/2} (a-b x)^{3/2}\right )+\frac {1}{6} x^{7/2} (a-b x)^{5/2}\) |
Input:
Int[x^(5/2)*(a - b*x)^(5/2),x]
Output:
(x^(7/2)*(a - b*x)^(5/2))/6 + (5*a*((x^(7/2)*(a - b*x)^(3/2))/5 + (3*a*((x ^(7/2)*Sqrt[a - b*x])/4 + (a*(-1/3*(x^(5/2)*Sqrt[a - b*x])/b + (5*a*(-1/2* (x^(3/2)*Sqrt[a - b*x])/b + (3*a*(-((Sqrt[x]*Sqrt[a - b*x])/b) + (a*ArcTan [(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/b^(3/2)))/(4*b)))/(6*b)))/8))/10))/12
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {\left (-256 b^{5} x^{5}+640 a \,b^{4} x^{4}-432 a^{2} b^{3} x^{3}+8 a^{3} b^{2} x^{2}+10 a^{4} b x +15 a^{5}\right ) \sqrt {x}\, \sqrt {-b x +a}}{1536 b^{3}}+\frac {5 a^{6} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{1024 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(124\) |
| default | \(-\frac {x^{\frac {5}{2}} \left (-b x +a \right )^{\frac {7}{2}}}{6 b}+\frac {5 a \left (-\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {7}{2}}}{5 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {7}{2}}}{4 b}+\frac {a \left (\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {5}{2}}}{3}+\frac {5 a \left (\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {3}{2}}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )}{12 b}\) | \(169\) |
Input:
int(x^(5/2)*(-b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/1536*(-256*b^5*x^5+640*a*b^4*x^4-432*a^2*b^3*x^3+8*a^3*b^2*x^2+10*a^4*b *x+15*a^5)/b^3*x^(1/2)*(-b*x+a)^(1/2)+5/1024*a^6/b^(7/2)*arctan(b^(1/2)*(x -1/2*a/b)/(-b*x^2+a*x)^(1/2))*(x*(-b*x+a))^(1/2)/x^(1/2)/(-b*x+a)^(1/2)
Time = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.24 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\left [-\frac {15 \, a^{6} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (256 \, b^{6} x^{5} - 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} - 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x - 15 \, a^{5} b\right )} \sqrt {-b x + a} \sqrt {x}}{3072 \, b^{4}}, -\frac {15 \, a^{6} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a} \sqrt {b} \sqrt {x}}{b x - a}\right ) - {\left (256 \, b^{6} x^{5} - 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} - 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x - 15 \, a^{5} b\right )} \sqrt {-b x + a} \sqrt {x}}{1536 \, b^{4}}\right ] \] Input:
integrate(x^(5/2)*(-b*x+a)^(5/2),x, algorithm="fricas")
Output:
[-1/3072*(15*a^6*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) - 2*(256*b^6*x^5 - 640*a*b^5*x^4 + 432*a^2*b^4*x^3 - 8*a^3*b^3*x^2 - 1 0*a^4*b^2*x - 15*a^5*b)*sqrt(-b*x + a)*sqrt(x))/b^4, -1/1536*(15*a^6*sqrt( b)*arctan(sqrt(-b*x + a)*sqrt(b)*sqrt(x)/(b*x - a)) - (256*b^6*x^5 - 640*a *b^5*x^4 + 432*a^2*b^4*x^3 - 8*a^3*b^3*x^2 - 10*a^4*b^2*x - 15*a^5*b)*sqrt (-b*x + a)*sqrt(x))/b^4]
Result contains complex when optimal does not.
Time = 150.80 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.49 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\begin {cases} \frac {5 i a^{\frac {11}{2}} \sqrt {x}}{512 b^{3} \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{\frac {9}{2}} x^{\frac {3}{2}}}{1536 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {7}{2}} x^{\frac {5}{2}}}{768 b \sqrt {-1 + \frac {b x}{a}}} - \frac {55 i a^{\frac {5}{2}} x^{\frac {7}{2}}}{192 \sqrt {-1 + \frac {b x}{a}}} + \frac {67 i a^{\frac {3}{2}} b x^{\frac {9}{2}}}{96 \sqrt {-1 + \frac {b x}{a}}} - \frac {7 i \sqrt {a} b^{2} x^{\frac {11}{2}}}{12 \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i a^{6} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{512 b^{\frac {7}{2}}} + \frac {i b^{3} x^{\frac {13}{2}}}{6 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {5 a^{\frac {11}{2}} \sqrt {x}}{512 b^{3} \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{\frac {9}{2}} x^{\frac {3}{2}}}{1536 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {7}{2}} x^{\frac {5}{2}}}{768 b \sqrt {1 - \frac {b x}{a}}} + \frac {55 a^{\frac {5}{2}} x^{\frac {7}{2}}}{192 \sqrt {1 - \frac {b x}{a}}} - \frac {67 a^{\frac {3}{2}} b x^{\frac {9}{2}}}{96 \sqrt {1 - \frac {b x}{a}}} + \frac {7 \sqrt {a} b^{2} x^{\frac {11}{2}}}{12 \sqrt {1 - \frac {b x}{a}}} + \frac {5 a^{6} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{512 b^{\frac {7}{2}}} - \frac {b^{3} x^{\frac {13}{2}}}{6 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**(5/2)*(-b*x+a)**(5/2),x)
Output:
Piecewise((5*I*a**(11/2)*sqrt(x)/(512*b**3*sqrt(-1 + b*x/a)) - 5*I*a**(9/2 )*x**(3/2)/(1536*b**2*sqrt(-1 + b*x/a)) - I*a**(7/2)*x**(5/2)/(768*b*sqrt( -1 + b*x/a)) - 55*I*a**(5/2)*x**(7/2)/(192*sqrt(-1 + b*x/a)) + 67*I*a**(3/ 2)*b*x**(9/2)/(96*sqrt(-1 + b*x/a)) - 7*I*sqrt(a)*b**2*x**(11/2)/(12*sqrt( -1 + b*x/a)) - 5*I*a**6*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(512*b**(7/2)) + I* b**3*x**(13/2)/(6*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-5*a**(11/2 )*sqrt(x)/(512*b**3*sqrt(1 - b*x/a)) + 5*a**(9/2)*x**(3/2)/(1536*b**2*sqrt (1 - b*x/a)) + a**(7/2)*x**(5/2)/(768*b*sqrt(1 - b*x/a)) + 55*a**(5/2)*x** (7/2)/(192*sqrt(1 - b*x/a)) - 67*a**(3/2)*b*x**(9/2)/(96*sqrt(1 - b*x/a)) + 7*sqrt(a)*b**2*x**(11/2)/(12*sqrt(1 - b*x/a)) + 5*a**6*asin(sqrt(b)*sqrt (x)/sqrt(a))/(512*b**(7/2)) - b**3*x**(13/2)/(6*sqrt(a)*sqrt(1 - b*x/a)), True))
Time = 0.11 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.38 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=-\frac {5 \, a^{6} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{512 \, b^{\frac {7}{2}}} + \frac {\frac {15 \, \sqrt {-b x + a} a^{6} b^{5}}{\sqrt {x}} + \frac {85 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{6} b^{4}}{x^{\frac {3}{2}}} + \frac {198 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{6} b^{3}}{x^{\frac {5}{2}}} - \frac {198 \, {\left (-b x + a\right )}^{\frac {7}{2}} a^{6} b^{2}}{x^{\frac {7}{2}}} - \frac {85 \, {\left (-b x + a\right )}^{\frac {9}{2}} a^{6} b}{x^{\frac {9}{2}}} - \frac {15 \, {\left (-b x + a\right )}^{\frac {11}{2}} a^{6}}{x^{\frac {11}{2}}}}{1536 \, {\left (b^{9} - \frac {6 \, {\left (b x - a\right )} b^{8}}{x} + \frac {15 \, {\left (b x - a\right )}^{2} b^{7}}{x^{2}} - \frac {20 \, {\left (b x - a\right )}^{3} b^{6}}{x^{3}} + \frac {15 \, {\left (b x - a\right )}^{4} b^{5}}{x^{4}} - \frac {6 \, {\left (b x - a\right )}^{5} b^{4}}{x^{5}} + \frac {{\left (b x - a\right )}^{6} b^{3}}{x^{6}}\right )}} \] Input:
integrate(x^(5/2)*(-b*x+a)^(5/2),x, algorithm="maxima")
Output:
-5/512*a^6*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/b^(7/2) + 1/1536*(15*s qrt(-b*x + a)*a^6*b^5/sqrt(x) + 85*(-b*x + a)^(3/2)*a^6*b^4/x^(3/2) + 198* (-b*x + a)^(5/2)*a^6*b^3/x^(5/2) - 198*(-b*x + a)^(7/2)*a^6*b^2/x^(7/2) - 85*(-b*x + a)^(9/2)*a^6*b/x^(9/2) - 15*(-b*x + a)^(11/2)*a^6/x^(11/2))/(b^ 9 - 6*(b*x - a)*b^8/x + 15*(b*x - a)^2*b^7/x^2 - 20*(b*x - a)^3*b^6/x^3 + 15*(b*x - a)^4*b^5/x^4 - 6*(b*x - a)^5*b^4/x^5 + (b*x - a)^6*b^3/x^6)
Timed out. \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\text {Timed out} \] Input:
integrate(x^(5/2)*(-b*x+a)^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\int x^{5/2}\,{\left (a-b\,x\right )}^{5/2} \,d x \] Input:
int(x^(5/2)*(a - b*x)^(5/2),x)
Output:
int(x^(5/2)*(a - b*x)^(5/2), x)
Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.81 \[ \int x^{5/2} (a-b x)^{5/2} \, dx=\frac {-15 \sqrt {x}\, \sqrt {-b x +a}\, a^{5} b -10 \sqrt {x}\, \sqrt {-b x +a}\, a^{4} b^{2} x -8 \sqrt {x}\, \sqrt {-b x +a}\, a^{3} b^{3} x^{2}+432 \sqrt {x}\, \sqrt {-b x +a}\, a^{2} b^{4} x^{3}-640 \sqrt {x}\, \sqrt {-b x +a}\, a \,b^{5} x^{4}+256 \sqrt {x}\, \sqrt {-b x +a}\, b^{6} x^{5}-15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {-b x +a}+\sqrt {x}\, \sqrt {b}\, i}{\sqrt {a}}\right ) a^{6} i}{1536 b^{4}} \] Input:
int(x^(5/2)*(-b*x+a)^(5/2),x)
Output:
( - 15*sqrt(x)*sqrt(a - b*x)*a**5*b - 10*sqrt(x)*sqrt(a - b*x)*a**4*b**2*x - 8*sqrt(x)*sqrt(a - b*x)*a**3*b**3*x**2 + 432*sqrt(x)*sqrt(a - b*x)*a**2 *b**4*x**3 - 640*sqrt(x)*sqrt(a - b*x)*a*b**5*x**4 + 256*sqrt(x)*sqrt(a - b*x)*b**6*x**5 - 15*sqrt(b)*log((sqrt(a - b*x) + sqrt(x)*sqrt(b)*i)/sqrt(a ))*a**6*i)/(1536*b**4)