Integrand size = 16, antiderivative size = 99 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\frac {2 a^2 \sqrt {x}}{3 b^3 (a-b x)^{3/2}}-\frac {14 a \sqrt {x}}{3 b^3 \sqrt {a-b x}}-\frac {\sqrt {x} \sqrt {a-b x}}{b^3}+\frac {5 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{7/2}} \] Output:
2/3*a^2*x^(1/2)/b^3/(-b*x+a)^(3/2)-14/3*a*x^(1/2)/b^3/(-b*x+a)^(1/2)-x^(1/ 2)*(-b*x+a)^(1/2)/b^3+5*a*arctan(b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))/b^(7/2)
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=-\frac {\sqrt {x} \left (15 a^2-20 a b x+3 b^2 x^2\right )}{3 b^3 (a-b x)^{3/2}}+\frac {10 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{b^{7/2}} \] Input:
Integrate[x^(5/2)/(a - b*x)^(5/2),x]
Output:
-1/3*(Sqrt[x]*(15*a^2 - 20*a*b*x + 3*b^2*x^2))/(b^3*(a - b*x)^(3/2)) + (10 *a*ArcTan[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a - b*x])])/b^(7/2)
Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {57, 57, 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.
\(\displaystyle \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {5 \int \frac {x^{3/2}}{(a-b x)^{3/2}}dx}{3 b}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {5 \left (\frac {2 x^{3/2}}{b \sqrt {a-b x}}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {a-b x}}dx}{b}\right )}{3 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {5 \left (\frac {2 x^{3/2}}{b \sqrt {a-b x}}-\frac {3 \left (\frac {a \int \frac {1}{\sqrt {x} \sqrt {a-b x}}dx}{2 b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{b}\right )}{3 b}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {5 \left (\frac {2 x^{3/2}}{b \sqrt {a-b x}}-\frac {3 \left (\frac {a \int \frac {1}{\frac {b x}{a-b x}+1}d\frac {\sqrt {x}}{\sqrt {a-b x}}}{b}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{b}\right )}{3 b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 x^{5/2}}{3 b (a-b x)^{3/2}}-\frac {5 \left (\frac {2 x^{3/2}}{b \sqrt {a-b x}}-\frac {3 \left (\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b}\right )}{b}\right )}{3 b}\) |
Input:
Int[x^(5/2)/(a - b*x)^(5/2),x]
Output:
(2*x^(5/2))/(3*b*(a - b*x)^(3/2)) - (5*((2*x^(3/2))/(b*Sqrt[a - b*x]) - (3 *(-((Sqrt[x]*Sqrt[a - b*x])/b) + (a*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x] ])/b^(3/2)))/b))/(3*b)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(159\) vs. \(2(75)=150\).
Time = 0.10 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.62
method | result | size |
risch | \(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b^{3}}+\frac {\left (\frac {5 a \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {7}{2}}}+\frac {14 a \sqrt {-b \left (x -\frac {a}{b}\right )^{2}-\left (x -\frac {a}{b}\right ) a}}{3 b^{4} \left (x -\frac {a}{b}\right )}+\frac {2 a^{2} \sqrt {-b \left (x -\frac {a}{b}\right )^{2}-\left (x -\frac {a}{b}\right ) a}}{3 b^{5} \left (x -\frac {a}{b}\right )^{2}}\right ) \sqrt {x \left (-b x +a \right )}}{\sqrt {x}\, \sqrt {-b x +a}}\) | \(160\) |
Input:
int(x^(5/2)/(-b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-x^(1/2)*(-b*x+a)^(1/2)/b^3+(5/2/b^(7/2)*a*arctan(b^(1/2)*(x-1/2*a/b)/(-b* x^2+a*x)^(1/2))+14/3/b^4*a/(x-a/b)*(-b*(x-a/b)^2-(x-a/b)*a)^(1/2)+2/3/b^5* a^2/(x-a/b)^2*(-b*(x-a/b)^2-(x-a/b)*a)^(1/2))*(x*(-b*x+a))^(1/2)/x^(1/2)/( -b*x+a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.26 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (3 \, b^{3} x^{2} - 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{6 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac {15 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a} \sqrt {b} \sqrt {x}}{b x - a}\right ) + {\left (3 \, b^{3} x^{2} - 20 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (b^{6} x^{2} - 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \] Input:
integrate(x^(5/2)/(-b*x+a)^(5/2),x, algorithm="fricas")
Output:
[-1/6*(15*(a*b^2*x^2 - 2*a^2*b*x + a^3)*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) + 2*(3*b^3*x^2 - 20*a*b^2*x + 15*a^2*b)*sqrt(-b *x + a)*sqrt(x))/(b^6*x^2 - 2*a*b^5*x + a^2*b^4), -1/3*(15*(a*b^2*x^2 - 2* a^2*b*x + a^3)*sqrt(b)*arctan(sqrt(-b*x + a)*sqrt(b)*sqrt(x)/(b*x - a)) + (3*b^3*x^2 - 20*a*b^2*x + 15*a^2*b)*sqrt(-b*x + a)*sqrt(x))/(b^6*x^2 - 2*a *b^5*x + a^2*b^4)]
Result contains complex when optimal does not.
Time = 3.63 (sec) , antiderivative size = 971, normalized size of antiderivative = 9.81 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(x**(5/2)/(-b*x+a)**(5/2),x)
Output:
Piecewise((-30*I*a**(81/2)*b**22*x**(51/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)* sqrt(x)/sqrt(a))/(6*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a** (77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) + 15*pi*a**(81/2)*b**22*x**(5 1/2)*sqrt(-1 + b*x/a)/(6*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) + 30*I*a**(79/2)*b**23*x **(53/2)*sqrt(-1 + b*x/a)*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**(79/2)*b**( 51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) - 15*pi*a**(79/2)*b**23*x**(53/2)*sqrt(-1 + b*x/a)/(6*a**(79/2) *b**(51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sq rt(-1 + b*x/a)) + 30*I*a**40*b**(45/2)*x**26/(6*a**(79/2)*b**(51/2)*x**(51 /2)*sqrt(-1 + b*x/a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) - 40*I*a**39*b**(47/2)*x**27/(6*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(-1 + b*x /a) - 6*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)) + 6*I*a**38*b**(49 /2)*x**28/(6*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(-1 + b*x/a) - 6*a**(77/2)* b**(53/2)*x**(53/2)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (15*a**(81/2)*b**2 2*x**(51/2)*sqrt(1 - b*x/a)*asin(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(79/2)*b** (51/2)*x**(51/2)*sqrt(1 - b*x/a) - 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 - b*x/a)) - 15*a**(79/2)*b**23*x**(53/2)*sqrt(1 - b*x/a)*asin(sqrt(b)*sqrt (x)/sqrt(a))/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 - b*x/a) - 3*a**(77/2 )*b**(53/2)*x**(53/2)*sqrt(1 - b*x/a)) - 15*a**40*b**(45/2)*x**26/(3*a*...
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.95 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\frac {2 \, a b^{2} + \frac {10 \, {\left (b x - a\right )} a b}{x} - \frac {15 \, {\left (b x - a\right )}^{2} a}{x^{2}}}{3 \, {\left (\frac {{\left (-b x + a\right )}^{\frac {3}{2}} b^{4}}{x^{\frac {3}{2}}} + \frac {{\left (-b x + a\right )}^{\frac {5}{2}} b^{3}}{x^{\frac {5}{2}}}\right )}} - \frac {5 \, a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {7}{2}}} \] Input:
integrate(x^(5/2)/(-b*x+a)^(5/2),x, algorithm="maxima")
Output:
1/3*(2*a*b^2 + 10*(b*x - a)*a*b/x - 15*(b*x - a)^2*a/x^2)/((-b*x + a)^(3/2 )*b^4/x^(3/2) + (-b*x + a)^(5/2)*b^3/x^(5/2)) - 5*a*arctan(sqrt(-b*x + a)/ (sqrt(b)*sqrt(x)))/b^(7/2)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (75) = 150\).
Time = 15.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.23 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\frac {{\left (\frac {15 \, a \log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b} b^{2}} - \frac {6 \, \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}}{b^{3}} - \frac {8 \, {\left (9 \, a^{2} {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{4} - 12 \, a^{3} {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} b + 7 \, a^{4} b^{2}\right )}}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} \sqrt {-b} b}\right )} {\left | b \right |}}{6 \, b^{2}} \] Input:
integrate(x^(5/2)/(-b*x+a)^(5/2),x, algorithm="giac")
Output:
1/6*(15*a*log((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2)/(sqrt (-b)*b^2) - 6*sqrt((b*x - a)*b + a*b)*sqrt(-b*x + a)/b^3 - 8*(9*a^2*(sqrt( -b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^4 - 12*a^3*(sqrt(-b*x + a)*s qrt(-b) - sqrt((b*x - a)*b + a*b))^2*b + 7*a^4*b^2)/(((sqrt(-b*x + a)*sqrt (-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)^3*sqrt(-b)*b))*abs(b)/b^2
Timed out. \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\int \frac {x^{5/2}}{{\left (a-b\,x\right )}^{5/2}} \,d x \] 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 = 150, normalized size of antiderivative = 1.52 \[ \int \frac {x^{5/2}}{(a-b x)^{5/2}} \, dx=\frac {-30 \sqrt {b}\, \sqrt {-b x +a}\, \mathrm {log}\left (\frac {\sqrt {-b x +a}+\sqrt {x}\, \sqrt {b}\, i}{\sqrt {a}}\right ) a^{2} i +30 \sqrt {b}\, \sqrt {-b x +a}\, \mathrm {log}\left (\frac {\sqrt {-b x +a}+\sqrt {x}\, \sqrt {b}\, i}{\sqrt {a}}\right ) a b i x -5 \sqrt {b}\, \sqrt {-b x +a}\, a^{2} i +5 \sqrt {b}\, \sqrt {-b x +a}\, a b i x -30 \sqrt {x}\, a^{2} b +40 \sqrt {x}\, a \,b^{2} x -6 \sqrt {x}\, b^{3} x^{2}}{6 \sqrt {-b x +a}\, b^{4} \left (-b x +a \right )} \] Input:
int(x^(5/2)/(-b*x+a)^(5/2),x)
Output:
( - 30*sqrt(b)*sqrt(a - b*x)*log((sqrt(a - b*x) + sqrt(x)*sqrt(b)*i)/sqrt( a))*a**2*i + 30*sqrt(b)*sqrt(a - b*x)*log((sqrt(a - b*x) + sqrt(x)*sqrt(b) *i)/sqrt(a))*a*b*i*x - 5*sqrt(b)*sqrt(a - b*x)*a**2*i + 5*sqrt(b)*sqrt(a - b*x)*a*b*i*x - 30*sqrt(x)*a**2*b + 40*sqrt(x)*a*b**2*x - 6*sqrt(x)*b**3*x **2)/(6*sqrt(a - b*x)*b**4*(a - b*x))