Integrand size = 16, antiderivative size = 88 \[ \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx=\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a-b x}}-\frac {16 \sqrt {a-b x}}{3 a^3 x^{3/2}}-\frac {32 b \sqrt {a-b x}}{3 a^4 \sqrt {x}} \]
2/3/a/x^(3/2)/(-b*x+a)^(3/2)+4/a^2/x^(3/2)/(-b*x+a)^(1/2)-16/3*(-b*x+a)^(1 /2)/a^3/x^(3/2)-32/3*b*(-b*x+a)^(1/2)/a^4/x^(1/2)
Time = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx=-\frac {2 \left (a^3+6 a^2 b x-24 a b^2 x^2+16 b^3 x^3\right )}{3 a^4 x^{3/2} (a-b x)^{3/2}} \]
Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {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.
\(\displaystyle \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2 \int \frac {1}{x^{5/2} (a-b x)^{3/2}}dx}{a}+\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2 \left (\frac {4 \int \frac {1}{x^{5/2} \sqrt {a-b x}}dx}{a}+\frac {2}{a x^{3/2} \sqrt {a-b x}}\right )}{a}+\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2 \left (\frac {4 \left (\frac {2 b \int \frac {1}{x^{3/2} \sqrt {a-b x}}dx}{3 a}-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}\right )}{a}+\frac {2}{a x^{3/2} \sqrt {a-b x}}\right )}{a}+\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {2 \left (\frac {4 \left (-\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}\right )}{a}+\frac {2}{a x^{3/2} \sqrt {a-b x}}\right )}{a}+\frac {2}{3 a x^{3/2} (a-b x)^{3/2}}\) |
2/(3*a*x^(3/2)*(a - b*x)^(3/2)) + (2*(2/(a*x^(3/2)*Sqrt[a - b*x]) + (4*((- 2*Sqrt[a - b*x])/(3*a*x^(3/2)) - (4*b*Sqrt[a - b*x])/(3*a^2*Sqrt[x])))/a)) /a
3.7.7.3.1 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])
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {2 \left (16 b^{3} x^{3}-24 a \,b^{2} x^{2}+6 a^{2} b x +a^{3}\right )}{3 x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}} a^{4}}\) | \(45\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (8 b x +a \right )}{3 a^{4} x^{\frac {3}{2}}}+\frac {2 b^{2} \left (-8 b x +9 a \right ) \sqrt {x}}{3 \left (-b x +a \right )^{\frac {3}{2}} a^{4}}\) | \(51\) |
default | \(-\frac {2}{3 a \,x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}}}+\frac {2 b \left (-\frac {2}{a \left (-b x +a \right )^{\frac {3}{2}} \sqrt {x}}+\frac {4 b \left (\frac {2 \sqrt {x}}{3 a \left (-b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {-b x +a}}\right )}{a}\right )}{a}\) | \(80\) |
Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx=-\frac {2 \, {\left (16 \, b^{3} x^{3} - 24 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} \]
-2/3*(16*b^3*x^3 - 24*a*b^2*x^2 + 6*a^2*b*x + a^3)*sqrt(-b*x + a)*sqrt(x)/ (a^4*b^2*x^4 - 2*a^5*b*x^3 + a^6*x^2)
Result contains complex when optimal does not.
Time = 5.77 (sec) , antiderivative size = 688, normalized size of antiderivative = 7.82 \[ \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx=\begin {cases} \frac {2 a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {10 a^{3} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac {60 a^{2} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {80 a b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac {32 b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} - 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {2 i a^{4} b^{\frac {19}{2}} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {10 i a^{3} b^{\frac {21}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac {60 i a^{2} b^{\frac {23}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {80 i a b^{\frac {25}{2}} x^{3} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} - \frac {32 i b^{\frac {27}{2}} x^{4} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} - 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} & \text {otherwise} \end {cases} \]
Piecewise((2*a**4*b**(19/2)*sqrt(a/(b*x) - 1)/(-3*a**7*b**9*x + 9*a**6*b** 10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4) + 10*a**3*b**(21/2)*x*sqr t(a/(b*x) - 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3 *a**4*b**12*x**4) - 60*a**2*b**(23/2)*x**2*sqrt(a/(b*x) - 1)/(-3*a**7*b**9 *x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4) + 80*a*b** (25/2)*x**3*sqrt(a/(b*x) - 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5 *b**11*x**3 + 3*a**4*b**12*x**4) - 32*b**(27/2)*x**4*sqrt(a/(b*x) - 1)/(-3 *a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4), Abs(a/(b*x)) > 1), (2*I*a**4*b**(19/2)*sqrt(-a/(b*x) + 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4) + 10*I*a**3* b**(21/2)*x*sqrt(-a/(b*x) + 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a** 5*b**11*x**3 + 3*a**4*b**12*x**4) - 60*I*a**2*b**(23/2)*x**2*sqrt(-a/(b*x) + 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b** 12*x**4) + 80*I*a*b**(25/2)*x**3*sqrt(-a/(b*x) + 1)/(-3*a**7*b**9*x + 9*a* *6*b**10*x**2 - 9*a**5*b**11*x**3 + 3*a**4*b**12*x**4) - 32*I*b**(27/2)*x* *4*sqrt(-a/(b*x) + 1)/(-3*a**7*b**9*x + 9*a**6*b**10*x**2 - 9*a**5*b**11*x **3 + 3*a**4*b**12*x**4), True))
Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx=-\frac {2 \, {\left (\frac {9 \, \sqrt {-b x + a} b}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{4}} + \frac {2 \, {\left (b^{3} - \frac {9 \, {\left (b x - a\right )} b^{2}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{4}} \]
-2/3*(9*sqrt(-b*x + a)*b/sqrt(x) + (-b*x + a)^(3/2)/x^(3/2))/a^4 + 2/3*(b^ 3 - 9*(b*x - a)*b^2/x)*x^(3/2)/((-b*x + a)^(3/2)*a^4)
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (66) = 132\).
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.35 \[ \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx=-\frac {2 \, \sqrt {-b x + a} {\left (\frac {8 \, {\left (b x - a\right )} b^{2} {\left | b \right |}}{a^{4}} + \frac {9 \, b^{2} {\left | b \right |}}{a^{3}}\right )}}{3 \, {\left ({\left (b x - a\right )} b + a b\right )}^{\frac {3}{2}}} - \frac {8 \, {\left (3 \, {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt {-b} b^{3} - 9 \, a {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} \sqrt {-b} b^{4} + 4 \, a^{2} \sqrt {-b} b^{5}\right )}}{3 \, {\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} a^{3} {\left | b \right |}} \]
-2/3*sqrt(-b*x + a)*(8*(b*x - a)*b^2*abs(b)/a^4 + 9*b^2*abs(b)/a^3)/((b*x - a)*b + a*b)^(3/2) - 8/3*(3*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^4*sqrt(-b)*b^3 - 9*a*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2*sqrt(-b)*b^4 + 4*a^2*sqrt(-b)*b^5)/(((sqrt(-b*x + a)*sqrt(-b) - sq rt((b*x - a)*b + a*b))^2 - a*b)^3*a^3*abs(b))
Time = 0.65 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^{5/2} (a-b x)^{5/2}} \, dx=\frac {2\,a^3\,\sqrt {a-b\,x}+32\,b^3\,x^3\,\sqrt {a-b\,x}+12\,a^2\,b\,x\,\sqrt {a-b\,x}-48\,a\,b^2\,x^2\,\sqrt {a-b\,x}}{x^{3/2}\,\left (x\,\left (6\,a^5\,b-3\,a^4\,b^2\,x\right )-3\,a^6\right )} \]