Integrand size = 16, antiderivative size = 71 \[ \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx=-\frac {2 \sqrt {a-b x}}{5 a x^{5/2}}-\frac {8 b \sqrt {a-b x}}{15 a^2 x^{3/2}}-\frac {16 b^2 \sqrt {a-b x}}{15 a^3 \sqrt {x}} \] Output:
-2/5*(-b*x+a)^(1/2)/a/x^(5/2)-8/15*b*(-b*x+a)^(1/2)/a^2/x^(3/2)-16/15*b^2* (-b*x+a)^(1/2)/a^3/x^(1/2)
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx=-\frac {2 \sqrt {a-b x} \left (3 a^2+4 a b x+8 b^2 x^2\right )}{15 a^3 x^{5/2}} \] Input:
Integrate[1/(x^(7/2)*Sqrt[a - b*x]),x]
Output:
(-2*Sqrt[a - b*x]*(3*a^2 + 4*a*b*x + 8*b^2*x^2))/(15*a^3*x^(5/2))
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {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.
\(\displaystyle \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4 b \int \frac {1}{x^{5/2} \sqrt {a-b x}}dx}{5 a}-\frac {2 \sqrt {a-b x}}{5 a x^{5/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4 b \left (\frac {2 b \int \frac {1}{x^{3/2} \sqrt {a-b x}}dx}{3 a}-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}\right )}{5 a}-\frac {2 \sqrt {a-b x}}{5 a x^{5/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {4 b \left (-\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}\right )}{5 a}-\frac {2 \sqrt {a-b x}}{5 a x^{5/2}}\) |
Input:
Int[1/(x^(7/2)*Sqrt[a - b*x]),x]
Output:
(-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)
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 = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {2 \sqrt {-b x +a}\, \left (8 b^{2} x^{2}+4 a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{3}}\) | \(36\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (8 b^{2} x^{2}+4 a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{3}}\) | \(36\) |
orering | \(-\frac {2 \sqrt {-b x +a}\, \left (8 b^{2} x^{2}+4 a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{3}}\) | \(36\) |
default | \(-\frac {2 \sqrt {-b x +a}}{5 a \,x^{\frac {5}{2}}}+\frac {4 b \left (-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\) | \(58\) |
Input:
int(1/x^(7/2)/(-b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/15*(-b*x+a)^(1/2)*(8*b^2*x^2+4*a*b*x+3*a^2)/x^(5/2)/a^3
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx=-\frac {2 \, {\left (8 \, b^{2} x^{2} + 4 \, a b x + 3 \, a^{2}\right )} \sqrt {-b x + a}}{15 \, a^{3} x^{\frac {5}{2}}} \] Input:
integrate(1/x^(7/2)/(-b*x+a)^(1/2),x, algorithm="fricas")
Output:
-2/15*(8*b^2*x^2 + 4*a*b*x + 3*a^2)*sqrt(-b*x + a)/(a^3*x^(5/2))
Result contains complex when optimal does not.
Time = 2.83 (sec) , antiderivative size = 586, normalized size of antiderivative = 8.25 \[ \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx=\begin {cases} - \frac {6 a^{4} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} - 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} + \frac {4 a^{3} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} - 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {6 a^{2} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} + \frac {24 a b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} - 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {16 b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} - 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {6 i a^{4} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} + \frac {4 i a^{3} b^{\frac {11}{2}} x \sqrt {- \frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {6 i a^{2} b^{\frac {13}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} + \frac {24 i a b^{\frac {15}{2}} x^{3} \sqrt {- \frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} - \frac {16 i b^{\frac {17}{2}} x^{4} \sqrt {- \frac {a}{b x} + 1}}{15 a^{5} b^{4} x^{2} - 30 a^{4} b^{5} x^{3} + 15 a^{3} b^{6} x^{4}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**(7/2)/(-b*x+a)**(1/2),x)
Output:
Piecewise((-6*a**4*b**(9/2)*sqrt(a/(b*x) - 1)/(15*a**5*b**4*x**2 - 30*a**4 *b**5*x**3 + 15*a**3*b**6*x**4) + 4*a**3*b**(11/2)*x*sqrt(a/(b*x) - 1)/(15 *a**5*b**4*x**2 - 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 6*a**2*b**(13/2 )*x**2*sqrt(a/(b*x) - 1)/(15*a**5*b**4*x**2 - 30*a**4*b**5*x**3 + 15*a**3* b**6*x**4) + 24*a*b**(15/2)*x**3*sqrt(a/(b*x) - 1)/(15*a**5*b**4*x**2 - 30 *a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 16*b**(17/2)*x**4*sqrt(a/(b*x) - 1) /(15*a**5*b**4*x**2 - 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4), Abs(a/(b*x)) > 1), (-6*I*a**4*b**(9/2)*sqrt(-a/(b*x) + 1)/(15*a**5*b**4*x**2 - 30*a**4 *b**5*x**3 + 15*a**3*b**6*x**4) + 4*I*a**3*b**(11/2)*x*sqrt(-a/(b*x) + 1)/ (15*a**5*b**4*x**2 - 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 6*I*a**2*b** (13/2)*x**2*sqrt(-a/(b*x) + 1)/(15*a**5*b**4*x**2 - 30*a**4*b**5*x**3 + 15 *a**3*b**6*x**4) + 24*I*a*b**(15/2)*x**3*sqrt(-a/(b*x) + 1)/(15*a**5*b**4* x**2 - 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4) - 16*I*b**(17/2)*x**4*sqrt(- a/(b*x) + 1)/(15*a**5*b**4*x**2 - 30*a**4*b**5*x**3 + 15*a**3*b**6*x**4), True))
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx=-\frac {2 \, {\left (\frac {15 \, \sqrt {-b x + a} b^{2}}{\sqrt {x}} + \frac {10 \, {\left (-b x + a\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {3 \, {\left (-b x + a\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}\right )}}{15 \, a^{3}} \] Input:
integrate(1/x^(7/2)/(-b*x+a)^(1/2),x, algorithm="maxima")
Output:
-2/15*(15*sqrt(-b*x + a)*b^2/sqrt(x) + 10*(-b*x + a)^(3/2)*b/x^(3/2) + 3*( -b*x + a)^(5/2)/x^(5/2))/a^3
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx=-\frac {2 \, {\left (\frac {15 \, b^{5}}{a} + 4 \, {\left (\frac {2 \, {\left (b x - a\right )} b^{5}}{a^{3}} + \frac {5 \, b^{5}}{a^{2}}\right )} {\left (b x - a\right )}\right )} \sqrt {-b x + a} b}{15 \, {\left ({\left (b x - a\right )} b + a b\right )}^{\frac {5}{2}} {\left | b \right |}} \] Input:
integrate(1/x^(7/2)/(-b*x+a)^(1/2),x, algorithm="giac")
Output:
-2/15*(15*b^5/a + 4*(2*(b*x - a)*b^5/a^3 + 5*b^5/a^2)*(b*x - a))*sqrt(-b*x + a)*b/(((b*x - a)*b + a*b)^(5/2)*abs(b))
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.52 \[ \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx=-\frac {\sqrt {a-b\,x}\,\left (\frac {2}{5\,a}+\frac {16\,b^2\,x^2}{15\,a^3}+\frac {8\,b\,x}{15\,a^2}\right )}{x^{5/2}} \] Input:
int(1/(x^(7/2)*(a - b*x)^(1/2)),x)
Output:
-((a - b*x)^(1/2)*(2/(5*a) + (16*b^2*x^2)/(15*a^3) + (8*b*x)/(15*a^2)))/x^ (5/2)
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^{7/2} \sqrt {a-b x}} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {-b x +a}\, a^{2}}{5}-\frac {8 \sqrt {x}\, \sqrt {-b x +a}\, a b x}{15}-\frac {16 \sqrt {x}\, \sqrt {-b x +a}\, b^{2} x^{2}}{15}+\frac {16 \sqrt {b}\, b^{2} i \,x^{3}}{15}}{a^{3} x^{3}} \] Input:
int(1/x^(7/2)/(-b*x+a)^(1/2),x)
Output:
(2*( - 3*sqrt(x)*sqrt(a - b*x)*a**2 - 4*sqrt(x)*sqrt(a - b*x)*a*b*x - 8*sq rt(x)*sqrt(a - b*x)*b**2*x**2 + 8*sqrt(b)*b**2*i*x**3))/(15*a**3*x**3)