Integrand size = 26, antiderivative size = 121 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 a^4 \sqrt {1-a x}}{9 (a x)^{9/2}}-\frac {34 a^4 \sqrt {1-a x}}{63 (a x)^{7/2}}-\frac {68 a^4 \sqrt {1-a x}}{105 (a x)^{5/2}}-\frac {272 a^4 \sqrt {1-a x}}{315 (a x)^{3/2}}-\frac {544 a^4 \sqrt {1-a x}}{315 \sqrt {a x}} \] Output:
-2/9*a^4*(-a*x+1)^(1/2)/(a*x)^(9/2)-34/63*a^4*(-a*x+1)^(1/2)/(a*x)^(7/2)-6 8/105*a^4*(-a*x+1)^(1/2)/(a*x)^(5/2)-272/315*a^4*(-a*x+1)^(1/2)/(a*x)^(3/2 )-544/315*a^4*(-a*x+1)^(1/2)/(a*x)^(1/2)
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.44 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \sqrt {-a x (-1+a x)} \left (35+85 a x+102 a^2 x^2+136 a^3 x^3+272 a^4 x^4\right )}{315 a x^5} \] Input:
Integrate[(1 + a*x)/(x^5*Sqrt[a*x]*Sqrt[1 - a*x]),x]
Output:
(-2*Sqrt[-(a*x*(-1 + a*x))]*(35 + 85*a*x + 102*a^2*x^2 + 136*a^3*x^3 + 272 *a^4*x^4))/(315*a*x^5)
Time = 0.32 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {8, 87, 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.
\(\displaystyle \int \frac {a x+1}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx\) |
\(\Big \downarrow \) 8 |
\(\displaystyle a^5 \int \frac {a x+1}{(a x)^{11/2} \sqrt {1-a x}}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle a^5 \left (\frac {17}{9} \int \frac {1}{(a x)^{9/2} \sqrt {1-a x}}dx-\frac {2 \sqrt {1-a x}}{9 a (a x)^{9/2}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle a^5 \left (\frac {17}{9} \left (\frac {6}{7} \int \frac {1}{(a x)^{7/2} \sqrt {1-a x}}dx-\frac {2 \sqrt {1-a x}}{7 a (a x)^{7/2}}\right )-\frac {2 \sqrt {1-a x}}{9 a (a x)^{9/2}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle a^5 \left (\frac {17}{9} \left (\frac {6}{7} \left (\frac {4}{5} \int \frac {1}{(a x)^{5/2} \sqrt {1-a x}}dx-\frac {2 \sqrt {1-a x}}{5 a (a x)^{5/2}}\right )-\frac {2 \sqrt {1-a x}}{7 a (a x)^{7/2}}\right )-\frac {2 \sqrt {1-a x}}{9 a (a x)^{9/2}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle a^5 \left (\frac {17}{9} \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{(a x)^{3/2} \sqrt {1-a x}}dx-\frac {2 \sqrt {1-a x}}{3 a (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a x}}{5 a (a x)^{5/2}}\right )-\frac {2 \sqrt {1-a x}}{7 a (a x)^{7/2}}\right )-\frac {2 \sqrt {1-a x}}{9 a (a x)^{9/2}}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle a^5 \left (\frac {17}{9} \left (\frac {6}{7} \left (\frac {4}{5} \left (-\frac {4 \sqrt {1-a x}}{3 a \sqrt {a x}}-\frac {2 \sqrt {1-a x}}{3 a (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a x}}{5 a (a x)^{5/2}}\right )-\frac {2 \sqrt {1-a x}}{7 a (a x)^{7/2}}\right )-\frac {2 \sqrt {1-a x}}{9 a (a x)^{9/2}}\right )\) |
Input:
Int[(1 + a*x)/(x^5*Sqrt[a*x]*Sqrt[1 - a*x]),x]
Output:
a^5*((-2*Sqrt[1 - a*x])/(9*a*(a*x)^(9/2)) + (17*((-2*Sqrt[1 - a*x])/(7*a*( a*x)^(7/2)) + (6*((-2*Sqrt[1 - a*x])/(5*a*(a*x)^(5/2)) + (4*((-2*Sqrt[1 - a*x])/(3*a*(a*x)^(3/2)) - (4*Sqrt[1 - a*x])/(3*a*Sqrt[a*x])))/5))/7))/9)
Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m Int[u*(a* x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
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 = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.40
method | result | size |
gosper | \(-\frac {2 \sqrt {-x a +1}\, \left (272 x^{4} a^{4}+136 a^{3} x^{3}+102 a^{2} x^{2}+85 x a +35\right )}{315 x^{4} \sqrt {x a}}\) | \(49\) |
default | \(-\frac {2 \sqrt {-x a +1}\, \operatorname {csgn}\left (a \right )^{2} \left (272 x^{4} a^{4}+136 a^{3} x^{3}+102 a^{2} x^{2}+85 x a +35\right )}{315 x^{4} \sqrt {x a}}\) | \(53\) |
orering | \(\frac {2 \left (x a -1\right ) \left (272 x^{4} a^{4}+136 a^{3} x^{3}+102 a^{2} x^{2}+85 x a +35\right )}{315 x^{4} \sqrt {x a}\, \sqrt {-x a +1}}\) | \(54\) |
risch | \(\frac {2 \sqrt {x a \left (-x a +1\right )}\, \left (272 a^{5} x^{5}-136 x^{4} a^{4}-34 a^{3} x^{3}-17 a^{2} x^{2}-50 x a -35\right )}{315 \sqrt {x a}\, \sqrt {-x a +1}\, x^{4} \sqrt {-x \left (x a -1\right ) a}}\) | \(79\) |
meijerg | \(-\frac {2 a \left (\frac {16}{5} a^{3} x^{3}+\frac {8}{5} a^{2} x^{2}+\frac {6}{5} x a +1\right ) \sqrt {-x a +1}}{7 \sqrt {x a}\, x^{3}}-\frac {2 \left (\frac {128}{35} x^{4} a^{4}+\frac {64}{35} a^{3} x^{3}+\frac {48}{35} a^{2} x^{2}+\frac {8}{7} x a +1\right ) \sqrt {-x a +1}}{9 \sqrt {x a}\, x^{4}}\) | \(91\) |
Input:
int((a*x+1)/x^5/(x*a)^(1/2)/(-a*x+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/315/x^4/(x*a)^(1/2)*(-a*x+1)^(1/2)*(272*a^4*x^4+136*a^3*x^3+102*a^2*x^2 +85*a*x+35)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.42 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \, {\left (272 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 85 \, a x + 35\right )} \sqrt {a x} \sqrt {-a x + 1}}{315 \, a x^{5}} \] Input:
integrate((a*x+1)/x^5/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="fricas")
Output:
-2/315*(272*a^4*x^4 + 136*a^3*x^3 + 102*a^2*x^2 + 85*a*x + 35)*sqrt(a*x)*s qrt(-a*x + 1)/(a*x^5)
Result contains complex when optimal does not.
Time = 8.69 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.97 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {32 a^{3} \sqrt {-1 + \frac {1}{a x}}}{35} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{35 x} - \frac {12 a \sqrt {-1 + \frac {1}{a x}}}{35 x^{2}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{7 x^{3}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {32 i a^{3} \sqrt {1 - \frac {1}{a x}}}{35} - \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{35 x} - \frac {12 i a \sqrt {1 - \frac {1}{a x}}}{35 x^{2}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{7 x^{3}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {256 a^{4} \sqrt {-1 + \frac {1}{a x}}}{315} - \frac {128 a^{3} \sqrt {-1 + \frac {1}{a x}}}{315 x} - \frac {32 a^{2} \sqrt {-1 + \frac {1}{a x}}}{105 x^{2}} - \frac {16 a \sqrt {-1 + \frac {1}{a x}}}{63 x^{3}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{9 x^{4}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {256 i a^{4} \sqrt {1 - \frac {1}{a x}}}{315} - \frac {128 i a^{3} \sqrt {1 - \frac {1}{a x}}}{315 x} - \frac {32 i a^{2} \sqrt {1 - \frac {1}{a x}}}{105 x^{2}} - \frac {16 i a \sqrt {1 - \frac {1}{a x}}}{63 x^{3}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{9 x^{4}} & \text {otherwise} \end {cases} \] Input:
integrate((a*x+1)/x**5/(a*x)**(1/2)/(-a*x+1)**(1/2),x)
Output:
a*Piecewise((-32*a**3*sqrt(-1 + 1/(a*x))/35 - 16*a**2*sqrt(-1 + 1/(a*x))/( 35*x) - 12*a*sqrt(-1 + 1/(a*x))/(35*x**2) - 2*sqrt(-1 + 1/(a*x))/(7*x**3), 1/Abs(a*x) > 1), (-32*I*a**3*sqrt(1 - 1/(a*x))/35 - 16*I*a**2*sqrt(1 - 1/ (a*x))/(35*x) - 12*I*a*sqrt(1 - 1/(a*x))/(35*x**2) - 2*I*sqrt(1 - 1/(a*x)) /(7*x**3), True)) + Piecewise((-256*a**4*sqrt(-1 + 1/(a*x))/315 - 128*a**3 *sqrt(-1 + 1/(a*x))/(315*x) - 32*a**2*sqrt(-1 + 1/(a*x))/(105*x**2) - 16*a *sqrt(-1 + 1/(a*x))/(63*x**3) - 2*sqrt(-1 + 1/(a*x))/(9*x**4), 1/Abs(a*x) > 1), (-256*I*a**4*sqrt(1 - 1/(a*x))/315 - 128*I*a**3*sqrt(1 - 1/(a*x))/(3 15*x) - 32*I*a**2*sqrt(1 - 1/(a*x))/(105*x**2) - 16*I*a*sqrt(1 - 1/(a*x))/ (63*x**3) - 2*I*sqrt(1 - 1/(a*x))/(9*x**4), True))
Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {544 \, \sqrt {-a^{2} x^{2} + a x} a^{3}}{315 \, x} - \frac {272 \, \sqrt {-a^{2} x^{2} + a x} a^{2}}{315 \, x^{2}} - \frac {68 \, \sqrt {-a^{2} x^{2} + a x} a}{105 \, x^{3}} - \frac {34 \, \sqrt {-a^{2} x^{2} + a x}}{63 \, x^{4}} - \frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{9 \, a x^{5}} \] Input:
integrate((a*x+1)/x^5/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="maxima")
Output:
-544/315*sqrt(-a^2*x^2 + a*x)*a^3/x - 272/315*sqrt(-a^2*x^2 + a*x)*a^2/x^2 - 68/105*sqrt(-a^2*x^2 + a*x)*a/x^3 - 34/63*sqrt(-a^2*x^2 + a*x)/x^4 - 2/ 9*sqrt(-a^2*x^2 + a*x)/(a*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (91) = 182\).
Time = 0.13 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.79 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\frac {35 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{9}}{\left (a x\right )^{\frac {9}{2}}} + \frac {585 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac {7}{2}}} + \frac {4032 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac {5}{2}}} + \frac {17640 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac {3}{2}}} + \frac {83790 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}}{\sqrt {a x}} - \frac {{\left (35 \, a^{5} + \frac {585 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{2}}{x} + \frac {4032 \, a^{3} {\left (\sqrt {-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac {17640 \, a^{2} {\left (\sqrt {-a x + 1} - 1\right )}^{6}}{x^{3}} + \frac {83790 \, a {\left (\sqrt {-a x + 1} - 1\right )}^{8}}{x^{4}}\right )} \left (a x\right )^{\frac {9}{2}}}{{\left (\sqrt {-a x + 1} - 1\right )}^{9}}}{80640 \, a} \] Input:
integrate((a*x+1)/x^5/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="giac")
Output:
-1/80640*(35*a^5*(sqrt(-a*x + 1) - 1)^9/(a*x)^(9/2) + 585*a^5*(sqrt(-a*x + 1) - 1)^7/(a*x)^(7/2) + 4032*a^5*(sqrt(-a*x + 1) - 1)^5/(a*x)^(5/2) + 176 40*a^5*(sqrt(-a*x + 1) - 1)^3/(a*x)^(3/2) + 83790*a^5*(sqrt(-a*x + 1) - 1) /sqrt(a*x) - (35*a^5 + 585*a^4*(sqrt(-a*x + 1) - 1)^2/x + 4032*a^3*(sqrt(- a*x + 1) - 1)^4/x^2 + 17640*a^2*(sqrt(-a*x + 1) - 1)^6/x^3 + 83790*a*(sqrt (-a*x + 1) - 1)^8/x^4)*(a*x)^(9/2)/(sqrt(-a*x + 1) - 1)^9)/a
Time = 22.73 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.40 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {1-a\,x}\,\left (\frac {544\,a^4\,x^4}{315}+\frac {272\,a^3\,x^3}{315}+\frac {68\,a^2\,x^2}{105}+\frac {34\,a\,x}{63}+\frac {2}{9}\right )}{x^4\,\sqrt {a\,x}} \] Input:
int((a*x + 1)/(x^5*(a*x)^(1/2)*(1 - a*x)^(1/2)),x)
Output:
-((1 - a*x)^(1/2)*((34*a*x)/63 + (68*a^2*x^2)/105 + (272*a^3*x^3)/315 + (5 44*a^4*x^4)/315 + 2/9))/(x^4*(a*x)^(1/2))
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.85 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=\frac {-\frac {544 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a^{4} x^{4}}{315}-\frac {272 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a^{3} x^{3}}{315}-\frac {68 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a^{2} x^{2}}{105}-\frac {34 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a x}{63}-\frac {2 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}}{9}+\frac {544 a^{5} i \,x^{5}}{315}}{a \,x^{5}} \] Input:
int((a*x+1)/x^5/(a*x)^(1/2)/(-a*x+1)^(1/2),x)
Output:
(2*( - 272*sqrt(x)*sqrt(a)*sqrt( - a*x + 1)*a**4*x**4 - 136*sqrt(x)*sqrt(a )*sqrt( - a*x + 1)*a**3*x**3 - 102*sqrt(x)*sqrt(a)*sqrt( - a*x + 1)*a**2*x **2 - 85*sqrt(x)*sqrt(a)*sqrt( - a*x + 1)*a*x - 35*sqrt(x)*sqrt(a)*sqrt( - a*x + 1) + 272*a**5*i*x**5))/(315*a*x**5)