Integrand size = 26, antiderivative size = 97 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}}-\frac {26 a^3 \sqrt {1-a x}}{35 (a x)^{5/2}}-\frac {104 a^3 \sqrt {1-a x}}{105 (a x)^{3/2}}-\frac {208 a^3 \sqrt {1-a x}}{105 \sqrt {a x}} \] Output:
-2/7*a^3*(-a*x+1)^(1/2)/(a*x)^(7/2)-26/35*a^3*(-a*x+1)^(1/2)/(a*x)^(5/2)-1 04/105*a^3*(-a*x+1)^(1/2)/(a*x)^(3/2)-208/105*a^3*(-a*x+1)^(1/2)/(a*x)^(1/ 2)
Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.46 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \sqrt {-a x (-1+a x)} \left (15+39 a x+52 a^2 x^2+104 a^3 x^3\right )}{105 a x^4} \] Input:
Integrate[(1 + a*x)/(x^4*Sqrt[a*x]*Sqrt[1 - a*x]),x]
Output:
(-2*Sqrt[-(a*x*(-1 + a*x))]*(15 + 39*a*x + 52*a^2*x^2 + 104*a^3*x^3))/(105 *a*x^4)
Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {8, 87, 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.
\(\displaystyle \int \frac {a x+1}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx\) |
\(\Big \downarrow \) 8 |
\(\displaystyle a^4 \int \frac {a x+1}{(a x)^{9/2} \sqrt {1-a x}}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle a^4 \left (\frac {13}{7} \int \frac {1}{(a x)^{7/2} \sqrt {1-a x}}dx-\frac {2 \sqrt {1-a x}}{7 a (a x)^{7/2}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle a^4 \left (\frac {13}{7} \left (\frac {4}{5} \int \frac {1}{(a x)^{5/2} \sqrt {1-a x}}dx-\frac {2 \sqrt {1-a x}}{5 a (a x)^{5/2}}\right )-\frac {2 \sqrt {1-a x}}{7 a (a x)^{7/2}}\right )\) |
\(\Big \downarrow \) 55 |
\(\displaystyle a^4 \left (\frac {13}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{(a x)^{3/2} \sqrt {1-a x}}dx-\frac {2 \sqrt {1-a x}}{3 a (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a x}}{5 a (a x)^{5/2}}\right )-\frac {2 \sqrt {1-a x}}{7 a (a x)^{7/2}}\right )\) |
\(\Big \downarrow \) 48 |
\(\displaystyle a^4 \left (\frac {13}{7} \left (\frac {4}{5} \left (-\frac {4 \sqrt {1-a x}}{3 a \sqrt {a x}}-\frac {2 \sqrt {1-a x}}{3 a (a x)^{3/2}}\right )-\frac {2 \sqrt {1-a x}}{5 a (a x)^{5/2}}\right )-\frac {2 \sqrt {1-a x}}{7 a (a x)^{7/2}}\right )\) |
Input:
Int[(1 + a*x)/(x^4*Sqrt[a*x]*Sqrt[1 - a*x]),x]
Output:
a^4*((-2*Sqrt[1 - a*x])/(7*a*(a*x)^(7/2)) + (13*((-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)
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 = 41, normalized size of antiderivative = 0.42
method | result | size |
gosper | \(-\frac {2 \sqrt {-x a +1}\, \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 x a +15\right )}{105 x^{3} \sqrt {x a}}\) | \(41\) |
default | \(-\frac {2 \sqrt {-x a +1}\, \operatorname {csgn}\left (a \right )^{2} \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 x a +15\right )}{105 x^{3} \sqrt {x a}}\) | \(45\) |
orering | \(\frac {2 \left (x a -1\right ) \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 x a +15\right )}{105 x^{3} \sqrt {x a}\, \sqrt {-x a +1}}\) | \(46\) |
risch | \(\frac {2 \sqrt {x a \left (-x a +1\right )}\, \left (104 x^{4} a^{4}-52 a^{3} x^{3}-13 a^{2} x^{2}-24 x a -15\right )}{105 \sqrt {x a}\, \sqrt {-x a +1}\, x^{3} \sqrt {-x \left (x a -1\right ) a}}\) | \(71\) |
meijerg | \(-\frac {2 a \left (\frac {8}{3} a^{2} x^{2}+\frac {4}{3} x a +1\right ) \sqrt {-x a +1}}{5 \sqrt {x a}\, x^{2}}-\frac {2 \left (\frac {16}{5} a^{3} x^{3}+\frac {8}{5} a^{2} x^{2}+\frac {6}{5} x a +1\right ) \sqrt {-x a +1}}{7 \sqrt {x a}\, x^{3}}\) | \(75\) |
Input:
int((a*x+1)/x^4/(x*a)^(1/2)/(-a*x+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/105/x^3/(x*a)^(1/2)*(-a*x+1)^(1/2)*(104*a^3*x^3+52*a^2*x^2+39*a*x+15)
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.44 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \, {\left (104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right )} \sqrt {a x} \sqrt {-a x + 1}}{105 \, a x^{4}} \] Input:
integrate((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="fricas")
Output:
-2/105*(104*a^3*x^3 + 52*a^2*x^2 + 39*a*x + 15)*sqrt(a*x)*sqrt(-a*x + 1)/( a*x^4)
Result contains complex when optimal does not.
Time = 5.22 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.82 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{15} - \frac {8 a \sqrt {-1 + \frac {1}{a x}}}{15 x} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{5 x^{2}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{15} - \frac {8 i a \sqrt {1 - \frac {1}{a x}}}{15 x} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{5 x^{2}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {32 a^{3} \sqrt {-1 + \frac {1}{a x}}}{35} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{35 x} - \frac {12 a \sqrt {-1 + \frac {1}{a x}}}{35 x^{2}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{7 x^{3}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {32 i a^{3} \sqrt {1 - \frac {1}{a x}}}{35} - \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{35 x} - \frac {12 i a \sqrt {1 - \frac {1}{a x}}}{35 x^{2}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{7 x^{3}} & \text {otherwise} \end {cases} \] Input:
integrate((a*x+1)/x**4/(a*x)**(1/2)/(-a*x+1)**(1/2),x)
Output:
a*Piecewise((-16*a**2*sqrt(-1 + 1/(a*x))/15 - 8*a*sqrt(-1 + 1/(a*x))/(15*x ) - 2*sqrt(-1 + 1/(a*x))/(5*x**2), 1/Abs(a*x) > 1), (-16*I*a**2*sqrt(1 - 1 /(a*x))/15 - 8*I*a*sqrt(1 - 1/(a*x))/(15*x) - 2*I*sqrt(1 - 1/(a*x))/(5*x** 2), True)) + 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))
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {208 \, \sqrt {-a^{2} x^{2} + a x} a^{2}}{105 \, x} - \frac {104 \, \sqrt {-a^{2} x^{2} + a x} a}{105 \, x^{2}} - \frac {26 \, \sqrt {-a^{2} x^{2} + a x}}{35 \, x^{3}} - \frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{7 \, a x^{4}} \] Input:
integrate((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="maxima")
Output:
-208/105*sqrt(-a^2*x^2 + a*x)*a^2/x - 104/105*sqrt(-a^2*x^2 + a*x)*a/x^2 - 26/35*sqrt(-a^2*x^2 + a*x)/x^3 - 2/7*sqrt(-a^2*x^2 + a*x)/(a*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (73) = 146\).
Time = 0.12 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.80 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\frac {15 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac {7}{2}}} + \frac {231 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac {5}{2}}} + \frac {1435 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac {3}{2}}} + \frac {7875 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}}{\sqrt {a x}} - \frac {{\left (15 \, a^{4} + \frac {231 \, a^{3} {\left (\sqrt {-a x + 1} - 1\right )}^{2}}{x} + \frac {1435 \, a^{2} {\left (\sqrt {-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac {7875 \, a {\left (\sqrt {-a x + 1} - 1\right )}^{6}}{x^{3}}\right )} \left (a x\right )^{\frac {7}{2}}}{{\left (\sqrt {-a x + 1} - 1\right )}^{7}}}{6720 \, a} \] Input:
integrate((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="giac")
Output:
-1/6720*(15*a^4*(sqrt(-a*x + 1) - 1)^7/(a*x)^(7/2) + 231*a^4*(sqrt(-a*x + 1) - 1)^5/(a*x)^(5/2) + 1435*a^4*(sqrt(-a*x + 1) - 1)^3/(a*x)^(3/2) + 7875 *a^4*(sqrt(-a*x + 1) - 1)/sqrt(a*x) - (15*a^4 + 231*a^3*(sqrt(-a*x + 1) - 1)^2/x + 1435*a^2*(sqrt(-a*x + 1) - 1)^4/x^2 + 7875*a*(sqrt(-a*x + 1) - 1) ^6/x^3)*(a*x)^(7/2)/(sqrt(-a*x + 1) - 1)^7)/a
Time = 22.64 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.41 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {1-a\,x}\,\left (\frac {208\,a^3\,x^3}{105}+\frac {104\,a^2\,x^2}{105}+\frac {26\,a\,x}{35}+\frac {2}{7}\right )}{x^3\,\sqrt {a\,x}} \] Input:
int((a*x + 1)/(x^4*(a*x)^(1/2)*(1 - a*x)^(1/2)),x)
Output:
-((1 - a*x)^(1/2)*((26*a*x)/35 + (104*a^2*x^2)/105 + (208*a^3*x^3)/105 + 2 /7))/(x^3*(a*x)^(1/2))
Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=\frac {-\frac {208 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a^{3} x^{3}}{105}-\frac {104 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a^{2} x^{2}}{105}-\frac {26 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a x}{35}-\frac {2 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}}{7}+\frac {208 a^{4} i \,x^{4}}{105}}{a \,x^{4}} \] Input:
int((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x)
Output:
(2*( - 104*sqrt(x)*sqrt(a)*sqrt( - a*x + 1)*a**3*x**3 - 52*sqrt(x)*sqrt(a) *sqrt( - a*x + 1)*a**2*x**2 - 39*sqrt(x)*sqrt(a)*sqrt( - a*x + 1)*a*x - 15 *sqrt(x)*sqrt(a)*sqrt( - a*x + 1) + 104*a**4*i*x**4))/(105*a*x**4)