\(\int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx\) [30]

Optimal result

Integrand size = 26, antiderivative size = 121 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 a^4 \sqrt {1-a x}}{9 (a x)^{9/2}}-\frac {34 a^4 \sqrt {1-a x}}{63 (a x)^{7/2}}-\frac {68 a^4 \sqrt {1-a x}}{105 (a x)^{5/2}}-\frac {272 a^4 \sqrt {1-a x}}{315 (a x)^{3/2}}-\frac {544 a^4 \sqrt {1-a x}}{315 \sqrt {a x}} \]

[Out]

-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)-68/105*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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {16, 79, 47, 37} \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {544 a^4 \sqrt {1-a x}}{315 \sqrt {a x}}-\frac {272 a^4 \sqrt {1-a x}}{315 (a x)^{3/2}}-\frac {68 a^4 \sqrt {1-a x}}{105 (a x)^{5/2}}-\frac {34 a^4 \sqrt {1-a x}}{63 (a x)^{7/2}}-\frac {2 a^4 \sqrt {1-a x}}{9 (a x)^{9/2}} \]

[In]

Int[(1 + a*x)/(x^5*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-2*a^4*Sqrt[1 - a*x])/(9*(a*x)^(9/2)) - (34*a^4*Sqrt[1 - a*x])/(63*(a*x)^(7/2)) - (68*a^4*Sqrt[1 - a*x])/(105
*(a*x)^(5/2)) - (272*a^4*Sqrt[1 - a*x])/(315*(a*x)^(3/2)) - (544*a^4*Sqrt[1 - a*x])/(315*Sqrt[a*x])

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 37

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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

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] - Dist[d*(Simplify[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] && NeQ[b*c - a*d, 0] && 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] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = a^5 \int \frac {1+a x}{(a x)^{11/2} \sqrt {1-a x}} \, dx \\ & = -\frac {2 a^4 \sqrt {1-a x}}{9 (a x)^{9/2}}+\frac {1}{9} \left (17 a^5\right ) \int \frac {1}{(a x)^{9/2} \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 {1}{21} \left (34 a^5\right ) \int \frac {1}{(a x)^{7/2} \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 {1}{105} \left (136 a^5\right ) \int \frac {1}{(a x)^{5/2} \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 {1}{315} \left (272 a^5\right ) \int \frac {1}{(a x)^{3/2} \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.44 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \sqrt {-a x (-1+a x)} \left (35+85 a x+102 a^2 x^2+136 a^3 x^3+272 a^4 x^4\right )}{315 a x^5} \]

[In]

Integrate[(1 + a*x)/(x^5*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-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)

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.40

method	result	size
gosper	\(-\frac {2 \sqrt {-a x +1}\, \left (272 a^{4} x^{4}+136 a^{3} x^{3}+102 a^{2} x^{2}+85 a x +35\right )}{315 x^{4} \sqrt {a x}}\)	\(49\)
default	\(-\frac {2 \sqrt {-a x +1}\, \operatorname {csgn}\left (a \right )^{2} \left (272 a^{4} x^{4}+136 a^{3} x^{3}+102 a^{2} x^{2}+85 a x +35\right )}{315 x^{4} \sqrt {a x}}\)	\(53\)
risch	\(\frac {2 \sqrt {a x \left (-a x +1\right )}\, \left (272 a^{5} x^{5}-136 a^{4} x^{4}-34 a^{3} x^{3}-17 a^{2} x^{2}-50 a x -35\right )}{315 \sqrt {a x}\, \sqrt {-a x +1}\, x^{4} \sqrt {-x \left (a x -1\right ) a}}\)	\(79\)
meijerg	\(-\frac {2 a \left (\frac {16}{5} a^{3} x^{3}+\frac {8}{5} a^{2} x^{2}+\frac {6}{5} a x +1\right ) \sqrt {-a x +1}}{7 \sqrt {a x}\, x^{3}}-\frac {2 \left (\frac {128}{35} a^{4} x^{4}+\frac {64}{35} a^{3} x^{3}+\frac {48}{35} a^{2} x^{2}+\frac {8}{7} a x +1\right ) \sqrt {-a x +1}}{9 \sqrt {a x}\, x^{4}}\)	\(91\)

[In]

int((a*x+1)/x^5/(a*x)^(1/2)/(-a*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/315/x^4/(a*x)^(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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.42 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \, {\left (272 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 85 \, a x + 35\right )} \sqrt {a x} \sqrt {-a x + 1}}{315 \, a x^{5}} \]

[In]

integrate((a*x+1)/x^5/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="fricas")

[Out]

-2/315*(272*a^4*x^4 + 136*a^3*x^3 + 102*a^2*x^2 + 85*a*x + 35)*sqrt(a*x)*sqrt(-a*x + 1)/(a*x^5)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.14 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.97 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {32 a^{3} \sqrt {-1 + \frac {1}{a x}}}{35} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{35 x} - \frac {12 a \sqrt {-1 + \frac {1}{a x}}}{35 x^{2}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{7 x^{3}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {32 i a^{3} \sqrt {1 - \frac {1}{a x}}}{35} - \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{35 x} - \frac {12 i a \sqrt {1 - \frac {1}{a x}}}{35 x^{2}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{7 x^{3}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {256 a^{4} \sqrt {-1 + \frac {1}{a x}}}{315} - \frac {128 a^{3} \sqrt {-1 + \frac {1}{a x}}}{315 x} - \frac {32 a^{2} \sqrt {-1 + \frac {1}{a x}}}{105 x^{2}} - \frac {16 a \sqrt {-1 + \frac {1}{a x}}}{63 x^{3}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{9 x^{4}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {256 i a^{4} \sqrt {1 - \frac {1}{a x}}}{315} - \frac {128 i a^{3} \sqrt {1 - \frac {1}{a x}}}{315 x} - \frac {32 i a^{2} \sqrt {1 - \frac {1}{a x}}}{105 x^{2}} - \frac {16 i a \sqrt {1 - \frac {1}{a x}}}{63 x^{3}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{9 x^{4}} & \text {otherwise} \end {cases} \]

[In]

integrate((a*x+1)/x**5/(a*x)**(1/2)/(-a*x+1)**(1/2),x)

[Out]

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((-25
6*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))/(315*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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {544 \, \sqrt {-a^{2} x^{2} + a x} a^{3}}{315 \, x} - \frac {272 \, \sqrt {-a^{2} x^{2} + a x} a^{2}}{315 \, x^{2}} - \frac {68 \, \sqrt {-a^{2} x^{2} + a x} a}{105 \, x^{3}} - \frac {34 \, \sqrt {-a^{2} x^{2} + a x}}{63 \, x^{4}} - \frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{9 \, a x^{5}} \]

[In]

integrate((a*x+1)/x^5/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="maxima")

[Out]

-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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (91) = 182\).

Time = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.79 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\frac {35 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{9}}{\left (a x\right )^{\frac {9}{2}}} + \frac {585 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac {7}{2}}} + \frac {4032 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac {5}{2}}} + \frac {17640 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac {3}{2}}} + \frac {83790 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}}{\sqrt {a x}} - \frac {{\left (35 \, a^{5} + \frac {585 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{2}}{x} + \frac {4032 \, a^{3} {\left (\sqrt {-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac {17640 \, a^{2} {\left (\sqrt {-a x + 1} - 1\right )}^{6}}{x^{3}} + \frac {83790 \, a {\left (\sqrt {-a x + 1} - 1\right )}^{8}}{x^{4}}\right )} \left (a x\right )^{\frac {9}{2}}}{{\left (\sqrt {-a x + 1} - 1\right )}^{9}}}{80640 \, a} \]

[In]

integrate((a*x+1)/x^5/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="giac")

[Out]

-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*(s
qrt(-a*x + 1) - 1)^5/(a*x)^(5/2) + 17640*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*(s
qrt(-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

Mupad [B] (verification not implemented)

Time = 3.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.40 \[ \int \frac {1+a x}{x^5 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {1-a\,x}\,\left (\frac {544\,a^4\,x^4}{315}+\frac {272\,a^3\,x^3}{315}+\frac {68\,a^2\,x^2}{105}+\frac {34\,a\,x}{63}+\frac {2}{9}\right )}{x^4\,\sqrt {a\,x}} \]

[In]

int((a*x + 1)/(x^5*(a*x)^(1/2)*(1 - a*x)^(1/2)),x)

[Out]

-((1 - a*x)^(1/2)*((34*a*x)/63 + (68*a^2*x^2)/105 + (272*a^3*x^3)/315 + (544*a^4*x^4)/315 + 2/9))/(x^4*(a*x)^(
1/2))