\(\int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx\) [29]

Optimal result

Integrand size = 26, antiderivative size = 97 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}}-\frac {26 a^3 \sqrt {1-a x}}{35 (a x)^{5/2}}-\frac {104 a^3 \sqrt {1-a x}}{105 (a x)^{3/2}}-\frac {208 a^3 \sqrt {1-a x}}{105 \sqrt {a x}} \]

[Out]

-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)-104/105*a^3*(-a*x+1)^(1/2)/(a*x)^(3/2
)-208/105*a^3*(-a*x+1)^(1/2)/(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {16, 79, 47, 37} \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {208 a^3 \sqrt {1-a x}}{105 \sqrt {a x}}-\frac {104 a^3 \sqrt {1-a x}}{105 (a x)^{3/2}}-\frac {26 a^3 \sqrt {1-a x}}{35 (a x)^{5/2}}-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}} \]

[In]

Int[(1 + a*x)/(x^4*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-2*a^3*Sqrt[1 - a*x])/(7*(a*x)^(7/2)) - (26*a^3*Sqrt[1 - a*x])/(35*(a*x)^(5/2)) - (104*a^3*Sqrt[1 - a*x])/(10
5*(a*x)^(3/2)) - (208*a^3*Sqrt[1 - a*x])/(105*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^4 \int \frac {1+a x}{(a x)^{9/2} \sqrt {1-a x}} \, dx \\ & = -\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}}+\frac {1}{7} \left (13 a^4\right ) \int \frac {1}{(a x)^{7/2} \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 {1}{35} \left (52 a^4\right ) \int \frac {1}{(a x)^{5/2} \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 {1}{105} \left (104 a^4\right ) \int \frac {1}{(a x)^{3/2} \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.46 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \sqrt {-a x (-1+a x)} \left (15+39 a x+52 a^2 x^2+104 a^3 x^3\right )}{105 a x^4} \]

[In]

Integrate[(1 + a*x)/(x^4*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-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)

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.42

method	result	size
gosper	\(-\frac {2 \sqrt {-a x +1}\, \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 a x +15\right )}{105 x^{3} \sqrt {a x}}\)	\(41\)
default	\(-\frac {2 \sqrt {-a x +1}\, \operatorname {csgn}\left (a \right )^{2} \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 a x +15\right )}{105 x^{3} \sqrt {a x}}\)	\(45\)
risch	\(\frac {2 \sqrt {a x \left (-a x +1\right )}\, \left (104 a^{4} x^{4}-52 a^{3} x^{3}-13 a^{2} x^{2}-24 a x -15\right )}{105 \sqrt {a x}\, \sqrt {-a x +1}\, x^{3} \sqrt {-x \left (a x -1\right ) a}}\)	\(71\)
meijerg	\(-\frac {2 a \left (\frac {8}{3} a^{2} x^{2}+\frac {4}{3} a x +1\right ) \sqrt {-a x +1}}{5 \sqrt {a x}\, x^{2}}-\frac {2 \left (\frac {16}{5} a^{3} x^{3}+\frac {8}{5} a^{2} x^{2}+\frac {6}{5} a x +1\right ) \sqrt {-a x +1}}{7 \sqrt {a x}\, x^{3}}\)	\(75\)

[In]

int((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/105/x^3/(a*x)^(1/2)*(-a*x+1)^(1/2)*(104*a^3*x^3+52*a^2*x^2+39*a*x+15)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.44 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {2 \, {\left (104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right )} \sqrt {a x} \sqrt {-a x + 1}}{105 \, a x^{4}} \]

[In]

integrate((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="fricas")

[Out]

-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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.00 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.82 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{15} - \frac {8 a \sqrt {-1 + \frac {1}{a x}}}{15 x} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{5 x^{2}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{15} - \frac {8 i a \sqrt {1 - \frac {1}{a x}}}{15 x} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{5 x^{2}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {32 a^{3} \sqrt {-1 + \frac {1}{a x}}}{35} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{35 x} - \frac {12 a \sqrt {-1 + \frac {1}{a x}}}{35 x^{2}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{7 x^{3}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {32 i a^{3} \sqrt {1 - \frac {1}{a x}}}{35} - \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{35 x} - \frac {12 i a \sqrt {1 - \frac {1}{a x}}}{35 x^{2}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{7 x^{3}} & \text {otherwise} \end {cases} \]

[In]

integrate((a*x+1)/x**4/(a*x)**(1/2)/(-a*x+1)**(1/2),x)

[Out]

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))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {208 \, \sqrt {-a^{2} x^{2} + a x} a^{2}}{105 \, x} - \frac {104 \, \sqrt {-a^{2} x^{2} + a x} a}{105 \, x^{2}} - \frac {26 \, \sqrt {-a^{2} x^{2} + a x}}{35 \, x^{3}} - \frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{7 \, a x^{4}} \]

[In]

integrate((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="maxima")

[Out]

-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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (73) = 146\).

Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.80 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\frac {15 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac {7}{2}}} + \frac {231 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac {5}{2}}} + \frac {1435 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac {3}{2}}} + \frac {7875 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}}{\sqrt {a x}} - \frac {{\left (15 \, a^{4} + \frac {231 \, a^{3} {\left (\sqrt {-a x + 1} - 1\right )}^{2}}{x} + \frac {1435 \, a^{2} {\left (\sqrt {-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac {7875 \, a {\left (\sqrt {-a x + 1} - 1\right )}^{6}}{x^{3}}\right )} \left (a x\right )^{\frac {7}{2}}}{{\left (\sqrt {-a x + 1} - 1\right )}^{7}}}{6720 \, a} \]

[In]

integrate((a*x+1)/x^4/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="giac")

[Out]

-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*(sq
rt(-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

Mupad [B] (verification not implemented)

Time = 3.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.41 \[ \int \frac {1+a x}{x^4 \sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {1-a\,x}\,\left (\frac {208\,a^3\,x^3}{105}+\frac {104\,a^2\,x^2}{105}+\frac {26\,a\,x}{35}+\frac {2}{7}\right )}{x^3\,\sqrt {a\,x}} \]

[In]

int((a*x + 1)/(x^4*(a*x)^(1/2)*(1 - a*x)^(1/2)),x)

[Out]

-((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))