3.1.23 \(\int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx\)

Optimal. Leaf size=87 \[ -\frac {\sqrt {1-a x} (a x)^{5/2}}{3 a^3}-\frac {11 \sqrt {1-a x} (a x)^{3/2}}{12 a^3}-\frac {11 \sqrt {1-a x} \sqrt {a x}}{8 a^3}-\frac {11 \sin ^{-1}(1-2 a x)}{16 a^3} \]

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Rubi [A]  time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {16, 80, 50, 53, 619, 216} \begin {gather*} -\frac {\sqrt {1-a x} (a x)^{5/2}}{3 a^3}-\frac {11 \sqrt {1-a x} (a x)^{3/2}}{12 a^3}-\frac {11 \sqrt {1-a x} \sqrt {a x}}{8 a^3}-\frac {11 \sin ^{-1}(1-2 a x)}{16 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-11*Sqrt[a*x]*Sqrt[1 - a*x])/(8*a^3) - (11*(a*x)^(3/2)*Sqrt[1 - a*x])/(12*a^3) - ((a*x)^(5/2)*Sqrt[1 - a*x])/
(3*a^3) - (11*ArcSin[1 - 2*a*x])/(16*a^3)

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx &=\frac {\int \frac {(a x)^{3/2} (1+a x)}{\sqrt {1-a x}} \, dx}{a^2}\\ &=-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a^3}+\frac {11 \int \frac {(a x)^{3/2}}{\sqrt {1-a x}} \, dx}{6 a^2}\\ &=-\frac {11 (a x)^{3/2} \sqrt {1-a x}}{12 a^3}-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a^3}+\frac {11 \int \frac {\sqrt {a x}}{\sqrt {1-a x}} \, dx}{8 a^2}\\ &=-\frac {11 \sqrt {a x} \sqrt {1-a x}}{8 a^3}-\frac {11 (a x)^{3/2} \sqrt {1-a x}}{12 a^3}-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a^3}+\frac {11 \int \frac {1}{\sqrt {a x} \sqrt {1-a x}} \, dx}{16 a^2}\\ &=-\frac {11 \sqrt {a x} \sqrt {1-a x}}{8 a^3}-\frac {11 (a x)^{3/2} \sqrt {1-a x}}{12 a^3}-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a^3}+\frac {11 \int \frac {1}{\sqrt {a x-a^2 x^2}} \, dx}{16 a^2}\\ &=-\frac {11 \sqrt {a x} \sqrt {1-a x}}{8 a^3}-\frac {11 (a x)^{3/2} \sqrt {1-a x}}{12 a^3}-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a^3}-\frac {11 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{16 a^4}\\ &=-\frac {11 \sqrt {a x} \sqrt {1-a x}}{8 a^3}-\frac {11 (a x)^{3/2} \sqrt {1-a x}}{12 a^3}-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a^3}-\frac {11 \sin ^{-1}(1-2 a x)}{16 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 81, normalized size = 0.93 \begin {gather*} \frac {\sqrt {a} x \left (8 a^3 x^3+14 a^2 x^2+11 a x-33\right )+33 \sqrt {x} \sqrt {1-a x} \sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{24 a^{5/2} \sqrt {-a x (a x-1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(Sqrt[a]*x*(-33 + 11*a*x + 14*a^2*x^2 + 8*a^3*x^3) + 33*Sqrt[x]*Sqrt[1 - a*x]*ArcSin[Sqrt[a]*Sqrt[x]])/(24*a^(
5/2)*Sqrt[-(a*x*(-1 + a*x))])

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IntegrateAlgebraic [A]  time = 0.09, size = 100, normalized size = 1.15 \begin {gather*} -\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x}}\right )}{8 a^3}-\frac {\sqrt {1-a x} \left (\frac {33 (1-a x)^2}{a^2 x^2}+\frac {88 (1-a x)}{a x}+63\right )}{24 a^3 \sqrt {a x} \left (\frac {1-a x}{a x}+1\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

-1/24*(Sqrt[1 - a*x]*(63 + (88*(1 - a*x))/(a*x) + (33*(1 - a*x)^2)/(a^2*x^2)))/(a^3*Sqrt[a*x]*(1 + (1 - a*x)/(
a*x))^3) - (11*ArcTan[Sqrt[1 - a*x]/Sqrt[a*x]])/(8*a^3)

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fricas [A]  time = 1.29, size = 57, normalized size = 0.66 \begin {gather*} -\frac {{\left (8 \, a^{2} x^{2} + 22 \, a x + 33\right )} \sqrt {a x} \sqrt {-a x + 1} + 33 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{24 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((8*a^2*x^2 + 22*a*x + 33)*sqrt(a*x)*sqrt(-a*x + 1) + 33*arctan(sqrt(a*x)*sqrt(-a*x + 1)/(a*x)))/a^3

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giac [A]  time = 1.34, size = 53, normalized size = 0.61 \begin {gather*} -\frac {{\left (2 \, a x {\left (\frac {4 \, x}{a} + \frac {11}{a^{2}}\right )} + \frac {33}{a^{2}}\right )} \sqrt {a x} \sqrt {-a x + 1} - \frac {33 \, \arcsin \left (\sqrt {a x}\right )}{a^{2}}}{24 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((2*a*x*(4*x/a + 11/a^2) + 33/a^2)*sqrt(a*x)*sqrt(-a*x + 1) - 33*arcsin(sqrt(a*x))/a^2)/a

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maple [C]  time = 0.02, size = 111, normalized size = 1.28 \begin {gather*} -\frac {\sqrt {-a x +1}\, \left (16 \sqrt {-\left (a x -1\right ) a x}\, a^{2} x^{2} \mathrm {csgn}\relax (a )+44 \sqrt {-\left (a x -1\right ) a x}\, a x \,\mathrm {csgn}\relax (a )-33 \arctan \left (\frac {\left (2 a x -1\right ) \mathrm {csgn}\relax (a )}{2 \sqrt {-\left (a x -1\right ) a x}}\right )+66 \sqrt {-\left (a x -1\right ) a x}\, \mathrm {csgn}\relax (a )\right ) x \,\mathrm {csgn}\relax (a )}{48 \sqrt {a x}\, \sqrt {-\left (a x -1\right ) a x}\, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(-a*x+1)^(1/2)*x*(16*(-(a*x-1)*a*x)^(1/2)*a^2*x^2*csgn(a)+44*(-(a*x-1)*a*x)^(1/2)*a*x*csgn(a)+66*(-(a*x-
1)*a*x)^(1/2)*csgn(a)-33*arctan(1/2*(2*a*x-1)/(-(a*x-1)*a*x)^(1/2)*csgn(a)))*csgn(a)/a^2/(a*x)^(1/2)/(-(a*x-1)
*a*x)^(1/2)

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maxima [A]  time = 0.99, size = 83, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + a x} x^{2}}{3 \, a} - \frac {11 \, \sqrt {-a^{2} x^{2} + a x} x}{12 \, a^{2}} - \frac {11 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{16 \, a^{3}} - \frac {11 \, \sqrt {-a^{2} x^{2} + a x}}{8 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(-a^2*x^2 + a*x)*x^2/a - 11/12*sqrt(-a^2*x^2 + a*x)*x/a^2 - 11/16*arcsin(-(2*a^2*x - a)/a)/a^3 - 11/8
*sqrt(-a^2*x^2 + a*x)/a^3

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mupad [B]  time = 5.92, size = 269, normalized size = 3.09 \begin {gather*} \frac {11\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{4\,a^3}-\frac {\frac {5\,\sqrt {a\,x}}{4\,\left (\sqrt {1-a\,x}-1\right )}+\frac {85\,{\left (a\,x\right )}^{3/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {33\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {33\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {85\,{\left (a\,x\right )}^{9/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {5\,{\left (a\,x\right )}^{11/2}}{4\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}}{a^3\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^6}-\frac {\frac {3\,\sqrt {a\,x}}{2\,\left (\sqrt {1-a\,x}-1\right )}+\frac {11\,{\left (a\,x\right )}^{3/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^3}-\frac {11\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {3\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}}{a^3\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11*atan((a*x)^(1/2)/((1 - a*x)^(1/2) - 1)))/(4*a^3) - ((5*(a*x)^(1/2))/(4*((1 - a*x)^(1/2) - 1)) + (85*(a*x)^
(3/2))/(12*((1 - a*x)^(1/2) - 1)^3) + (33*(a*x)^(5/2))/(2*((1 - a*x)^(1/2) - 1)^5) - (33*(a*x)^(7/2))/(2*((1 -
a*x)^(1/2) - 1)^7) - (85*(a*x)^(9/2))/(12*((1 - a*x)^(1/2) - 1)^9) - (5*(a*x)^(11/2))/(4*((1 - a*x)^(1/2) - 1
)^11))/(a^3*((a*x)/((1 - a*x)^(1/2) - 1)^2 + 1)^6) - ((3*(a*x)^(1/2))/(2*((1 - a*x)^(1/2) - 1)) + (11*(a*x)^(3
/2))/(2*((1 - a*x)^(1/2) - 1)^3) - (11*(a*x)^(5/2))/(2*((1 - a*x)^(1/2) - 1)^5) - (3*(a*x)^(7/2))/(2*((1 - a*x
)^(1/2) - 1)^7))/(a^3*((a*x)/((1 - a*x)^(1/2) - 1)^2 + 1)^4)

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sympy [C]  time = 25.60, size = 393, normalized size = 4.52 \begin {gather*} a \left (\begin {cases} - \frac {5 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {5 i x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {a x - 1}} + \frac {5 i \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {5 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {5 x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {- a x + 1}} - \frac {5 \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {a x - 1}} + \frac {3 i \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {3 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {- a x + 1}} - \frac {3 \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*Piecewise((-5*I*acosh(sqrt(a)*sqrt(x))/(8*a**4) - I*x**(7/2)/(3*sqrt(a)*sqrt(a*x - 1)) - I*x**(5/2)/(12*a**(
3/2)*sqrt(a*x - 1)) - 5*I*x**(3/2)/(24*a**(5/2)*sqrt(a*x - 1)) + 5*I*sqrt(x)/(8*a**(7/2)*sqrt(a*x - 1)), Abs(a
*x) > 1), (5*asin(sqrt(a)*sqrt(x))/(8*a**4) + x**(7/2)/(3*sqrt(a)*sqrt(-a*x + 1)) + x**(5/2)/(12*a**(3/2)*sqrt
(-a*x + 1)) + 5*x**(3/2)/(24*a**(5/2)*sqrt(-a*x + 1)) - 5*sqrt(x)/(8*a**(7/2)*sqrt(-a*x + 1)), True)) + Piecew
ise((-3*I*acosh(sqrt(a)*sqrt(x))/(4*a**3) - I*x**(5/2)/(2*sqrt(a)*sqrt(a*x - 1)) - I*x**(3/2)/(4*a**(3/2)*sqrt
(a*x - 1)) + 3*I*sqrt(x)/(4*a**(5/2)*sqrt(a*x - 1)), Abs(a*x) > 1), (3*asin(sqrt(a)*sqrt(x))/(4*a**3) + x**(5/
2)/(2*sqrt(a)*sqrt(-a*x + 1)) + x**(3/2)/(4*a**(3/2)*sqrt(-a*x + 1)) - 3*sqrt(x)/(4*a**(5/2)*sqrt(-a*x + 1)),
True))

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