3.1.22 \(\int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx\)

Optimal. Leaf size=111 \[ -\frac {\sqrt {1-a x} (a x)^{7/2}}{4 a^4}-\frac {5 \sqrt {1-a x} (a x)^{5/2}}{8 a^4}-\frac {25 \sqrt {1-a x} (a x)^{3/2}}{32 a^4}-\frac {75 \sqrt {1-a x} \sqrt {a x}}{64 a^4}-\frac {75 \sin ^{-1}(1-2 a x)}{128 a^4} \]

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Rubi [A]  time = 0.04, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, 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)^{7/2}}{4 a^4}-\frac {5 \sqrt {1-a x} (a x)^{5/2}}{8 a^4}-\frac {25 \sqrt {1-a x} (a x)^{3/2}}{32 a^4}-\frac {75 \sqrt {1-a x} \sqrt {a x}}{64 a^4}-\frac {75 \sin ^{-1}(1-2 a x)}{128 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-75*Sqrt[a*x]*Sqrt[1 - a*x])/(64*a^4) - (25*(a*x)^(3/2)*Sqrt[1 - a*x])/(32*a^4) - (5*(a*x)^(5/2)*Sqrt[1 - a*x
])/(8*a^4) - ((a*x)^(7/2)*Sqrt[1 - a*x])/(4*a^4) - (75*ArcSin[1 - 2*a*x])/(128*a^4)

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^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx &=\frac {\int \frac {(a x)^{5/2} (1+a x)}{\sqrt {1-a x}} \, dx}{a^3}\\ &=-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {15 \int \frac {(a x)^{5/2}}{\sqrt {1-a x}} \, dx}{8 a^3}\\ &=-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {25 \int \frac {(a x)^{3/2}}{\sqrt {1-a x}} \, dx}{16 a^3}\\ &=-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {\sqrt {a x}}{\sqrt {1-a x}} \, dx}{64 a^3}\\ &=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {1}{\sqrt {a x} \sqrt {1-a x}} \, dx}{128 a^3}\\ &=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {1}{\sqrt {a x-a^2 x^2}} \, dx}{128 a^3}\\ &=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}-\frac {75 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{128 a^5}\\ &=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}-\frac {75 \sin ^{-1}(1-2 a x)}{128 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 89, normalized size = 0.80 \begin {gather*} \frac {\sqrt {a} x \left (16 a^4 x^4+24 a^3 x^3+10 a^2 x^2+25 a x-75\right )+75 \sqrt {x} \sqrt {1-a x} \sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{64 a^{7/2} \sqrt {-a x (a x-1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(Sqrt[a]*x*(-75 + 25*a*x + 10*a^2*x^2 + 24*a^3*x^3 + 16*a^4*x^4) + 75*Sqrt[x]*Sqrt[1 - a*x]*ArcSin[Sqrt[a]*Sqr
t[x]])/(64*a^(7/2)*Sqrt[-(a*x*(-1 + a*x))])

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IntegrateAlgebraic [A]  time = 0.10, size = 116, normalized size = 1.05 \begin {gather*} -\frac {75 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x}}\right )}{64 a^4}-\frac {\sqrt {1-a x} \left (\frac {75 (1-a x)^3}{a^3 x^3}+\frac {275 (1-a x)^2}{a^2 x^2}+\frac {365 (1-a x)}{a x}+181\right )}{64 a^4 \sqrt {a x} \left (\frac {1-a x}{a x}+1\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*(1 + a*x))/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

-1/64*(Sqrt[1 - a*x]*(181 + (365*(1 - a*x))/(a*x) + (275*(1 - a*x)^2)/(a^2*x^2) + (75*(1 - a*x)^3)/(a^3*x^3)))
/(a^4*Sqrt[a*x]*(1 + (1 - a*x)/(a*x))^4) - (75*ArcTan[Sqrt[1 - a*x]/Sqrt[a*x]])/(64*a^4)

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fricas [A]  time = 1.32, size = 65, normalized size = 0.59 \begin {gather*} -\frac {{\left (16 \, a^{3} x^{3} + 40 \, a^{2} x^{2} + 50 \, a x + 75\right )} \sqrt {a x} \sqrt {-a x + 1} + 75 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{64 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="fricas")

[Out]

-1/64*((16*a^3*x^3 + 40*a^2*x^2 + 50*a*x + 75)*sqrt(a*x)*sqrt(-a*x + 1) + 75*arctan(sqrt(a*x)*sqrt(-a*x + 1)/(
a*x)))/a^4

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giac [A]  time = 1.25, size = 63, normalized size = 0.57 \begin {gather*} -\frac {{\left (2 \, {\left (4 \, a x {\left (\frac {2 \, x}{a^{2}} + \frac {5}{a^{3}}\right )} + \frac {25}{a^{3}}\right )} a x + \frac {75}{a^{3}}\right )} \sqrt {a x} \sqrt {-a x + 1} - \frac {75 \, \arcsin \left (\sqrt {a x}\right )}{a^{3}}}{64 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="giac")

[Out]

-1/64*((2*(4*a*x*(2*x/a^2 + 5/a^3) + 25/a^3)*a*x + 75/a^3)*sqrt(a*x)*sqrt(-a*x + 1) - 75*arcsin(sqrt(a*x))/a^3
)/a

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maple [C]  time = 0.04, size = 132, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {-a x +1}\, \left (32 \sqrt {-\left (a x -1\right ) a x}\, a^{3} x^{3} \mathrm {csgn}\relax (a )+80 \sqrt {-\left (a x -1\right ) a x}\, a^{2} x^{2} \mathrm {csgn}\relax (a )+100 \sqrt {-\left (a x -1\right ) a x}\, a x \,\mathrm {csgn}\relax (a )-75 \arctan \left (\frac {\left (2 a x -1\right ) \mathrm {csgn}\relax (a )}{2 \sqrt {-\left (a x -1\right ) a x}}\right )+150 \sqrt {-\left (a x -1\right ) a x}\, \mathrm {csgn}\relax (a )\right ) x \,\mathrm {csgn}\relax (a )}{128 \sqrt {a x}\, \sqrt {-\left (a x -1\right ) a x}\, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x)

[Out]

-1/128*(-a*x+1)^(1/2)*x*(32*csgn(a)*x^3*a^3*(-x*(a*x-1)*a)^(1/2)+80*csgn(a)*x^2*a^2*(-x*(a*x-1)*a)^(1/2)+100*c
sgn(a)*(-x*(a*x-1)*a)^(1/2)*x*a+150*csgn(a)*(-x*(a*x-1)*a)^(1/2)-75*arctan(1/2*csgn(a)*(2*a*x-1)/(-x*(a*x-1)*a
)^(1/2)))*csgn(a)/a^3/(a*x)^(1/2)/(-x*(a*x-1)*a)^(1/2)

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maxima [A]  time = 0.97, size = 105, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + a x} x^{3}}{4 \, a} - \frac {5 \, \sqrt {-a^{2} x^{2} + a x} x^{2}}{8 \, a^{2}} - \frac {25 \, \sqrt {-a^{2} x^{2} + a x} x}{32 \, a^{3}} - \frac {75 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{128 \, a^{4}} - \frac {75 \, \sqrt {-a^{2} x^{2} + a x}}{64 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(-a^2*x^2 + a*x)*x^3/a - 5/8*sqrt(-a^2*x^2 + a*x)*x^2/a^2 - 25/32*sqrt(-a^2*x^2 + a*x)*x/a^3 - 75/128
*arcsin(-(2*a^2*x - a)/a)/a^4 - 75/64*sqrt(-a^2*x^2 + a*x)/a^4

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mupad [B]  time = 7.78, size = 345, normalized size = 3.11 \begin {gather*} \frac {75\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{32\,a^4}-\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^4\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^6}-\frac {\frac {35\,\sqrt {a\,x}}{32\,\left (\sqrt {1-a\,x}-1\right )}+\frac {805\,{\left (a\,x\right )}^{3/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {2681\,{\left (a\,x\right )}^{5/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^5}+\frac {5053\,{\left (a\,x\right )}^{7/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {5053\,{\left (a\,x\right )}^{9/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {2681\,{\left (a\,x\right )}^{11/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}-\frac {805\,{\left (a\,x\right )}^{13/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^{13}}-\frac {35\,{\left (a\,x\right )}^{15/2}}{32\,{\left (\sqrt {1-a\,x}-1\right )}^{15}}}{a^4\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a*x + 1))/((a*x)^(1/2)*(1 - a*x)^(1/2)),x)

[Out]

(75*atan((a*x)^(1/2)/((1 - a*x)^(1/2) - 1)))/(32*a^4) - ((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^4*((a*x)/((1 - a*x)^(1/2) - 1)^2 + 1)^6) - ((35*(a*x)^(1/2))/(32*((1 - a*x)^(1/2) - 1)) + (805*(a*x
)^(3/2))/(96*((1 - a*x)^(1/2) - 1)^3) + (2681*(a*x)^(5/2))/(96*((1 - a*x)^(1/2) - 1)^5) + (5053*(a*x)^(7/2))/(
96*((1 - a*x)^(1/2) - 1)^7) - (5053*(a*x)^(9/2))/(96*((1 - a*x)^(1/2) - 1)^9) - (2681*(a*x)^(11/2))/(96*((1 -
a*x)^(1/2) - 1)^11) - (805*(a*x)^(13/2))/(96*((1 - a*x)^(1/2) - 1)^13) - (35*(a*x)^(15/2))/(32*((1 - a*x)^(1/2
) - 1)^15))/(a^4*((a*x)/((1 - a*x)^(1/2) - 1)^2 + 1)^8)

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sympy [C]  time = 35.80, size = 484, normalized size = 4.36 \begin {gather*} a \left (\begin {cases} - \frac {35 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{64 a^{5}} - \frac {i x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {7}{2}}}{24 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {7 i x^{\frac {5}{2}}}{96 a^{\frac {5}{2}} \sqrt {a x - 1}} - \frac {35 i x^{\frac {3}{2}}}{192 a^{\frac {7}{2}} \sqrt {a x - 1}} + \frac {35 i \sqrt {x}}{64 a^{\frac {9}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {35 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{64 a^{5}} + \frac {x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {7}{2}}}{24 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {7 x^{\frac {5}{2}}}{96 a^{\frac {5}{2}} \sqrt {- a x + 1}} + \frac {35 x^{\frac {3}{2}}}{192 a^{\frac {7}{2}} \sqrt {- a x + 1}} - \frac {35 \sqrt {x}}{64 a^{\frac {9}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a*x+1)/(a*x)**(1/2)/(-a*x+1)**(1/2),x)

[Out]

a*Piecewise((-35*I*acosh(sqrt(a)*sqrt(x))/(64*a**5) - I*x**(9/2)/(4*sqrt(a)*sqrt(a*x - 1)) - I*x**(7/2)/(24*a*
*(3/2)*sqrt(a*x - 1)) - 7*I*x**(5/2)/(96*a**(5/2)*sqrt(a*x - 1)) - 35*I*x**(3/2)/(192*a**(7/2)*sqrt(a*x - 1))
+ 35*I*sqrt(x)/(64*a**(9/2)*sqrt(a*x - 1)), Abs(a*x) > 1), (35*asin(sqrt(a)*sqrt(x))/(64*a**5) + x**(9/2)/(4*s
qrt(a)*sqrt(-a*x + 1)) + x**(7/2)/(24*a**(3/2)*sqrt(-a*x + 1)) + 7*x**(5/2)/(96*a**(5/2)*sqrt(-a*x + 1)) + 35*
x**(3/2)/(192*a**(7/2)*sqrt(-a*x + 1)) - 35*sqrt(x)/(64*a**(9/2)*sqrt(-a*x + 1)), True)) + Piecewise((-5*I*aco
sh(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(sqr
t(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))

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