3.8.1 \(\int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx\)

Optimal. Leaf size=61 \[ -\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {1}{3} \sqrt {1-x} (x+1)^{3/2}-\sqrt {1-x} \sqrt {x+1}+\sin ^{-1}(x) \]

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Rubi [A]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {80, 50, 41, 216} \begin {gather*} -\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {1}{3} \sqrt {1-x} (x+1)^{3/2}-\sqrt {1-x} \sqrt {x+1}+\sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(1 + x)^(3/2))/Sqrt[1 - x],x]

[Out]

-(Sqrt[1 - x]*Sqrt[1 + x]) - (Sqrt[1 - x]*(1 + x)^(3/2))/3 - (Sqrt[1 - x]*(1 + x)^(5/2))/3 + ArcSin[x]

Rule 41

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

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 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]

Rubi steps

\begin {align*} \int \frac {x (1+x)^{3/2}}{\sqrt {1-x}} \, dx &=-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {2}{3} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}-\frac {1}{3} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 42, normalized size = 0.69 \begin {gather*} -\frac {1}{3} \sqrt {1-x^2} \left (x^2+3 x+5\right )-2 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*(1 + x)^(3/2))/Sqrt[1 - x],x]

[Out]

-1/3*(Sqrt[1 - x^2]*(5 + 3*x + x^2)) - 2*ArcSin[Sqrt[1 - x]/Sqrt[2]]

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IntegrateAlgebraic [A]  time = 0.08, size = 84, normalized size = 1.38 \begin {gather*} -\frac {2 \sqrt {1-x} \left (\frac {3 (1-x)^2}{(x+1)^2}+\frac {8 (1-x)}{x+1}+9\right )}{3 \sqrt {x+1} \left (\frac {1-x}{x+1}+1\right )^3}-2 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*(1 + x)^(3/2))/Sqrt[1 - x],x]

[Out]

(-2*Sqrt[1 - x]*(9 + (3*(1 - x)^2)/(1 + x)^2 + (8*(1 - x))/(1 + x)))/(3*Sqrt[1 + x]*(1 + (1 - x)/(1 + x))^3) -
2*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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fricas [A]  time = 1.38, size = 45, normalized size = 0.74 \begin {gather*} -\frac {1}{3} \, {\left (x^{2} + 3 \, x + 5\right )} \sqrt {x + 1} \sqrt {-x + 1} - 2 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(x^2 + 3*x + 5)*sqrt(x + 1)*sqrt(-x + 1) - 2*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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giac [A]  time = 1.13, size = 37, normalized size = 0.61 \begin {gather*} -\frac {1}{3} \, {\left ({\left (x + 2\right )} {\left (x + 1\right )} + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((x + 2)*(x + 1) + 3)*sqrt(x + 1)*sqrt(-x + 1) + 2*arcsin(1/2*sqrt(2)*sqrt(x + 1))

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maple [A]  time = 0.01, size = 66, normalized size = 1.08 \begin {gather*} \frac {\sqrt {x +1}\, \sqrt {-x +1}\, \left (-\sqrt {-x^{2}+1}\, x^{2}-3 \sqrt {-x^{2}+1}\, x +3 \arcsin \relax (x )-5 \sqrt {-x^{2}+1}\right )}{3 \sqrt {-x^{2}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x+1)^(1/2)*(-x+1)^(1/2)*(-(-x^2+1)^(1/2)*x^2-3*(-x^2+1)^(1/2)*x+3*arcsin(x)-5*(-x^2+1)^(1/2))/(-x^2+1)^(1
/2)

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maxima [A]  time = 1.97, size = 40, normalized size = 0.66 \begin {gather*} -\frac {1}{3} \, \sqrt {-x^{2} + 1} x^{2} - \sqrt {-x^{2} + 1} x - \frac {5}{3} \, \sqrt {-x^{2} + 1} + \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(-x^2 + 1)*x^2 - sqrt(-x^2 + 1)*x - 5/3*sqrt(-x^2 + 1) + arcsin(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,{\left (x+1\right )}^{3/2}}{\sqrt {1-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 65.98, size = 129, normalized size = 2.11 \begin {gather*} - 2 \left (\begin {cases} - \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \sqrt {1 - x} \sqrt {x + 1} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) + 2 \left (\begin {cases} - \frac {3 x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} - 2 \sqrt {1 - x} \sqrt {x + 1} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*Piecewise((-x*sqrt(1 - x)*sqrt(x + 1)/4 - sqrt(1 - x)*sqrt(x + 1) + 3*asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >=
-1) & (x < 1))) + 2*Piecewise((-3*x*sqrt(1 - x)*sqrt(x + 1)/4 + (1 - x)**(3/2)*(x + 1)**(3/2)/6 - 2*sqrt(1 - x
)*sqrt(x + 1) + 5*asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1)))

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