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

Optimal. Leaf size=69 \[ \frac{1}{4} (x-1)^{3/2} \sqrt{x+1} x^2+\frac{1}{24} (7-2 x) (x-1)^{3/2} \sqrt{x+1}-\frac{3}{8} \sqrt{x-1} \sqrt{x+1}+\frac{3}{8} \cosh ^{-1}(x) \]

[Out]

(-3*Sqrt[-1 + x]*Sqrt[1 + x])/8 + ((7 - 2*x)*(-1 + x)^(3/2)*Sqrt[1 + x])/24 + ((-1 + x)^(3/2)*x^2*Sqrt[1 + x])
/4 + (3*ArcCosh[x])/8

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Rubi [A]  time = 0.012763, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {100, 147, 50, 52} \[ \frac{1}{4} (x-1)^{3/2} \sqrt{x+1} x^2+\frac{1}{24} (7-2 x) (x-1)^{3/2} \sqrt{x+1}-\frac{3}{8} \sqrt{x-1} \sqrt{x+1}+\frac{3}{8} \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + x]*x^3)/Sqrt[1 + x],x]

[Out]

(-3*Sqrt[-1 + x]*Sqrt[1 + x])/8 + ((7 - 2*x)*(-1 + x)^(3/2)*Sqrt[1 + x])/24 + ((-1 + x)^(3/2)*x^2*Sqrt[1 + x])
/4 + (3*ArcCosh[x])/8

Rule 100

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

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 52

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

Rubi steps

\begin{align*} \int \frac{\sqrt{-1+x} x^3}{\sqrt{1+x}} \, dx &=\frac{1}{4} (-1+x)^{3/2} x^2 \sqrt{1+x}+\frac{1}{4} \int \frac{(2-x) \sqrt{-1+x} x}{\sqrt{1+x}} \, dx\\ &=\frac{1}{24} (7-2 x) (-1+x)^{3/2} \sqrt{1+x}+\frac{1}{4} (-1+x)^{3/2} x^2 \sqrt{1+x}-\frac{3}{8} \int \frac{\sqrt{-1+x}}{\sqrt{1+x}} \, dx\\ &=-\frac{3}{8} \sqrt{-1+x} \sqrt{1+x}+\frac{1}{24} (7-2 x) (-1+x)^{3/2} \sqrt{1+x}+\frac{1}{4} (-1+x)^{3/2} x^2 \sqrt{1+x}+\frac{3}{8} \int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx\\ &=-\frac{3}{8} \sqrt{-1+x} \sqrt{1+x}+\frac{1}{24} (7-2 x) (-1+x)^{3/2} \sqrt{1+x}+\frac{1}{4} (-1+x)^{3/2} x^2 \sqrt{1+x}+\frac{3}{8} \cosh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0741051, size = 76, normalized size = 1.1 \[ \frac{\sqrt{\frac{x-1}{x+1}} \left (6 x^5-8 x^4+3 x^3-8 x^2-18 \sqrt{1-x^2} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )-9 x+16\right )}{24 (x-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(-1 + x)/(1 + x)]*(16 - 9*x - 8*x^2 + 3*x^3 - 8*x^4 + 6*x^5 - 18*Sqrt[1 - x^2]*ArcSin[Sqrt[1 - x]/Sqrt[2
]]))/(24*(-1 + x))

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Maple [A]  time = 0.01, size = 76, normalized size = 1.1 \begin{align*}{\frac{1}{24}\sqrt{x-1}\sqrt{1+x} \left ( 6\,{x}^{3}\sqrt{{x}^{2}-1}-8\,{x}^{2}\sqrt{{x}^{2}-1}+9\,x\sqrt{{x}^{2}-1}+9\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ) -16\,\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(x-1)^(1/2)*(1+x)^(1/2)*(6*x^3*(x^2-1)^(1/2)-8*x^2*(x^2-1)^(1/2)+9*x*(x^2-1)^(1/2)+9*ln(x+(x^2-1)^(1/2))-
16*(x^2-1)^(1/2))/(x^2-1)^(1/2)

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Maxima [A]  time = 1.00543, size = 74, normalized size = 1.07 \begin{align*} \frac{1}{4} \,{\left (x^{2} - 1\right )}^{\frac{3}{2}} x - \frac{1}{3} \,{\left (x^{2} - 1\right )}^{\frac{3}{2}} + \frac{5}{8} \, \sqrt{x^{2} - 1} x - \sqrt{x^{2} - 1} + \frac{3}{8} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x^2 - 1)^(3/2)*x - 1/3*(x^2 - 1)^(3/2) + 5/8*sqrt(x^2 - 1)*x - sqrt(x^2 - 1) + 3/8*log(2*x + 2*sqrt(x^2 -
1))

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Fricas [A]  time = 1.71221, size = 130, normalized size = 1.88 \begin{align*} \frac{1}{24} \,{\left (6 \, x^{3} - 8 \, x^{2} + 9 \, x - 16\right )} \sqrt{x + 1} \sqrt{x - 1} - \frac{3}{8} \, \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*x^3 - 8*x^2 + 9*x - 16)*sqrt(x + 1)*sqrt(x - 1) - 3/8*log(sqrt(x + 1)*sqrt(x - 1) - x)

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Sympy [A]  time = 14.2676, size = 83, normalized size = 1.2 \begin{align*} \frac{\left (x - 1\right )^{\frac{7}{2}} \sqrt{x + 1}}{4} + \frac{5 \left (x - 1\right )^{\frac{5}{2}} \sqrt{x + 1}}{12} + \frac{11 \left (x - 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{24} - \frac{3 \sqrt{x - 1} \sqrt{x + 1}}{8} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{x - 1}}{2} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 1)**(7/2)*sqrt(x + 1)/4 + 5*(x - 1)**(5/2)*sqrt(x + 1)/12 + 11*(x - 1)**(3/2)*sqrt(x + 1)/24 - 3*sqrt(x -
1)*sqrt(x + 1)/8 + 3*asinh(sqrt(2)*sqrt(x - 1)/2)/4

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Giac [A]  time = 1.18659, size = 65, normalized size = 0.94 \begin{align*} \frac{1}{24} \,{\left ({\left (2 \,{\left (3 \, x - 10\right )}{\left (x + 1\right )} + 43\right )}{\left (x + 1\right )} - 39\right )} \sqrt{x + 1} \sqrt{x - 1} - \frac{3}{4} \, \log \left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(x - 1) - 3/4*log(abs(-sqrt(x + 1) + sqrt(x -
1)))