3.1076 \(\int (1-x)^{7/2} (1+x)^{3/2} \, dx\)

Optimal. Leaf size=89 \[ \frac{1}{6} (x+1)^{5/2} (1-x)^{7/2}+\frac{7}{30} (x+1)^{5/2} (1-x)^{5/2}+\frac{7}{24} x (x+1)^{3/2} (1-x)^{3/2}+\frac{7}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{7}{16} \sin ^{-1}(x) \]

[Out]

(7*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (7*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + (7*(1 - x)^(5/2)*(1 + x)^(5/2))/30 +
((1 - x)^(7/2)*(1 + x)^(5/2))/6 + (7*ArcSin[x])/16

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Rubi [A]  time = 0.0133954, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ \frac{1}{6} (x+1)^{5/2} (1-x)^{7/2}+\frac{7}{30} (x+1)^{5/2} (1-x)^{5/2}+\frac{7}{24} x (x+1)^{3/2} (1-x)^{3/2}+\frac{7}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{7}{16} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(7/2)*(1 + x)^(3/2),x]

[Out]

(7*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (7*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + (7*(1 - x)^(5/2)*(1 + x)^(5/2))/30 +
((1 - x)^(7/2)*(1 + x)^(5/2))/6 + (7*ArcSin[x])/16

Rule 49

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[(2*c*n)/(m + n + 1), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x]
&& EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]

Rule 38

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

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 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 (1-x)^{7/2} (1+x)^{3/2} \, dx &=\frac{1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac{7}{6} \int (1-x)^{5/2} (1+x)^{3/2} \, dx\\ &=\frac{7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac{1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac{7}{6} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac{7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac{7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac{1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac{7}{8} \int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{7}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac{7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac{1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac{7}{16} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{7}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac{7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac{1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac{7}{16} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{7}{16} \sqrt{1-x} x \sqrt{1+x}+\frac{7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac{7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac{1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac{7}{16} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0542176, size = 61, normalized size = 0.69 \[ \frac{1}{240} \sqrt{1-x^2} \left (-40 x^5+96 x^4+10 x^3-192 x^2+135 x+96\right )-\frac{7}{8} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(7/2)*(1 + x)^(3/2),x]

[Out]

(Sqrt[1 - x^2]*(96 + 135*x - 192*x^2 + 10*x^3 + 96*x^4 - 40*x^5))/240 - (7*ArcSin[Sqrt[1 - x]/Sqrt[2]])/8

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Maple [A]  time = 0.005, size = 113, normalized size = 1.3 \begin{align*}{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{7}{30} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{7}{24} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{7}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{7}{48}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{7}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{7\,\arcsin \left ( x \right ) }{16}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(7/2)*(1+x)^(3/2),x)

[Out]

1/6*(1-x)^(7/2)*(1+x)^(5/2)+7/30*(1-x)^(5/2)*(1+x)^(5/2)+7/24*(1-x)^(3/2)*(1+x)^(5/2)+7/24*(1-x)^(1/2)*(1+x)^(
5/2)-7/48*(1-x)^(1/2)*(1+x)^(3/2)-7/16*(1-x)^(1/2)*(1+x)^(1/2)+7/16*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2
)*arcsin(x)

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Maxima [A]  time = 1.53828, size = 70, normalized size = 0.79 \begin{align*} -\frac{1}{6} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{2}{5} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} + \frac{7}{24} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{7}{16} \, \sqrt{-x^{2} + 1} x + \frac{7}{16} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)*(1+x)^(3/2),x, algorithm="maxima")

[Out]

-1/6*(-x^2 + 1)^(5/2)*x + 2/5*(-x^2 + 1)^(5/2) + 7/24*(-x^2 + 1)^(3/2)*x + 7/16*sqrt(-x^2 + 1)*x + 7/16*arcsin
(x)

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Fricas [A]  time = 1.55404, size = 176, normalized size = 1.98 \begin{align*} -\frac{1}{240} \,{\left (40 \, x^{5} - 96 \, x^{4} - 10 \, x^{3} + 192 \, x^{2} - 135 \, x - 96\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{7}{8} \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)*(1+x)^(3/2),x, algorithm="fricas")

[Out]

-1/240*(40*x^5 - 96*x^4 - 10*x^3 + 192*x^2 - 135*x - 96)*sqrt(x + 1)*sqrt(-x + 1) - 7/8*arctan((sqrt(x + 1)*sq
rt(-x + 1) - 1)/x)

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Sympy [A]  time = 137.498, size = 289, normalized size = 3.25 \begin{align*} \begin{cases} - \frac{7 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{8} - \frac{i \left (x + 1\right )^{\frac{13}{2}}}{6 \sqrt{x - 1}} + \frac{47 i \left (x + 1\right )^{\frac{11}{2}}}{30 \sqrt{x - 1}} - \frac{683 i \left (x + 1\right )^{\frac{9}{2}}}{120 \sqrt{x - 1}} + \frac{1151 i \left (x + 1\right )^{\frac{7}{2}}}{120 \sqrt{x - 1}} - \frac{1543 i \left (x + 1\right )^{\frac{5}{2}}}{240 \sqrt{x - 1}} - \frac{7 i \left (x + 1\right )^{\frac{3}{2}}}{48 \sqrt{x - 1}} + \frac{7 i \sqrt{x + 1}}{8 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{7 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{8} + \frac{\left (x + 1\right )^{\frac{13}{2}}}{6 \sqrt{1 - x}} - \frac{47 \left (x + 1\right )^{\frac{11}{2}}}{30 \sqrt{1 - x}} + \frac{683 \left (x + 1\right )^{\frac{9}{2}}}{120 \sqrt{1 - x}} - \frac{1151 \left (x + 1\right )^{\frac{7}{2}}}{120 \sqrt{1 - x}} + \frac{1543 \left (x + 1\right )^{\frac{5}{2}}}{240 \sqrt{1 - x}} + \frac{7 \left (x + 1\right )^{\frac{3}{2}}}{48 \sqrt{1 - x}} - \frac{7 \sqrt{x + 1}}{8 \sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(7/2)*(1+x)**(3/2),x)

[Out]

Piecewise((-7*I*acosh(sqrt(2)*sqrt(x + 1)/2)/8 - I*(x + 1)**(13/2)/(6*sqrt(x - 1)) + 47*I*(x + 1)**(11/2)/(30*
sqrt(x - 1)) - 683*I*(x + 1)**(9/2)/(120*sqrt(x - 1)) + 1151*I*(x + 1)**(7/2)/(120*sqrt(x - 1)) - 1543*I*(x +
1)**(5/2)/(240*sqrt(x - 1)) - 7*I*(x + 1)**(3/2)/(48*sqrt(x - 1)) + 7*I*sqrt(x + 1)/(8*sqrt(x - 1)), Abs(x + 1
)/2 > 1), (7*asin(sqrt(2)*sqrt(x + 1)/2)/8 + (x + 1)**(13/2)/(6*sqrt(1 - x)) - 47*(x + 1)**(11/2)/(30*sqrt(1 -
x)) + 683*(x + 1)**(9/2)/(120*sqrt(1 - x)) - 1151*(x + 1)**(7/2)/(120*sqrt(1 - x)) + 1543*(x + 1)**(5/2)/(240
*sqrt(1 - x)) + 7*(x + 1)**(3/2)/(48*sqrt(1 - x)) - 7*sqrt(x + 1)/(8*sqrt(1 - x)), True))

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Giac [A]  time = 1.0971, size = 161, normalized size = 1.81 \begin{align*} \frac{2}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} - \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{7}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)*(1+x)^(3/2),x, algorithm="giac")

[Out]

2/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x + 1) - 2/3*(x + 1)^(3/2)*(x - 1)*sqrt(-x +
1) - 1/48*((2*((4*(x + 1)*(x - 4) + 39)*(x + 1) - 37)*(x + 1) + 31)*(x + 1) - 3)*sqrt(x + 1)*sqrt(-x + 1) + 1/
2*sqrt(x + 1)*x*sqrt(-x + 1) + 7/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))