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

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

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Rubi [A]  time = 0.01, antiderivative size = 89, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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) \end {gather*}

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 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 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 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*}

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Mathematica [A]  time = 0.06, size = 61, normalized size = 0.69 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.13, size = 151, normalized size = 1.70 \begin {gather*} \frac {-\frac {105 (1-x)^{11/2}}{(x+1)^{11/2}}-\frac {595 (1-x)^{9/2}}{(x+1)^{9/2}}+\frac {1686 (1-x)^{7/2}}{(x+1)^{7/2}}+\frac {1386 (1-x)^{5/2}}{(x+1)^{5/2}}+\frac {595 (1-x)^{3/2}}{(x+1)^{3/2}}+\frac {105 \sqrt {1-x}}{\sqrt {x+1}}}{120 \left (\frac {1-x}{x+1}+1\right )^6}-\frac {7}{8} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-105*(1 - x)^(11/2))/(1 + x)^(11/2) - (595*(1 - x)^(9/2))/(1 + x)^(9/2) + (1686*(1 - x)^(7/2))/(1 + x)^(7/2)
+ (1386*(1 - x)^(5/2))/(1 + x)^(5/2) + (595*(1 - x)^(3/2))/(1 + x)^(3/2) + (105*Sqrt[1 - x])/Sqrt[1 + x])/(12
0*(1 + (1 - x)/(1 + x))^6) - (7*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]])/8

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fricas [A]  time = 1.15, size = 62, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 1.33, size = 185, normalized size = 2.08 \begin {gather*} -\frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{12} \, {\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 {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {7}{8} \, \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((1-x)^(7/2)*(1+x)^(3/2),x, algorithm="giac")

[Out]

-1/240*((2*((4*(5*x - 26)*(x + 1) + 321)*(x + 1) - 451)*(x + 1) + 745)*(x + 1) - 405)*sqrt(x + 1)*sqrt(-x + 1)
+ 1/120*((2*(3*(4*x - 17)*(x + 1) + 133)*(x + 1) - 295)*(x + 1) + 195)*sqrt(x + 1)*sqrt(-x + 1) + 1/12*((2*(3
*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) - 1/3*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-
x + 1) - 1/2*sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 7/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))

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maple [A]  time = 0.00, size = 113, normalized size = 1.27 \begin {gather*} \frac {7 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{16 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {7}{2}} \left (x +1\right )^{\frac {5}{2}}}{6}+\frac {7 \left (-x +1\right )^{\frac {5}{2}} \left (x +1\right )^{\frac {5}{2}}}{30}+\frac {7 \left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {5}{2}}}{24}+\frac {7 \sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{24}-\frac {7 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{48}-\frac {7 \sqrt {-x +1}\, \sqrt {x +1}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.90, size = 52, normalized size = 0.58 \begin {gather*} -\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 \relax (x) \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{7/2}\,{\left (x+1\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(7/2)*(x + 1)^(3/2),x)

[Out]

int((1 - x)^(7/2)*(x + 1)^(3/2), x)

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sympy [A]  time = 32.99, size = 289, normalized size = 3.25 \begin {gather*} \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 {gather*}

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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