3.11.21 \(\int (1-x)^{7/2} (1+x)^{5/2} \, dx\)

Optimal. Leaf size=90 \[ \frac {1}{7} (1-x)^{7/2} (x+1)^{7/2}+\frac {1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac {5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac {5}{16} \sqrt {1-x} x \sqrt {x+1}+\frac {5}{16} \sin ^{-1}(x) \]

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Rubi [A]  time = 0.01, antiderivative size = 90, 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}{7} (1-x)^{7/2} (x+1)^{7/2}+\frac {1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac {5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac {5}{16} \sqrt {1-x} x \sqrt {x+1}+\frac {5}{16} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(7/2)*(1 + x)^(5/2),x]

[Out]

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

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Mathematica [A]  time = 0.06, size = 66, normalized size = 0.73 \begin {gather*} \frac {1}{336} \sqrt {1-x^2} \left (-48 x^6+56 x^5+144 x^4-182 x^3-144 x^2+231 x+48\right )-\frac {5}{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)^(5/2),x]

[Out]

(Sqrt[1 - x^2]*(48 + 231*x - 144*x^2 - 182*x^3 + 144*x^4 + 56*x^5 - 48*x^6))/336 - (5*ArcSin[Sqrt[1 - x]/Sqrt[
2]])/8

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IntegrateAlgebraic [A]  time = 0.14, size = 169, normalized size = 1.88 \begin {gather*} \frac {-\frac {105 (1-x)^{13/2}}{(x+1)^{13/2}}-\frac {700 (1-x)^{11/2}}{(x+1)^{11/2}}-\frac {1981 (1-x)^{9/2}}{(x+1)^{9/2}}+\frac {3072 (1-x)^{7/2}}{(x+1)^{7/2}}+\frac {1981 (1-x)^{5/2}}{(x+1)^{5/2}}+\frac {700 (1-x)^{3/2}}{(x+1)^{3/2}}+\frac {105 \sqrt {1-x}}{\sqrt {x+1}}}{168 \left (\frac {1-x}{x+1}+1\right )^7}-\frac {5}{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)^(5/2),x]

[Out]

((-105*(1 - x)^(13/2))/(1 + x)^(13/2) - (700*(1 - x)^(11/2))/(1 + x)^(11/2) - (1981*(1 - x)^(9/2))/(1 + x)^(9/
2) + (3072*(1 - x)^(7/2))/(1 + x)^(7/2) + (1981*(1 - x)^(5/2))/(1 + x)^(5/2) + (700*(1 - x)^(3/2))/(1 + x)^(3/
2) + (105*Sqrt[1 - x])/Sqrt[1 + x])/(168*(1 + (1 - x)/(1 + x))^7) - (5*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]])/8

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fricas [A]  time = 1.32, size = 67, normalized size = 0.74 \begin {gather*} -\frac {1}{336} \, {\left (48 \, x^{6} - 56 \, x^{5} - 144 \, x^{4} + 182 \, x^{3} + 144 \, x^{2} - 231 \, x - 48\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {5}{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)^(5/2),x, algorithm="fricas")

[Out]

-1/336*(48*x^6 - 56*x^5 - 144*x^4 + 182*x^3 + 144*x^2 - 231*x - 48)*sqrt(x + 1)*sqrt(-x + 1) - 5/8*arctan((sqr
t(x + 1)*sqrt(-x + 1) - 1)/x)

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giac [B]  time = 1.14, size = 143, normalized size = 1.59 \begin {gather*} -\frac {1}{1680} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, x - 37\right )} {\left (x + 1\right )} + 661\right )} {\left (x + 1\right )} - 4551\right )} {\left (x + 1\right )} + 4781\right )} {\left (x + 1\right )} - 6335\right )} {\left (x + 1\right )} + 2835\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{40} \, {\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}{2} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {5}{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)^(5/2),x, algorithm="giac")

[Out]

-1/1680*((2*((4*(5*(6*x - 37)*(x + 1) + 661)*(x + 1) - 4551)*(x + 1) + 4781)*(x + 1) - 6335)*(x + 1) + 2835)*s
qrt(x + 1)*sqrt(-x + 1) + 1/40*((2*(3*(4*x - 17)*(x + 1) + 133)*(x + 1) - 295)*(x + 1) + 195)*sqrt(x + 1)*sqrt
(-x + 1) - 1/2*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 5/8*arcsin(1/2*sq
rt(2)*sqrt(x + 1))

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maple [A]  time = 0.00, size = 127, normalized size = 1.41 \begin {gather*} \frac {5 \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 {7}{2}}}{7}+\frac {\left (-x +1\right )^{\frac {5}{2}} \left (x +1\right )^{\frac {7}{2}}}{6}+\frac {\left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {7}{2}}}{6}+\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {7}{2}}}{8}-\frac {\sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{24}-\frac {5 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{48}-\frac {5 \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)^(5/2),x)

[Out]

1/7*(-x+1)^(7/2)*(x+1)^(7/2)+1/6*(-x+1)^(5/2)*(x+1)^(7/2)+1/6*(-x+1)^(3/2)*(x+1)^(7/2)+1/8*(-x+1)^(1/2)*(x+1)^
(7/2)-1/24*(-x+1)^(1/2)*(x+1)^(5/2)-5/48*(-x+1)^(1/2)*(x+1)^(3/2)-5/16*(-x+1)^(1/2)*(x+1)^(1/2)+5/16*((x+1)*(-
x+1))^(1/2)/(x+1)^(1/2)/(-x+1)^(1/2)*arcsin(x)

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maxima [A]  time = 3.01, size = 52, normalized size = 0.58 \begin {gather*} \frac {1}{7} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} + \frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {5}{24} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {5}{16} \, \sqrt {-x^{2} + 1} x + \frac {5}{16} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)*(1+x)^(5/2),x, algorithm="maxima")

[Out]

1/7*(-x^2 + 1)^(7/2) + 1/6*(-x^2 + 1)^(5/2)*x + 5/24*(-x^2 + 1)^(3/2)*x + 5/16*sqrt(-x^2 + 1)*x + 5/16*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 )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(7/2)*(x + 1)^(5/2),x)

[Out]

int((1 - x)^(7/2)*(x + 1)^(5/2), x)

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sympy [A]  time = 53.58, size = 321, normalized size = 3.57 \begin {gather*} \begin {cases} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {i \left (x + 1\right )^{\frac {15}{2}}}{7 \sqrt {x - 1}} + \frac {55 i \left (x + 1\right )^{\frac {13}{2}}}{42 \sqrt {x - 1}} - \frac {193 i \left (x + 1\right )^{\frac {11}{2}}}{42 \sqrt {x - 1}} + \frac {1237 i \left (x + 1\right )^{\frac {9}{2}}}{168 \sqrt {x - 1}} - \frac {769 i \left (x + 1\right )^{\frac {7}{2}}}{168 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{48 \sqrt {x - 1}} - \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {x - 1}} + \frac {5 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {\left (x + 1\right )^{\frac {15}{2}}}{7 \sqrt {1 - x}} - \frac {55 \left (x + 1\right )^{\frac {13}{2}}}{42 \sqrt {1 - x}} + \frac {193 \left (x + 1\right )^{\frac {11}{2}}}{42 \sqrt {1 - x}} - \frac {1237 \left (x + 1\right )^{\frac {9}{2}}}{168 \sqrt {1 - x}} + \frac {769 \left (x + 1\right )^{\frac {7}{2}}}{168 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{48 \sqrt {1 - x}} + \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {1 - x}} - \frac {5 \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)**(5/2),x)

[Out]

Piecewise((-5*I*acosh(sqrt(2)*sqrt(x + 1)/2)/8 - I*(x + 1)**(15/2)/(7*sqrt(x - 1)) + 55*I*(x + 1)**(13/2)/(42*
sqrt(x - 1)) - 193*I*(x + 1)**(11/2)/(42*sqrt(x - 1)) + 1237*I*(x + 1)**(9/2)/(168*sqrt(x - 1)) - 769*I*(x + 1
)**(7/2)/(168*sqrt(x - 1)) - I*(x + 1)**(5/2)/(48*sqrt(x - 1)) - 5*I*(x + 1)**(3/2)/(48*sqrt(x - 1)) + 5*I*sqr
t(x + 1)/(8*sqrt(x - 1)), Abs(x + 1)/2 > 1), (5*asin(sqrt(2)*sqrt(x + 1)/2)/8 + (x + 1)**(15/2)/(7*sqrt(1 - x)
) - 55*(x + 1)**(13/2)/(42*sqrt(1 - x)) + 193*(x + 1)**(11/2)/(42*sqrt(1 - x)) - 1237*(x + 1)**(9/2)/(168*sqrt
(1 - x)) + 769*(x + 1)**(7/2)/(168*sqrt(1 - x)) + (x + 1)**(5/2)/(48*sqrt(1 - x)) + 5*(x + 1)**(3/2)/(48*sqrt(
1 - x)) - 5*sqrt(x + 1)/(8*sqrt(1 - x)), True))

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