3.1090 \(\int (1-x)^{7/2} (1+x)^{5/2} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0119287, antiderivative size = 90, 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} \[ \frac{1}{7} (1-x)^{7/2} (x+1)^{7/2}+\frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]

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 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)^{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*}

Mathematica [A]  time = 0.0630693, size = 66, normalized size = 0.73 \[ \frac{1}{336} \sqrt{1-x^2} \left (-48 x^6+56 x^5+144 x^4-182 x^3-144 x^2+231 x+48\right )-\frac{5}{8} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 127, normalized size = 1.4 \begin{align*}{\frac{1}{7} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{48}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\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)^(5/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.52674, size = 70, normalized size = 0.78 \begin{align*} \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 \left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.56064, size = 190, normalized size = 2.11 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(7/2)*(1+x)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.17025, size = 259, normalized size = 2.88 \begin{align*} -\frac{1}{105} \,{\left ({\left (3 \,{\left ({\left (5 \,{\left (x + 1\right )}{\left (x - 5\right )} + 74\right )}{\left (x + 1\right )} - 96\right )}{\left (x + 1\right )} + 203\right )}{\left (x + 1\right )} - 70\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} + \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{1}{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}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{5}{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)^(5/2),x, algorithm="giac")

[Out]

-1/105*((3*((5*(x + 1)*(x - 5) + 74)*(x + 1) - 96)*(x + 1) + 203)*(x + 1) - 70)*(x + 1)^(3/2)*sqrt(-x + 1) + 2
/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x + 1) - 1/3*(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) - 1/4
*((2*(x + 1)*(x - 2) + 5)*(x + 1) - 1)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt(-x + 1) + 5/8*arcsin(
1/2*sqrt(2)*sqrt(x + 1))