3.1088 \(\int (1-x)^{11/2} (1+x)^{5/2} \, dx\)

Optimal. Leaf size=130 \[ \frac{1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac{11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac{11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac{11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac{55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac{55}{128} x \sqrt{x+1} \sqrt{1-x}+\frac{55}{128} \sin ^{-1}(x) \]

[Out]

(55*Sqrt[1 - x]*x*Sqrt[1 + x])/128 + (55*(1 - x)^(3/2)*x*(1 + x)^(3/2))/192 + (11*(1 - x)^(5/2)*x*(1 + x)^(5/2
))/48 + (11*(1 - x)^(7/2)*(1 + x)^(7/2))/56 + (11*(1 - x)^(9/2)*(1 + x)^(7/2))/72 + ((1 - x)^(11/2)*(1 + x)^(7
/2))/9 + (55*ArcSin[x])/128

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Rubi [A]  time = 0.0254404, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {49, 38, 41, 216} \[ \frac{1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac{11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac{11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac{11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac{55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac{55}{128} x \sqrt{x+1} \sqrt{1-x}+\frac{55}{128} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(11/2)*(1 + x)^(5/2),x]

[Out]

(55*Sqrt[1 - x]*x*Sqrt[1 + x])/128 + (55*(1 - x)^(3/2)*x*(1 + x)^(3/2))/192 + (11*(1 - x)^(5/2)*x*(1 + x)^(5/2
))/48 + (11*(1 - x)^(7/2)*(1 + x)^(7/2))/56 + (11*(1 - x)^(9/2)*(1 + x)^(7/2))/72 + ((1 - x)^(11/2)*(1 + x)^(7
/2))/9 + (55*ArcSin[x])/128

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)^{11/2} (1+x)^{5/2} \, dx &=\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{11}{9} \int (1-x)^{9/2} (1+x)^{5/2} \, dx\\ &=\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{11}{8} \int (1-x)^{7/2} (1+x)^{5/2} \, dx\\ &=\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{11}{8} \int (1-x)^{5/2} (1+x)^{5/2} \, dx\\ &=\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{48} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac{55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{64} \int \sqrt{1-x} \sqrt{1+x} \, dx\\ &=\frac{55}{128} \sqrt{1-x} x \sqrt{1+x}+\frac{55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{128} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{55}{128} \sqrt{1-x} x \sqrt{1+x}+\frac{55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{128} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{55}{128} \sqrt{1-x} x \sqrt{1+x}+\frac{55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac{11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac{11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac{11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac{1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac{55}{128} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0706632, size = 75, normalized size = 0.58 \[ \frac{\sqrt{1-x^2} \left (-896 x^8+3024 x^7-1024 x^6-7224 x^5+8448 x^4+3066 x^3-10240 x^2+4599 x+3712\right )-6930 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )}{8064} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(11/2)*(1 + x)^(5/2),x]

[Out]

(Sqrt[1 - x^2]*(3712 + 4599*x - 10240*x^2 + 3066*x^3 + 8448*x^4 - 7224*x^5 - 1024*x^6 + 3024*x^7 - 896*x^8) -
6930*ArcSin[Sqrt[1 - x]/Sqrt[2]])/8064

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Maple [A]  time = 0.003, size = 155, normalized size = 1.2 \begin{align*}{\frac{1}{9} \left ( 1-x \right ) ^{{\frac{11}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{72} \left ( 1-x \right ) ^{{\frac{9}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{56} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{48} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{48} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{64}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{11}{192}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{55}{384}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{55}{128}\sqrt{1-x}\sqrt{1+x}}+{\frac{55\,\arcsin \left ( x \right ) }{128}\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)^(11/2)*(1+x)^(5/2),x)

[Out]

1/9*(1-x)^(11/2)*(1+x)^(7/2)+11/72*(1-x)^(9/2)*(1+x)^(7/2)+11/56*(1-x)^(7/2)*(1+x)^(7/2)+11/48*(1-x)^(5/2)*(1+
x)^(7/2)+11/48*(1-x)^(3/2)*(1+x)^(7/2)+11/64*(1-x)^(1/2)*(1+x)^(7/2)-11/192*(1-x)^(1/2)*(1+x)^(5/2)-55/384*(1-
x)^(1/2)*(1+x)^(3/2)-55/128*(1-x)^(1/2)*(1+x)^(1/2)+55/128*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(
x)

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Maxima [A]  time = 1.55367, size = 105, normalized size = 0.81 \begin{align*} \frac{1}{9} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} x^{2} - \frac{3}{8} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} x + \frac{29}{63} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} + \frac{11}{48} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{55}{192} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{55}{128} \, \sqrt{-x^{2} + 1} x + \frac{55}{128} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(11/2)*(1+x)^(5/2),x, algorithm="maxima")

[Out]

1/9*(-x^2 + 1)^(7/2)*x^2 - 3/8*(-x^2 + 1)^(7/2)*x + 29/63*(-x^2 + 1)^(7/2) + 11/48*(-x^2 + 1)^(5/2)*x + 55/192
*(-x^2 + 1)^(3/2)*x + 55/128*sqrt(-x^2 + 1)*x + 55/128*arcsin(x)

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Fricas [A]  time = 1.5462, size = 238, normalized size = 1.83 \begin{align*} -\frac{1}{8064} \,{\left (896 \, x^{8} - 3024 \, x^{7} + 1024 \, x^{6} + 7224 \, x^{5} - 8448 \, x^{4} - 3066 \, x^{3} + 10240 \, x^{2} - 4599 \, x - 3712\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{55}{64} \, \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)^(11/2)*(1+x)^(5/2),x, algorithm="fricas")

[Out]

-1/8064*(896*x^8 - 3024*x^7 + 1024*x^6 + 7224*x^5 - 8448*x^4 - 3066*x^3 + 10240*x^2 - 4599*x - 3712)*sqrt(x +
1)*sqrt(-x + 1) - 55/64*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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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)**(11/2)*(1+x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.21076, size = 409, normalized size = 3.15 \begin{align*} -\frac{1}{315} \,{\left ({\left ({\left ({\left (5 \,{\left ({\left (7 \,{\left (x + 1\right )}{\left (x - 7\right )} + 195\right )}{\left (x + 1\right )} - 386\right )}{\left (x + 1\right )} + 2369\right )}{\left (x + 1\right )} - 1836\right )}{\left (x + 1\right )} + 861\right )}{\left (x + 1\right )} - 210\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \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{1}{3} \,{\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} -{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{128} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (6 \,{\left (x + 1\right )}{\left (x - 6\right )} + 125\right )}{\left (x + 1\right )} - 205\right )}{\left (x + 1\right )} + 795\right )}{\left (x + 1\right )} - 449\right )}{\left (x + 1\right )} + 251\right )}{\left (x + 1\right )} - 15\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{5}{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}{8} \,{\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{55}{64} \, \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)^(11/2)*(1+x)^(5/2),x, algorithm="giac")

[Out]

-1/315*((((5*((7*(x + 1)*(x - 7) + 195)*(x + 1) - 386)*(x + 1) + 2369)*(x + 1) - 1836)*(x + 1) + 861)*(x + 1)
- 210)*(x + 1)^(3/2)*sqrt(-x + 1) - 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) + 1/3*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x + 1) - (x
+ 1)^(3/2)*(x - 1)*sqrt(-x + 1) + 1/128*((2*((4*((6*(x + 1)*(x - 6) + 125)*(x + 1) - 205)*(x + 1) + 795)*(x +
1) - 449)*(x + 1) + 251)*(x + 1) - 15)*sqrt(x + 1)*sqrt(-x + 1) - 5/48*((2*((4*(x + 1)*(x - 4) + 39)*(x + 1) -
37)*(x + 1) + 31)*(x + 1) - 3)*sqrt(x + 1)*sqrt(-x + 1) + 1/8*((2*(x + 1)*(x - 2) + 5)*(x + 1) - 1)*sqrt(x +
1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt(-x + 1) + 55/64*arcsin(1/2*sqrt(2)*sqrt(x + 1))