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

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

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Rubi [A]  time = 0.03, antiderivative size = 130, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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) \end {gather*}

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 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)^{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*}

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Mathematica [A]  time = 0.07, size = 75, normalized size = 0.58 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.19, size = 205, normalized size = 1.58 \begin {gather*} \frac {-\frac {3465 (1-x)^{17/2}}{(x+1)^{17/2}}-\frac {30030 (1-x)^{15/2}}{(x+1)^{15/2}}-\frac {115038 (1-x)^{13/2}}{(x+1)^{13/2}}+\frac {334602 (1-x)^{11/2}}{(x+1)^{11/2}}+\frac {360448 (1-x)^{9/2}}{(x+1)^{9/2}}+\frac {255222 (1-x)^{7/2}}{(x+1)^{7/2}}+\frac {115038 (1-x)^{5/2}}{(x+1)^{5/2}}+\frac {30030 (1-x)^{3/2}}{(x+1)^{3/2}}+\frac {3465 \sqrt {1-x}}{\sqrt {x+1}}}{4032 \left (\frac {1-x}{x+1}+1\right )^9}-\frac {55}{64} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3465*(1 - x)^(17/2))/(1 + x)^(17/2) - (30030*(1 - x)^(15/2))/(1 + x)^(15/2) - (115038*(1 - x)^(13/2))/(1 +
x)^(13/2) + (334602*(1 - x)^(11/2))/(1 + x)^(11/2) + (360448*(1 - x)^(9/2))/(1 + x)^(9/2) + (255222*(1 - x)^(7
/2))/(1 + x)^(7/2) + (115038*(1 - x)^(5/2))/(1 + x)^(5/2) + (30030*(1 - x)^(3/2))/(1 + x)^(3/2) + (3465*Sqrt[1
- x])/Sqrt[1 + x])/(4032*(1 + (1 - x)/(1 + x))^9) - (55*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]])/64

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fricas [A]  time = 0.77, size = 77, normalized size = 0.59 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 1.40, size = 323, normalized size = 2.48 \begin {gather*} -\frac {1}{40320} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, {\left (8 \, x - 65\right )} {\left (x + 1\right )} + 2073\right )} {\left (x + 1\right )} - 9833\right )} {\left (x + 1\right )} + 75293\right )} {\left (x + 1\right )} - 310203\right )} {\left (x + 1\right )} + 216993\right )} {\left (x + 1\right )} - 205275\right )} {\left (x + 1\right )} + 69615\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{6720} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, x - 50\right )} {\left (x + 1\right )} + 1219\right )} {\left (x + 1\right )} - 12463\right )} {\left (x + 1\right )} + 64233\right )} {\left (x + 1\right )} - 53963\right )} {\left (x + 1\right )} + 59465\right )} {\left (x + 1\right )} - 23205\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{840} \, {\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 ({\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}{4} \, {\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} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {55}{64} \, \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)^(11/2)*(1+x)^(5/2),x, algorithm="giac")

[Out]

-1/40320*((2*((4*(5*(2*(7*(8*x - 65)*(x + 1) + 2073)*(x + 1) - 9833)*(x + 1) + 75293)*(x + 1) - 310203)*(x + 1
) + 216993)*(x + 1) - 205275)*(x + 1) + 69615)*sqrt(x + 1)*sqrt(-x + 1) + 1/6720*((2*((4*(5*(6*(7*x - 50)*(x +
1) + 1219)*(x + 1) - 12463)*(x + 1) + 64233)*(x + 1) - 53963)*(x + 1) + 59465)*(x + 1) - 23205)*sqrt(x + 1)*s
qrt(-x + 1) + 1/840*((2*((4*(5*(6*x - 37)*(x + 1) + 661)*(x + 1) - 4551)*(x + 1) + 4781)*(x + 1) - 6335)*(x +
1) + 2835)*sqrt(x + 1)*sqrt(-x + 1) - 1/40*((2*((4*(5*x - 26)*(x + 1) + 321)*(x + 1) - 451)*(x + 1) + 745)*(x
+ 1) - 405)*sqrt(x + 1)*sqrt(-x + 1) + 1/4*((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) - sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(
-x + 1) + 55/64*arcsin(1/2*sqrt(2)*sqrt(x + 1))

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maple [A]  time = 0.01, size = 155, normalized size = 1.19 \begin {gather*} \frac {55 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{128 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {11}{2}} \left (x +1\right )^{\frac {7}{2}}}{9}+\frac {11 \left (-x +1\right )^{\frac {9}{2}} \left (x +1\right )^{\frac {7}{2}}}{72}+\frac {11 \left (-x +1\right )^{\frac {7}{2}} \left (x +1\right )^{\frac {7}{2}}}{56}+\frac {11 \left (-x +1\right )^{\frac {5}{2}} \left (x +1\right )^{\frac {7}{2}}}{48}+\frac {11 \left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {7}{2}}}{48}+\frac {11 \sqrt {-x +1}\, \left (x +1\right )^{\frac {7}{2}}}{64}-\frac {11 \sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{192}-\frac {55 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{384}-\frac {55 \sqrt {-x +1}\, \sqrt {x +1}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x+1)^(11/2)*(x+1)^(5/2),x)

[Out]

1/9*(-x+1)^(11/2)*(x+1)^(7/2)+11/72*(-x+1)^(9/2)*(x+1)^(7/2)+11/56*(-x+1)^(7/2)*(x+1)^(7/2)+11/48*(-x+1)^(5/2)
*(x+1)^(7/2)+11/48*(-x+1)^(3/2)*(x+1)^(7/2)+11/64*(-x+1)^(1/2)*(x+1)^(7/2)-11/192*(-x+1)^(1/2)*(x+1)^(5/2)-55/
384*(-x+1)^(1/2)*(x+1)^(3/2)-55/128*(-x+1)^(1/2)*(x+1)^(1/2)+55/128*((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 = 78, normalized size = 0.60 \begin {gather*} \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 \relax (x) \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{11/2}\,{\left (x+1\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(11/2)*(x + 1)^(5/2),x)

[Out]

int((1 - x)^(11/2)*(x + 1)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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