3.11.75 \(\int (1-x)^{9/2} (1+x)^{3/2} \, dx\) [1075]

Optimal. Leaf size=109 \[ \frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \sin ^{-1}(x) \]

[Out]

3/8*(1-x)^(3/2)*x*(1+x)^(3/2)+3/10*(1-x)^(5/2)*(1+x)^(5/2)+3/14*(1-x)^(7/2)*(1+x)^(5/2)+1/7*(1-x)^(9/2)*(1+x)^
(5/2)+9/16*arcsin(x)+9/16*x*(1-x)^(1/2)*(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222} \begin {gather*} \frac {1}{7} (x+1)^{5/2} (1-x)^{9/2}+\frac {3}{14} (x+1)^{5/2} (1-x)^{7/2}+\frac {3}{10} (x+1)^{5/2} (1-x)^{5/2}+\frac {3}{8} x (x+1)^{3/2} (1-x)^{3/2}+\frac {9}{16} x \sqrt {x+1} \sqrt {1-x}+\frac {9}{16} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(9/2)*(1 + x)^(3/2),x]

[Out]

(9*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (3*(1 - x)^(3/2)*x*(1 + x)^(3/2))/8 + (3*(1 - x)^(5/2)*(1 + x)^(5/2))/10 +
(3*(1 - x)^(7/2)*(1 + x)^(5/2))/14 + ((1 - x)^(9/2)*(1 + x)^(5/2))/7 + (9*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 51

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 222

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)^{9/2} (1+x)^{3/2} \, dx &=\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{7} \int (1-x)^{7/2} (1+x)^{3/2} \, dx\\ &=\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {3}{2} \int (1-x)^{5/2} (1+x)^{3/2} \, dx\\ &=\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {3}{2} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{8} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {9}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {3}{8} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{10} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{14} (1-x)^{7/2} (1+x)^{5/2}+\frac {1}{7} (1-x)^{9/2} (1+x)^{5/2}+\frac {9}{16} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 78, normalized size = 0.72 \begin {gather*} \frac {\sqrt {1-x} \left (368+613 x-411 x^2-306 x^3+558 x^4-72 x^5-200 x^6+80 x^7\right )}{560 \sqrt {1+x}}-\frac {9}{8} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(9/2)*(1 + x)^(3/2),x]

[Out]

(Sqrt[1 - x]*(368 + 613*x - 411*x^2 - 306*x^3 + 558*x^4 - 72*x^5 - 200*x^6 + 80*x^7))/(560*Sqrt[1 + x]) - (9*A
rcTan[Sqrt[1 - x]/Sqrt[1 + x]])/8

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

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 - x)^(9/2)*(1 + x)^(3/2),x]')

[Out]

Timed out

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Maple [A]
time = 0.14, size = 127, normalized size = 1.17

method	result	size
risch	\(-\frac {\left (80 x^{6}-280 x^{5}+208 x^{4}+350 x^{3}-656 x^{2}+245 x +368\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{560 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {9 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\)	\(97\)
default	\(\frac {\left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {5}{2}}}{7}+\frac {3 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {5}{2}}}{14}+\frac {3 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {5}{2}}}{10}+\frac {3 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {5}{2}}}{8}+\frac {3 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{8}-\frac {3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{16}-\frac {9 \sqrt {1-x}\, \sqrt {1+x}}{16}+\frac {9 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(9/2)*(1+x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/7*(1-x)^(9/2)*(1+x)^(5/2)+3/14*(1-x)^(7/2)*(1+x)^(5/2)+3/10*(1-x)^(5/2)*(1+x)^(5/2)+3/8*(1-x)^(3/2)*(1+x)^(5
/2)+3/8*(1-x)^(1/2)*(1+x)^(5/2)-3/16*(1-x)^(1/2)*(1+x)^(3/2)-9/16*(1-x)^(1/2)*(1+x)^(1/2)+9/16*((1+x)*(1-x))^(
1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]
time = 0.35, size = 66, normalized size = 0.61 \begin {gather*} \frac {1}{7} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x^{2} - \frac {1}{2} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {23}{35} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {3}{8} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {9}{16} \, \sqrt {-x^{2} + 1} x + \frac {9}{16} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(9/2)*(1+x)^(3/2),x, algorithm="maxima")

[Out]

1/7*(-x^2 + 1)^(5/2)*x^2 - 1/2*(-x^2 + 1)^(5/2)*x + 23/35*(-x^2 + 1)^(5/2) + 3/8*(-x^2 + 1)^(3/2)*x + 9/16*sqr
t(-x^2 + 1)*x + 9/16*arcsin(x)

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Fricas [A]
time = 0.30, size = 67, normalized size = 0.61 \begin {gather*} \frac {1}{560} \, {\left (80 \, x^{6} - 280 \, x^{5} + 208 \, x^{4} + 350 \, x^{3} - 656 \, x^{2} + 245 \, x + 368\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {9}{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)^(9/2)*(1+x)^(3/2),x, algorithm="fricas")

[Out]

1/560*(80*x^6 - 280*x^5 + 208*x^4 + 350*x^3 - 656*x^2 + 245*x + 368)*sqrt(x + 1)*sqrt(-x + 1) - 9/8*arctan((sq
rt(x + 1)*sqrt(-x + 1) - 1)/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)**(9/2)*(1+x)**(3/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (77) = 154\).
time = 0.04, size = 647, normalized size = 5.94 \begin {gather*} 2 \left (2 \left (\left (\left (\left (\left (\left (\frac {1}{28} \sqrt {-x+1} \sqrt {-x+1}-\frac {43}{168}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {661}{840}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {1517}{1120}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {683}{480}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {181}{192}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {27}{64}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {5}{16} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )-8 \left (2 \left (\left (\left (\left (\left (\frac {31}{120}-\frac {1}{24} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {107}{160}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {451}{480}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {149}{192}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {27}{64}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {5}{16} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )+10 \left (2 \left (\left (\left (\left (\frac {1}{20} \sqrt {-x+1} \sqrt {-x+1}-\frac {21}{80}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {133}{240}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {59}{96}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {13}{32}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {3}{8} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )-10 \left (2 \left (\left (\frac {1}{12} \sqrt {-x+1} \sqrt {-x+1}-\frac {7}{24}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {3}{8}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )+8 \left (2 \left (\frac {3}{8}-\frac {1}{8} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )-2 \left (\frac {1}{2} \sqrt {-x+1} \sqrt {x+1}+\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(9/2)*(1+x)^(3/2),x)

[Out]

1/1680*((2*((4*(5*(6*x + 37)*(x - 1) + 661)*(x - 1) + 4551)*(x - 1) + 4781)*(x - 1) + 6335)*(x - 1) + 2835)*sq
rt(x + 1)*sqrt(-x + 1) - 1/60*((2*((4*(5*x + 26)*(x - 1) + 321)*(x - 1) + 451)*(x - 1) + 745)*(x - 1) + 405)*s
qrt(x + 1)*sqrt(-x + 1) + 1/24*((2*(3*(4*x + 17)*(x - 1) + 133)*(x - 1) + 295)*(x - 1) + 195)*sqrt(x + 1)*sqrt
(-x + 1) - 5/6*((2*x + 5)*(x - 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) + 2*(x + 2)*sqrt(x + 1)*sqrt(-x + 1) - sqrt(x
+ 1)*sqrt(-x + 1) - 9/8*arcsin(1/2*sqrt(2)*sqrt(-x + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{9/2}\,{\left (x+1\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(9/2)*(x + 1)^(3/2),x)

[Out]

int((1 - x)^(9/2)*(x + 1)^(3/2), x)

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