3.11.64 \(\int (1-x)^{7/2} \sqrt {1+x} \, dx\) [1064]

Optimal. Leaf size=88 \[ \frac {7}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac {1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac {7}{8} \sin ^{-1}(x) \]

[Out]

7/12*(1-x)^(3/2)*(1+x)^(3/2)+7/20*(1-x)^(5/2)*(1+x)^(3/2)+1/5*(1-x)^(7/2)*(1+x)^(3/2)+7/8*arcsin(x)+7/8*x*(1-x
)^(1/2)*(1+x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 88, 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 = {51, 38, 41, 222} \begin {gather*} \frac {1}{5} (x+1)^{3/2} (1-x)^{7/2}+\frac {7}{20} (x+1)^{3/2} (1-x)^{5/2}+\frac {7}{12} (x+1)^{3/2} (1-x)^{3/2}+\frac {7}{8} x \sqrt {x+1} \sqrt {1-x}+\frac {7}{8} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(7/2)*Sqrt[1 + x],x]

[Out]

(7*Sqrt[1 - x]*x*Sqrt[1 + x])/8 + (7*(1 - x)^(3/2)*(1 + x)^(3/2))/12 + (7*(1 - x)^(5/2)*(1 + x)^(3/2))/20 + ((
1 - x)^(7/2)*(1 + x)^(3/2))/5 + (7*ArcSin[x])/8

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)^{7/2} \sqrt {1+x} \, dx &=\frac {1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac {7}{5} \int (1-x)^{5/2} \sqrt {1+x} \, dx\\ &=\frac {7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac {1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac {7}{4} \int (1-x)^{3/2} \sqrt {1+x} \, dx\\ &=\frac {7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac {1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac {7}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {7}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac {1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {7}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac {1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {7}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac {1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac {7}{8} \sin ^{-1}(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.07, size = 68, normalized size = 0.77 \begin {gather*} \frac {\sqrt {1+x} \left (136-121 x-127 x^2+202 x^3-114 x^4+24 x^5\right )}{120 \sqrt {1-x}}+\frac {7}{4} \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(7/2)*Sqrt[1 + x],x]

[Out]

(Sqrt[1 + x]*(136 - 121*x - 127*x^2 + 202*x^3 - 114*x^4 + 24*x^5))/(120*Sqrt[1 - x]) + (7*ArcTan[Sqrt[1 + x]/S
qrt[1 - x]])/4

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 24.89, size = 175, normalized size = 1.99 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-1315 \left (1+x\right )^{\frac {3}{2}}-898 \left (1+x\right )^{\frac {7}{2}}-210 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}-24 \left (1+x\right )^{\frac {11}{2}}+210 \sqrt {1+x}+234 \left (1+x\right )^{\frac {9}{2}}+1657 \left (1+x\right )^{\frac {5}{2}}\right )}{120 \sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-1657 \left (1+x\right )^{\frac {5}{2}}}{120 \sqrt {1-x}}-\frac {39 \left (1+x\right )^{\frac {9}{2}}}{20 \sqrt {1-x}}-\frac {7 \sqrt {1+x}}{4 \sqrt {1-x}}+\frac {\left (1+x\right )^{\frac {11}{2}}}{5 \sqrt {1-x}}+\frac {7 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]}{4}+\frac {449 \left (1+x\right )^{\frac {7}{2}}}{60 \sqrt {1-x}}+\frac {263 \left (1+x\right )^{\frac {3}{2}}}{24 \sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 - x)^(7/2)*(1 + x)^(1/2),x]')

[Out]

Piecewise[{{I / 120 (-1315 (1 + x) ^ (3 / 2) - 898 (1 + x) ^ (7 / 2) - 210 ArcCosh[Sqrt[2] Sqrt[1 + x] / 2] Sq
rt[-1 + x] - 24 (1 + x) ^ (11 / 2) + 210 Sqrt[1 + x] + 234 (1 + x) ^ (9 / 2) + 1657 (1 + x) ^ (5 / 2)) / Sqrt[
-1 + x], Abs[1 + x] > 2}}, -1657 (1 + x) ^ (5 / 2) / (120 Sqrt[1 - x]) - 39 (1 + x) ^ (9 / 2) / (20 Sqrt[1 - x
]) - 7 Sqrt[1 + x] / (4 Sqrt[1 - x]) + (1 + x) ^ (11 / 2) / (5 Sqrt[1 - x]) + 7 ArcSin[Sqrt[2] Sqrt[1 + x] / 2
] / 4 + 449 (1 + x) ^ (7 / 2) / (60 Sqrt[1 - x]) + 263 (1 + x) ^ (3 / 2) / (24 Sqrt[1 - x])]

________________________________________________________________________________________

Maple [A]
time = 0.15, size = 99, normalized size = 1.12

method	result	size
risch	\(\frac {\left (24 x^{4}-90 x^{3}+112 x^{2}-15 x -136\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{120 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {7 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(87\)
default	\(\frac {\left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}{5}+\frac {7 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}{20}+\frac {7 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}{12}+\frac {7 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{8}-\frac {7 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {7 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(7/2)*(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5*(1-x)^(7/2)*(1+x)^(3/2)+7/20*(1-x)^(5/2)*(1+x)^(3/2)+7/12*(1-x)^(3/2)*(1+x)^(3/2)+7/8*(1-x)^(1/2)*(1+x)^(3
/2)-7/8*(1-x)^(1/2)*(1+x)^(1/2)+7/8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

________________________________________________________________________________________

Maxima [A]
time = 0.37, size = 54, normalized size = 0.61 \begin {gather*} \frac {1}{5} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - \frac {3}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {17}{15} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {7}{8} \, \sqrt {-x^{2} + 1} x + \frac {7}{8} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(-x^2 + 1)^(3/2)*x^2 - 3/4*(-x^2 + 1)^(3/2)*x + 17/15*(-x^2 + 1)^(3/2) + 7/8*sqrt(-x^2 + 1)*x + 7/8*arcsin
(x)

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 57, normalized size = 0.65 \begin {gather*} -\frac {1}{120} \, {\left (24 \, x^{4} - 90 \, x^{3} + 112 \, x^{2} - 15 \, x - 136\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {7}{4} \, \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)^(1/2),x, algorithm="fricas")

[Out]

-1/120*(24*x^4 - 90*x^3 + 112*x^2 - 15*x - 136)*sqrt(x + 1)*sqrt(-x + 1) - 7/4*arctan((sqrt(x + 1)*sqrt(-x + 1
) - 1)/x)

________________________________________________________________________________________

Sympy [A]
time = 30.39, size = 252, normalized size = 2.86 \begin {gather*} \begin {cases} - \frac {7 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {x - 1}} + \frac {39 i \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {x - 1}} - \frac {449 i \left (x + 1\right )^{\frac {7}{2}}}{60 \sqrt {x - 1}} + \frac {1657 i \left (x + 1\right )^{\frac {5}{2}}}{120 \sqrt {x - 1}} - \frac {263 i \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {x - 1}} + \frac {7 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {7 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {1 - x}} - \frac {39 \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {1 - x}} + \frac {449 \left (x + 1\right )^{\frac {7}{2}}}{60 \sqrt {1 - x}} - \frac {1657 \left (x + 1\right )^{\frac {5}{2}}}{120 \sqrt {1 - x}} + \frac {263 \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {1 - x}} - \frac {7 \sqrt {x + 1}}{4 \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)**(1/2),x)

[Out]

Piecewise((-7*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4 - I*(x + 1)**(11/2)/(5*sqrt(x - 1)) + 39*I*(x + 1)**(9/2)/(20*s
qrt(x - 1)) - 449*I*(x + 1)**(7/2)/(60*sqrt(x - 1)) + 1657*I*(x + 1)**(5/2)/(120*sqrt(x - 1)) - 263*I*(x + 1)*
*(3/2)/(24*sqrt(x - 1)) + 7*I*sqrt(x + 1)/(4*sqrt(x - 1)), Abs(x + 1) > 2), (7*asin(sqrt(2)*sqrt(x + 1)/2)/4 +
(x + 1)**(11/2)/(5*sqrt(1 - x)) - 39*(x + 1)**(9/2)/(20*sqrt(1 - x)) + 449*(x + 1)**(7/2)/(60*sqrt(1 - x)) -
1657*(x + 1)**(5/2)/(120*sqrt(1 - x)) + 263*(x + 1)**(3/2)/(24*sqrt(1 - x)) - 7*sqrt(x + 1)/(4*sqrt(1 - x)), T
rue))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (62) = 124\).
time = 0.02, size = 429, normalized size = 4.88 \begin {gather*} -2 \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 )+8 \left (2 \left (\left (\left (\frac {13}{48}-\frac {1}{16} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {43}{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 )-12 \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)^(7/2)*(1+x)^(1/2),x)

[Out]

-1/120*((2*(3*(4*x + 17)*(x - 1) + 133)*(x - 1) + 295)*(x - 1) + 195)*sqrt(x + 1)*sqrt(-x + 1) + 1/6*((2*(3*x
+ 10)*(x - 1) + 43)*(x - 1) + 39)*sqrt(x + 1)*sqrt(-x + 1) - ((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) - 7/4*arcsin(1/2*sqrt(2)*sqrt(-x + 1))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{7/2}\,\sqrt {x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(7/2)*(x + 1)^(1/2),x)

[Out]

int((1 - x)^(7/2)*(x + 1)^(1/2), x)

________________________________________________________________________________________