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

Optimal result

Integrand size = 17, antiderivative size = 88 \[ \int (1-x)^{7/2} \sqrt {1+x} \, dx=\frac {7}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{12} (1-x)^{3/2} (1+x)^{3/2}+\frac {7}{20} (1-x)^{5/2} (1+x)^{3/2}+\frac {1}{5} (1-x)^{7/2} (1+x)^{3/2}+\frac {7 \arcsin (x)}{8} \]

[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] (verified)

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222} \[ \int (1-x)^{7/2} \sqrt {1+x} \, dx=\frac {7 \arcsin (x)}{8}+\frac {1}{5} (x+1)^{3/2} (1-x)^{7/2}+\frac {7}{20} (x+1)^{3/2} (1-x)^{5/2}+\frac {7}{12} (x+1)^{3/2} (1-x)^{3/2}+\frac {7}{8} x \sqrt {x+1} \sqrt {1-x} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int (1-x)^{7/2} \sqrt {1+x} \, dx=\frac {1}{120} \sqrt {1-x^2} \left (136+15 x-112 x^2+90 x^3-24 x^4\right )-\frac {7}{4} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]

[In]

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

[Out]

(Sqrt[1 - x^2]*(136 + 15*x - 112*x^2 + 90*x^3 - 24*x^4))/120 - (7*ArcTan[Sqrt[1 - x^2]/(-1 + x)])/4

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.65 \[ \int (1-x)^{7/2} \sqrt {1+x} \, dx=-\frac {1}{120} \, {\left (24 \, x^{4} - 90 \, x^{3} + 112 \, x^{2} - 15 \, x - 136\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {7}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]

[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 [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 40.03 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.86 \[ \int (1-x)^{7/2} \sqrt {1+x} \, dx=\begin {cases} - \frac {7 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {x - 1}} + \frac {39 i \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {x - 1}} - \frac {449 i \left (x + 1\right )^{\frac {7}{2}}}{60 \sqrt {x - 1}} + \frac {1657 i \left (x + 1\right )^{\frac {5}{2}}}{120 \sqrt {x - 1}} - \frac {263 i \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {x - 1}} + \frac {7 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {7 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {1 - x}} - \frac {39 \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {1 - x}} + \frac {449 \left (x + 1\right )^{\frac {7}{2}}}{60 \sqrt {1 - x}} - \frac {1657 \left (x + 1\right )^{\frac {5}{2}}}{120 \sqrt {1 - x}} + \frac {263 \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {1 - x}} - \frac {7 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int (1-x)^{7/2} \sqrt {1+x} \, dx=\frac {1}{5} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - \frac {3}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {17}{15} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {7}{8} \, \sqrt {-x^{2} + 1} x + \frac {7}{8} \, \arcsin \left (x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.31 \[ \int (1-x)^{7/2} \sqrt {1+x} \, dx=-\frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {7}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[In]

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

[Out]

-1/120*((2*(3*(4*x - 17)*(x + 1) + 133)*(x + 1) - 295)*(x + 1) + 195)*sqrt(x + 1)*sqrt(-x + 1) + 1/12*((2*(3*x
- 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) - sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + sqrt(x + 1)*s
qrt(-x + 1) + 7/4*arcsin(1/2*sqrt(2)*sqrt(x + 1))

Mupad [F(-1)]

Timed out. \[ \int (1-x)^{7/2} \sqrt {1+x} \, dx=\int {\left (1-x\right )}^{7/2}\,\sqrt {x+1} \,d x \]

[In]

int((1 - x)^(7/2)*(x + 1)^(1/2),x)

[Out]

int((1 - x)^(7/2)*(x + 1)^(1/2), x)