Integrand size = 17, antiderivative size = 87 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35 \arcsin (x)}{8} \] Output:
35/8*(1-x)^(1/2)*(1+x)^(1/2)+35/24*(1-x)^(3/2)*(1+x)^(1/2)+7/12*(1-x)^(5/2 )*(1+x)^(1/2)+1/4*(1-x)^(7/2)*(1+x)^(1/2)+35/8*arcsin(x)
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.61 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\frac {1}{24} \sqrt {1-x^2} \left (160-81 x+32 x^2-6 x^3\right )-\frac {35}{4} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \] Input:
Integrate[(1 - x)^(7/2)/Sqrt[1 + x],x]
Output:
(Sqrt[1 - x^2]*(160 - 81*x + 32*x^2 - 6*x^3))/24 - (35*ArcTan[Sqrt[1 - x^2 ]/(-1 + x)])/4
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {60, 60, 60, 50, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(1-x)^{7/2}}{\sqrt {x+1}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7}{4} \int \frac {(1-x)^{5/2}}{\sqrt {x+1}}dx+\frac {1}{4} \sqrt {x+1} (1-x)^{7/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7}{4} \left (\frac {5}{3} \int \frac {(1-x)^{3/2}}{\sqrt {x+1}}dx+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}\right )+\frac {1}{4} \sqrt {x+1} (1-x)^{7/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \int \frac {\sqrt {1-x}}{\sqrt {x+1}}dx+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}\right )+\frac {1}{4} \sqrt {x+1} (1-x)^{7/2}\) |
\(\Big \downarrow \) 50 |
\(\displaystyle \frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {1-x^2}}dx+\sqrt {1-x^2}\right )+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}\right )+\frac {1}{4} \sqrt {x+1} (1-x)^{7/2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {7}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (\arcsin (x)+\sqrt {1-x^2}\right )+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}\right )+\frac {1}{4} \sqrt {x+1} (1-x)^{7/2}\) |
Input:
Int[(1 - x)^(7/2)/Sqrt[1 + x],x]
Output:
((1 - x)^(7/2)*Sqrt[1 + x])/4 + (7*(((1 - x)^(5/2)*Sqrt[1 + x])/3 + (5*((( 1 - x)^(3/2)*Sqrt[1 + x])/2 + (3*(Sqrt[1 - x^2] + ArcSin[x]))/2))/3))/4
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a *c + b*d*x^2)^m/(2*d*m), x] + Simp[a Int[(a*c + b*d*x^2)^n, x], x] /; Fre eQ[{a, b, c, d, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 1] && GtQ[m, 0 ] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
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] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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]
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {\left (6 x^{3}-32 x^{2}+81 x -160\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{24 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(82\) |
default | \(\frac {\left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}{4}+\frac {7 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}{12}+\frac {35 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}{24}+\frac {35 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(85\) |
Input:
int((1-x)^(7/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/24*(6*x^3-32*x^2+81*x-160)*(1+x)^(1/2)*(-1+x)/(-(1+x)*(-1+x))^(1/2)*((1+ x)*(1-x))^(1/2)/(1-x)^(1/2)+35/8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/ 2)*arcsin(x)
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=-\frac {1}{24} \, {\left (6 \, x^{3} - 32 \, x^{2} + 81 \, x - 160\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {35}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \] Input:
integrate((1-x)^(7/2)/(1+x)^(1/2),x, algorithm="fricas")
Output:
-1/24*(6*x^3 - 32*x^2 + 81*x - 160)*sqrt(x + 1)*sqrt(-x + 1) - 35/4*arctan ((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)
Result contains complex when optimal does not.
Time = 9.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.26 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\begin {cases} - \frac {i \sqrt {x - 1} \left (x + 1\right )^{\frac {7}{2}}}{4} + \frac {25 i \sqrt {x - 1} \left (x + 1\right )^{\frac {5}{2}}}{12} - \frac {163 i \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}}{24} + \frac {93 i \sqrt {x - 1} \sqrt {x + 1}}{8} - \frac {35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {1 - x}} - \frac {31 \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {1 - x}} + \frac {263 \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {1 - x}} - \frac {605 \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {1 - x}} + \frac {93 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \] Input:
integrate((1-x)**(7/2)/(1+x)**(1/2),x)
Output:
Piecewise((-I*sqrt(x - 1)*(x + 1)**(7/2)/4 + 25*I*sqrt(x - 1)*(x + 1)**(5/ 2)/12 - 163*I*sqrt(x - 1)*(x + 1)**(3/2)/24 + 93*I*sqrt(x - 1)*sqrt(x + 1) /8 - 35*I*acosh(sqrt(2)*sqrt(x + 1)/2)/4, Abs(x + 1) > 2), (35*asin(sqrt(2 )*sqrt(x + 1)/2)/4 + (x + 1)**(9/2)/(4*sqrt(1 - x)) - 31*(x + 1)**(7/2)/(1 2*sqrt(1 - x)) + 263*(x + 1)**(5/2)/(24*sqrt(1 - x)) - 605*(x + 1)**(3/2)/ (24*sqrt(1 - x)) + 93*sqrt(x + 1)/(4*sqrt(1 - x)), True))
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=-\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} + \frac {4}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {27}{8} \, \sqrt {-x^{2} + 1} x + \frac {20}{3} \, \sqrt {-x^{2} + 1} + \frac {35}{8} \, \arcsin \left (x\right ) \] Input:
integrate((1-x)^(7/2)/(1+x)^(1/2),x, algorithm="maxima")
Output:
-1/4*sqrt(-x^2 + 1)*x^3 + 4/3*sqrt(-x^2 + 1)*x^2 - 27/8*sqrt(-x^2 + 1)*x + 20/3*sqrt(-x^2 + 1) + 35/8*arcsin(x)
Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=-\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {35}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \] Input:
integrate((1-x)^(7/2)/(1+x)^(1/2),x, algorithm="giac")
Output:
-1/24*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) - 3/2*sqrt(x + 1)*( x - 2)*sqrt(-x + 1) + sqrt(x + 1)*sqrt(-x + 1) + 35/4*arcsin(1/2*sqrt(2)*s qrt(x + 1))
Timed out. \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=\int \frac {{\left (1-x\right )}^{7/2}}{\sqrt {x+1}} \,d x \] Input:
int((1 - x)^(7/2)/(x + 1)^(1/2),x)
Output:
int((1 - x)^(7/2)/(x + 1)^(1/2), x)
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx=-\frac {35 \mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{4}-\frac {\sqrt {x +1}\, \sqrt {1-x}\, x^{3}}{4}+\frac {4 \sqrt {x +1}\, \sqrt {1-x}\, x^{2}}{3}-\frac {27 \sqrt {x +1}\, \sqrt {1-x}\, x}{8}+\frac {20 \sqrt {x +1}\, \sqrt {1-x}}{3} \] Input:
int((1-x)^(7/2)/(1+x)^(1/2),x)
Output:
( - 210*asin(sqrt( - x + 1)/sqrt(2)) - 6*sqrt(x + 1)*sqrt( - x + 1)*x**3 + 32*sqrt(x + 1)*sqrt( - x + 1)*x**2 - 81*sqrt(x + 1)*sqrt( - x + 1)*x + 16 0*sqrt(x + 1)*sqrt( - x + 1))/24