3.12.16 \(\int \frac {(1-x)^{7/2}}{(1+x)^{3/2}} \, dx\) [1116]

Optimal. Leaf size=85 \[ -\frac {2 (1-x)^{7/2}}{\sqrt {1+x}}-\frac {35}{2} \sqrt {1-x} \sqrt {1+x}-\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}-\frac {7}{3} (1-x)^{5/2} \sqrt {1+x}-\frac {35}{2} \sin ^{-1}(x) \]

[Out]

-35/2*arcsin(x)-2*(1-x)^(7/2)/(1+x)^(1/2)-35/6*(1-x)^(3/2)*(1+x)^(1/2)-7/3*(1-x)^(5/2)*(1+x)^(1/2)-35/2*(1-x)^
(1/2)*(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 85, 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 = {49, 52, 41, 222} \begin {gather*} -\frac {2 (1-x)^{7/2}}{\sqrt {x+1}}-\frac {7}{3} \sqrt {x+1} (1-x)^{5/2}-\frac {35}{6} \sqrt {x+1} (1-x)^{3/2}-\frac {35}{2} \sqrt {x+1} \sqrt {1-x}-\frac {35}{2} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(7/2)/(1 + x)^(3/2),x]

[Out]

(-2*(1 - x)^(7/2))/Sqrt[1 + x] - (35*Sqrt[1 - x]*Sqrt[1 + x])/2 - (35*(1 - x)^(3/2)*Sqrt[1 + x])/6 - (7*(1 - x
)^(5/2)*Sqrt[1 + x])/3 - (35*ArcSin[x])/2

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 + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 52

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[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 \frac {(1-x)^{7/2}}{(1+x)^{3/2}} \, dx &=-\frac {2 (1-x)^{7/2}}{\sqrt {1+x}}-7 \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{7/2}}{\sqrt {1+x}}-\frac {7}{3} (1-x)^{5/2} \sqrt {1+x}-\frac {35}{3} \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{7/2}}{\sqrt {1+x}}-\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}-\frac {7}{3} (1-x)^{5/2} \sqrt {1+x}-\frac {35}{2} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{7/2}}{\sqrt {1+x}}-\frac {35}{2} \sqrt {1-x} \sqrt {1+x}-\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}-\frac {7}{3} (1-x)^{5/2} \sqrt {1+x}-\frac {35}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {2 (1-x)^{7/2}}{\sqrt {1+x}}-\frac {35}{2} \sqrt {1-x} \sqrt {1+x}-\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}-\frac {7}{3} (1-x)^{5/2} \sqrt {1+x}-\frac {35}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 (1-x)^{7/2}}{\sqrt {1+x}}-\frac {35}{2} \sqrt {1-x} \sqrt {1+x}-\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}-\frac {7}{3} (1-x)^{5/2} \sqrt {1+x}-\frac {35}{2} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 56, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {1-x} \left (166+55 x-13 x^2+2 x^3\right )}{6 \sqrt {1+x}}+35 \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(7/2)/(1 + x)^(3/2),x]

[Out]

-1/6*(Sqrt[1 - x]*(166 + 55*x - 13*x^2 + 2*x^3))/Sqrt[1 + x] + 35*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 14.87, size = 154, normalized size = 1.81 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-55 x \sqrt {-1+x}+13 x^2 \sqrt {-1+x}-2 x^3 \sqrt {-1+x}-166 \sqrt {-1+x}+210 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {1+x}\right )}{6 \sqrt {1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},-35 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-\frac {32}{\sqrt {1+x} \sqrt {1-x}}-\frac {13 \sqrt {1+x}}{\sqrt {1-x}}-\frac {23 \left (1+x\right )^{\frac {5}{2}}}{6 \sqrt {1-x}}+\frac {\left (1+x\right )^{\frac {7}{2}}}{3 \sqrt {1-x}}+\frac {125 \left (1+x\right )^{\frac {3}{2}}}{6 \sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 - x)^(7/2)/(1 + x)^(3/2),x]')

[Out]

Piecewise[{{I / 6 (-55 x Sqrt[-1 + x] + 13 x ^ 2 Sqrt[-1 + x] - 2 x ^ 3 Sqrt[-1 + x] - 166 Sqrt[-1 + x] + 210
ArcCosh[Sqrt[2] Sqrt[1 + x] / 2] Sqrt[1 + x]) / Sqrt[1 + x], Abs[1 + x] > 2}}, -35 ArcSin[Sqrt[2] Sqrt[1 + x]
/ 2] - 32 / (Sqrt[1 + x] Sqrt[1 - x]) - 13 Sqrt[1 + x] / Sqrt[1 - x] - 23 (1 + x) ^ (5 / 2) / (6 Sqrt[1 - x])
+ (1 + x) ^ (7 / 2) / (3 Sqrt[1 - x]) + 125 (1 + x) ^ (3 / 2) / (6 Sqrt[1 - x])]

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Maple [A]
time = 0.14, size = 84, normalized size = 0.99

method	result	size
risch	\(\frac {\left (2 x^{4}-15 x^{3}+68 x^{2}+111 x -166\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(7/2)/(1+x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(2*x^4-15*x^3+68*x^2+111*x-166)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)/(1+x)^(1/2)-35/2*((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 = 70, normalized size = 0.82 \begin {gather*} \frac {x^{4}}{3 \, \sqrt {-x^{2} + 1}} - \frac {5 \, x^{3}}{2 \, \sqrt {-x^{2} + 1}} + \frac {34 \, x^{2}}{3 \, \sqrt {-x^{2} + 1}} + \frac {37 \, x}{2 \, \sqrt {-x^{2} + 1}} - \frac {83}{3 \, \sqrt {-x^{2} + 1}} - \frac {35}{2} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

1/3*x^4/sqrt(-x^2 + 1) - 5/2*x^3/sqrt(-x^2 + 1) + 34/3*x^2/sqrt(-x^2 + 1) + 37/2*x/sqrt(-x^2 + 1) - 83/3/sqrt(
-x^2 + 1) - 35/2*arcsin(x)

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Fricas [A]
time = 0.31, size = 65, normalized size = 0.76 \begin {gather*} -\frac {{\left (2 \, x^{3} - 13 \, x^{2} + 55 \, x + 166\right )} \sqrt {x + 1} \sqrt {-x + 1} - 210 \, {\left (x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 166 \, x + 166}{6 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

-1/6*((2*x^3 - 13*x^2 + 55*x + 166)*sqrt(x + 1)*sqrt(-x + 1) - 210*(x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) -
1)/x) + 166*x + 166)/(x + 1)

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Sympy [A]
time = 16.72, size = 206, normalized size = 2.42 \begin {gather*} \begin {cases} 35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} + \frac {23 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} - \frac {125 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {13 i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {32 i}{\sqrt {x - 1} \sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} - \frac {23 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} + \frac {125 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {13 \sqrt {x + 1}}{\sqrt {1 - x}} - \frac {32}{\sqrt {1 - x} \sqrt {x + 1}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(7/2)/(1+x)**(3/2),x)

[Out]

Piecewise((35*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(7/2)/(3*sqrt(x - 1)) + 23*I*(x + 1)**(5/2)/(6*sqrt(
x - 1)) - 125*I*(x + 1)**(3/2)/(6*sqrt(x - 1)) + 13*I*sqrt(x + 1)/sqrt(x - 1) + 32*I/(sqrt(x - 1)*sqrt(x + 1))
, Abs(x + 1) > 2), (-35*asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)**(7/2)/(3*sqrt(1 - x)) - 23*(x + 1)**(5/2)/(6*sq
rt(1 - x)) + 125*(x + 1)**(3/2)/(6*sqrt(1 - x)) - 13*sqrt(x + 1)/sqrt(1 - x) - 32/(sqrt(1 - x)*sqrt(x + 1)), T
rue))

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Giac [A]
time = 0.01, size = 108, normalized size = 1.27 \begin {gather*} \frac {2 \left (\left (\left (\frac {1}{6} \sqrt {-x+1} \sqrt {-x+1}+\frac {7}{12}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {35}{12}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {35}{2}\right ) \sqrt {-x+1} \sqrt {x+1}}{x+1}+35 \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(7/2)/(1+x)^(3/2),x)

[Out]

-1/6*(((2*x - 9)*(x - 1) + 35)*(x - 1) + 210)*sqrt(-x + 1)/sqrt(x + 1) + 35*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 \frac {{\left (1-x\right )}^{7/2}}{{\left (x+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(7/2)/(x + 1)^(3/2),x)

[Out]

int((1 - x)^(7/2)/(x + 1)^(3/2), x)

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