3.12.36 \(\int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx\) [1136]

Optimal. Leaf size=103 \[ \frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8 x}{63 (1-x)^{3/2} (1+x)^{3/2}}+\frac {16 x}{63 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

1/9/(1-x)^(9/2)/(1+x)^(3/2)+2/21/(1-x)^(7/2)/(1+x)^(3/2)+2/21/(1-x)^(5/2)/(1+x)^(3/2)+8/63*x/(1-x)^(3/2)/(1+x)
^(3/2)+16/63*x/(1-x)^(1/2)/(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 40, 39} \begin {gather*} \frac {16 x}{63 \sqrt {1-x} \sqrt {x+1}}+\frac {8 x}{63 (1-x)^{3/2} (x+1)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (x+1)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (x+1)^{3/2}}+\frac {1}{9 (1-x)^{9/2} (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(11/2)*(1 + x)^(5/2)),x]

[Out]

1/(9*(1 - x)^(9/2)*(1 + x)^(3/2)) + 2/(21*(1 - x)^(7/2)*(1 + x)^(3/2)) + 2/(21*(1 - x)^(5/2)*(1 + x)^(3/2)) +
(8*x)/(63*(1 - x)^(3/2)*(1 + x)^(3/2)) + (16*x)/(63*Sqrt[1 - x]*Sqrt[1 + x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-x)*(a + b*x)^(m + 1)*((c + d*x)^(m
+ 1)/(2*a*c*(m + 1))), x] + Dist[(2*m + 3)/(2*a*c*(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] && ILtQ[m + 3/2, 0]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1}{(1-x)^{11/2} (1+x)^{5/2}} \, dx &=\frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{3} \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {10}{21} \int \frac {1}{(1-x)^{7/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8}{21} \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8 x}{63 (1-x)^{3/2} (1+x)^{3/2}}+\frac {16}{63} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{7/2} (1+x)^{3/2}}+\frac {2}{21 (1-x)^{5/2} (1+x)^{3/2}}+\frac {8 x}{63 (1-x)^{3/2} (1+x)^{3/2}}+\frac {16 x}{63 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 50, normalized size = 0.49 \begin {gather*} \frac {19+6 x-66 x^2+56 x^3+24 x^4-48 x^5+16 x^6}{63 (1-x)^{9/2} (1+x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(11/2)*(1 + x)^(5/2)),x]

[Out]

(19 + 6*x - 66*x^2 + 56*x^3 + 24*x^4 - 48*x^5 + 16*x^6)/(63*(1 - x)^(9/2)*(1 + x)^(3/2))

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 167.48, size = 495, normalized size = 4.81 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-19-6 x+66 x^2-56 x^3-24 x^4+48 x^5-16 x^6\right ) \sqrt {\frac {1-x}{1+x}}}{63 \left (-1+4 x-5 x^2+5 x^4-4 x^5+x^6\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-630 I \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{-2016-2016 x-5040 \left (1+x\right )^3-630 \left (1+x\right )^5+63 \left (1+x\right )^6+2520 \left (1+x\right )^4+5040 \left (1+x\right )^2}-\frac {504 I \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}}{-2016-2016 x-5040 \left (1+x\right )^3-630 \left (1+x\right )^5+63 \left (1+x\right )^6+2520 \left (1+x\right )^4+5040 \left (1+x\right )^2}-\frac {16 I \left (1+x\right )^6 \sqrt {1-\frac {2}{1+x}}}{-2016-2016 x-5040 \left (1+x\right )^3-630 \left (1+x\right )^5+63 \left (1+x\right )^6+2520 \left (1+x\right )^4+5040 \left (1+x\right )^2}+\frac {I 21 \sqrt {1-\frac {2}{1+x}}}{-2016-2016 x-5040 \left (1+x\right )^3-630 \left (1+x\right )^5+63 \left (1+x\right )^6+2520 \left (1+x\right )^4+5040 \left (1+x\right )^2}+\frac {I 126 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{-2016-2016 x-5040 \left (1+x\right )^3-630 \left (1+x\right )^5+63 \left (1+x\right )^6+2520 \left (1+x\right )^4+5040 \left (1+x\right )^2}+\frac {I 144 \left (1+x\right )^5 \sqrt {1-\frac {2}{1+x}}}{-2016-2016 x-5040 \left (1+x\right )^3-630 \left (1+x\right )^5+63 \left (1+x\right )^6+2520 \left (1+x\right )^4+5040 \left (1+x\right )^2}+\frac {I 840 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}}{-2016-2016 x-5040 \left (1+x\right )^3-630 \left (1+x\right )^5+63 \left (1+x\right )^6+2520 \left (1+x\right )^4+5040 \left (1+x\right )^2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((1 - x)^(11/2)*(1 + x)^(5/2)),x]')

[Out]

Piecewise[{{(-19 - 6 x + 66 x ^ 2 - 56 x ^ 3 - 24 x ^ 4 + 48 x ^ 5 - 16 x ^ 6) Sqrt[(1 - x) / (1 + x)] / (63 (
-1 + 4 x - 5 x ^ 2 + 5 x ^ 4 - 4 x ^ 5 + x ^ 6)), 1 / Abs[1 + x] > 1 / 2}}, -630 I (1 + x) ^ 2 Sqrt[1 - 2 / (1
+ x)] / (-2016 - 2016 x - 5040 (1 + x) ^ 3 - 630 (1 + x) ^ 5 + 63 (1 + x) ^ 6 + 2520 (1 + x) ^ 4 + 5040 (1 +
x) ^ 2) - 504 I (1 + x) ^ 4 Sqrt[1 - 2 / (1 + x)] / (-2016 - 2016 x - 5040 (1 + x) ^ 3 - 630 (1 + x) ^ 5 + 63
(1 + x) ^ 6 + 2520 (1 + x) ^ 4 + 5040 (1 + x) ^ 2) - 16 I (1 + x) ^ 6 Sqrt[1 - 2 / (1 + x)] / (-2016 - 2016 x
- 5040 (1 + x) ^ 3 - 630 (1 + x) ^ 5 + 63 (1 + x) ^ 6 + 2520 (1 + x) ^ 4 + 5040 (1 + x) ^ 2) + I 21 Sqrt[1 - 2
/ (1 + x)] / (-2016 - 2016 x - 5040 (1 + x) ^ 3 - 630 (1 + x) ^ 5 + 63 (1 + x) ^ 6 + 2520 (1 + x) ^ 4 + 5040
(1 + x) ^ 2) + I 126 (1 + x) Sqrt[1 - 2 / (1 + x)] / (-2016 - 2016 x - 5040 (1 + x) ^ 3 - 630 (1 + x) ^ 5 + 63
(1 + x) ^ 6 + 2520 (1 + x) ^ 4 + 5040 (1 + x) ^ 2) + I 144 (1 + x) ^ 5 Sqrt[1 - 2 / (1 + x)] / (-2016 - 2016
x - 5040 (1 + x) ^ 3 - 630 (1 + x) ^ 5 + 63 (1 + x) ^ 6 + 2520 (1 + x) ^ 4 + 5040 (1 + x) ^ 2) + I 840 (1 + x)
^ 3 Sqrt[1 - 2 / (1 + x)] / (-2016 - 2016 x - 5040 (1 + x) ^ 3 - 630 (1 + x) ^ 5 + 63 (1 + x) ^ 6 + 2520 (1 +
x) ^ 4 + 5040 (1 + x) ^ 2)]

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Maple [A]
time = 0.16, size = 100, normalized size = 0.97

method	result	size
gosper	\(\frac {16 x^{6}-48 x^{5}+24 x^{4}+56 x^{3}-66 x^{2}+6 x +19}{63 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {9}{2}}}\)	\(45\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (16 x^{6}-48 x^{5}+24 x^{4}+56 x^{3}-66 x^{2}+6 x +19\right )}{63 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \left (-1+x \right )^{4} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(71\)
default	\(\frac {1}{9 \left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {2}{21 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {2}{21 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {8}{63 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {8}{21 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {16 \sqrt {1-x}}{63 \left (1+x \right )^{\frac {3}{2}}}-\frac {16 \sqrt {1-x}}{63 \sqrt {1+x}}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(11/2)/(1+x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/9/(1-x)^(9/2)/(1+x)^(3/2)+2/21/(1-x)^(7/2)/(1+x)^(3/2)+2/21/(1-x)^(5/2)/(1+x)^(3/2)+8/63/(1-x)^(3/2)/(1+x)^(
3/2)+8/21/(1-x)^(1/2)/(1+x)^(3/2)-16/63*(1-x)^(1/2)/(1+x)^(3/2)-16/63*(1-x)^(1/2)/(1+x)^(1/2)

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Maxima [A]
time = 0.26, size = 146, normalized size = 1.42 \begin {gather*} \frac {16 \, x}{63 \, \sqrt {-x^{2} + 1}} + \frac {8 \, x}{63 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {1}{9 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{3} - 3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} + 3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} + \frac {2}{21 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} - \frac {2}{21 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

16/63*x/sqrt(-x^2 + 1) + 8/63*x/(-x^2 + 1)^(3/2) - 1/9/((-x^2 + 1)^(3/2)*x^3 - 3*(-x^2 + 1)^(3/2)*x^2 + 3*(-x^
2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2)) + 2/21/((-x^2 + 1)^(3/2)*x^2 - 2*(-x^2 + 1)^(3/2)*x + (-x^2 + 1)^(3/2)) - 2
/21/((-x^2 + 1)^(3/2)*x - (-x^2 + 1)^(3/2))

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Fricas [A]
time = 0.30, size = 114, normalized size = 1.11 \begin {gather*} \frac {19 \, x^{7} - 57 \, x^{6} + 19 \, x^{5} + 95 \, x^{4} - 95 \, x^{3} - 19 \, x^{2} - {\left (16 \, x^{6} - 48 \, x^{5} + 24 \, x^{4} + 56 \, x^{3} - 66 \, x^{2} + 6 \, x + 19\right )} \sqrt {x + 1} \sqrt {-x + 1} + 57 \, x - 19}{63 \, {\left (x^{7} - 3 \, x^{6} + x^{5} + 5 \, x^{4} - 5 \, x^{3} - x^{2} + 3 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

1/63*(19*x^7 - 57*x^6 + 19*x^5 + 95*x^4 - 95*x^3 - 19*x^2 - (16*x^6 - 48*x^5 + 24*x^4 + 56*x^3 - 66*x^2 + 6*x
+ 19)*sqrt(x + 1)*sqrt(-x + 1) + 57*x - 19)/(x^7 - 3*x^6 + x^5 + 5*x^4 - 5*x^3 - x^2 + 3*x - 1)

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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/(1-x)**(11/2)/(1+x)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (73) = 146\).
time = 0.02, size = 414, normalized size = 4.02 \begin {gather*} 2 \left (\frac {\frac {1}{9}\cdot 340282366920938463463374607431768211456 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{9}+\frac {1}{7}\cdot 5784800237655953878877368326340059594752 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}+9527906273786276976974489008089509920768 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}+\frac {1}{3}\cdot 254531210456861970670604206358962622169088 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}-\frac {499194232273016725900770549102403966205952 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{22300745198530623141535718272648361505980416}+\frac {-184842 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{8}-15708 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{6}-1764 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}-153 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}-7}{4128768 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{9}}+\frac {2 \left (\frac {17}{768} \sqrt {-x+1} \sqrt {-x+1}-\frac {3}{64}\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(11/2)/(1+x)^(5/2),x)

[Out]

-1/294912*(sqrt(2) - sqrt(x + 1))^9/(-x + 1)^(9/2) - 17/229376*(sqrt(2) - sqrt(x + 1))^7/(-x + 1)^(7/2) - 7/81
92*(sqrt(2) - sqrt(x + 1))^5/(-x + 1)^(5/2) - 187/24576*(sqrt(2) - sqrt(x + 1))^3/(-x + 1)^(3/2) - 1467/16384*
(sqrt(2) - sqrt(x + 1))/sqrt(-x + 1) - 1/192*(17*x + 19)*sqrt(-x + 1)/(x + 1)^(3/2) + 1/2064384*(184842*(sqrt(
2) - sqrt(x + 1))^8/(x - 1)^4 - 15708*(sqrt(2) - sqrt(x + 1))^6/(x - 1)^3 + 1764*(sqrt(2) - sqrt(x + 1))^4/(x
- 1)^2 - 153*(sqrt(2) - sqrt(x + 1))^2/(x - 1) + 7)*(-x + 1)^(9/2)/(sqrt(2) - sqrt(x + 1))^9

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Mupad [B]
time = 0.42, size = 99, normalized size = 0.96 \begin {gather*} -\frac {6\,x\,\sqrt {1-x}+19\,\sqrt {1-x}-66\,x^2\,\sqrt {1-x}+56\,x^3\,\sqrt {1-x}+24\,x^4\,\sqrt {1-x}-48\,x^5\,\sqrt {1-x}+16\,x^6\,\sqrt {1-x}}{\left (63\,x+63\right )\,{\left (x-1\right )}^5\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - x)^(11/2)*(x + 1)^(5/2)),x)

[Out]

-(6*x*(1 - x)^(1/2) + 19*(1 - x)^(1/2) - 66*x^2*(1 - x)^(1/2) + 56*x^3*(1 - x)^(1/2) + 24*x^4*(1 - x)^(1/2) -
48*x^5*(1 - x)^(1/2) + 16*x^6*(1 - x)^(1/2))/((63*x + 63)*(x - 1)^5*(x + 1)^(1/2))

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