Optimal. Leaf size=81 \[ \frac {(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac {(1+x)^{5/2}}{33 (1-x)^{9/2}}+\frac {2 (1+x)^{5/2}}{231 (1-x)^{7/2}}+\frac {2 (1+x)^{5/2}}{1155 (1-x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37}
\begin {gather*} \frac {2 (x+1)^{5/2}}{1155 (1-x)^{5/2}}+\frac {2 (x+1)^{5/2}}{231 (1-x)^{7/2}}+\frac {(x+1)^{5/2}}{33 (1-x)^{9/2}}+\frac {(x+1)^{5/2}}{11 (1-x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2}}{(1-x)^{13/2}} \, dx &=\frac {(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac {3}{11} \int \frac {(1+x)^{3/2}}{(1-x)^{11/2}} \, dx\\ &=\frac {(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac {(1+x)^{5/2}}{33 (1-x)^{9/2}}+\frac {2}{33} \int \frac {(1+x)^{3/2}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac {(1+x)^{5/2}}{33 (1-x)^{9/2}}+\frac {2 (1+x)^{5/2}}{231 (1-x)^{7/2}}+\frac {2}{231} \int \frac {(1+x)^{3/2}}{(1-x)^{7/2}} \, dx\\ &=\frac {(1+x)^{5/2}}{11 (1-x)^{11/2}}+\frac {(1+x)^{5/2}}{33 (1-x)^{9/2}}+\frac {2 (1+x)^{5/2}}{231 (1-x)^{7/2}}+\frac {2 (1+x)^{5/2}}{1155 (1-x)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 35, normalized size = 0.43 \begin {gather*} \frac {(1+x)^{5/2} \left (152-61 x+16 x^2-2 x^3\right )}{1155 (1-x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 107.60, size = 975, normalized size = 12.04 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (152+243 x+46 x^2-31 x^3+12 x^4-2 x^5\right ) \sqrt {1+x}}{1155 \sqrt {-1+x} \left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right )},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-3564 \left (1+x\right )^{\frac {7}{2}}}{-2069760 \left (1+x\right )^3 \sqrt {1-x}-1182720 \left (1+x\right ) \sqrt {1-x}-517440 \left (1+x\right )^5 \sqrt {1-x}-18480 \left (1+x\right )^7 \sqrt {1-x}+1155 \left (1+x\right )^8 \sqrt {1-x}+129360 \left (1+x\right )^6 \sqrt {1-x}+295680 \sqrt {1-x}+1293600 \left (1+x\right )^4 \sqrt {1-x}+2069760 \left (1+x\right )^2 \sqrt {1-x}}-\frac {1105 \left (1+x\right )^{\frac {11}{2}}}{-2069760 \left (1+x\right )^3 \sqrt {1-x}-1182720 \left (1+x\right ) \sqrt {1-x}-517440 \left (1+x\right )^5 \sqrt {1-x}-18480 \left (1+x\right )^7 \sqrt {1-x}+1155 \left (1+x\right )^8 \sqrt {1-x}+129360 \left (1+x\right )^6 \sqrt {1-x}+295680 \sqrt {1-x}+1293600 \left (1+x\right )^4 \sqrt {1-x}+2069760 \left (1+x\right )^2 \sqrt {1-x}}-\frac {34 \left (1+x\right )^{\frac {15}{2}}}{-2069760 \left (1+x\right )^3 \sqrt {1-x}-1182720 \left (1+x\right ) \sqrt {1-x}-517440 \left (1+x\right )^5 \sqrt {1-x}-18480 \left (1+x\right )^7 \sqrt {1-x}+1155 \left (1+x\right )^8 \sqrt {1-x}+129360 \left (1+x\right )^6 \sqrt {1-x}+295680 \sqrt {1-x}+1293600 \left (1+x\right )^4 \sqrt {1-x}+2069760 \left (1+x\right )^2 \sqrt {1-x}}+\frac {2 \left (1+x\right )^{\frac {17}{2}}}{-2069760 \left (1+x\right )^3 \sqrt {1-x}-1182720 \left (1+x\right ) \sqrt {1-x}-517440 \left (1+x\right )^5 \sqrt {1-x}-18480 \left (1+x\right )^7 \sqrt {1-x}+1155 \left (1+x\right )^8 \sqrt {1-x}+129360 \left (1+x\right )^6 \sqrt {1-x}+295680 \sqrt {1-x}+1293600 \left (1+x\right )^4 \sqrt {1-x}+2069760 \left (1+x\right )^2 \sqrt {1-x}}+\frac {255 \left (1+x\right )^{\frac {13}{2}}}{-2069760 \left (1+x\right )^3 \sqrt {1-x}-1182720 \left (1+x\right ) \sqrt {1-x}-517440 \left (1+x\right )^5 \sqrt {1-x}-18480 \left (1+x\right )^7 \sqrt {1-x}+1155 \left (1+x\right )^8 \sqrt {1-x}+129360 \left (1+x\right )^6 \sqrt {1-x}+295680 \sqrt {1-x}+1293600 \left (1+x\right )^4 \sqrt {1-x}+2069760 \left (1+x\right )^2 \sqrt {1-x}}+\frac {1848 \left (1+x\right )^{\frac {5}{2}}}{-2069760 \left (1+x\right )^3 \sqrt {1-x}-1182720 \left (1+x\right ) \sqrt {1-x}-517440 \left (1+x\right )^5 \sqrt {1-x}-18480 \left (1+x\right )^7 \sqrt {1-x}+1155 \left (1+x\right )^8 \sqrt {1-x}+129360 \left (1+x\right )^6 \sqrt {1-x}+295680 \sqrt {1-x}+1293600 \left (1+x\right )^4 \sqrt {1-x}+2069760 \left (1+x\right )^2 \sqrt {1-x}}+\frac {2750 \left (1+x\right )^{\frac {9}{2}}}{-2069760 \left (1+x\right )^3 \sqrt {1-x}-1182720 \left (1+x\right ) \sqrt {1-x}-517440 \left (1+x\right )^5 \sqrt {1-x}-18480 \left (1+x\right )^7 \sqrt {1-x}+1155 \left (1+x\right )^8 \sqrt {1-x}+129360 \left (1+x\right )^6 \sqrt {1-x}+295680 \sqrt {1-x}+1293600 \left (1+x\right )^4 \sqrt {1-x}+2069760 \left (1+x\right )^2 \sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 100, normalized size = 1.23
method | result | size |
gosper | \(-\frac {\left (1+x \right )^{\frac {5}{2}} \left (2 x^{3}-16 x^{2}+61 x -152\right )}{1155 \left (1-x \right )^{\frac {11}{2}}}\) | \(30\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{6}-10 x^{5}+19 x^{4}-15 x^{3}-289 x^{2}-395 x -152\right )}{1155 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{5} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(71\) |
default | \(\frac {\left (1+x \right )^{\frac {3}{2}}}{4 \left (1-x \right )^{\frac {11}{2}}}-\frac {3 \sqrt {1+x}}{22 \left (1-x \right )^{\frac {11}{2}}}+\frac {\sqrt {1+x}}{132 \left (1-x \right )^{\frac {9}{2}}}+\frac {\sqrt {1+x}}{231 \left (1-x \right )^{\frac {7}{2}}}+\frac {\sqrt {1+x}}{385 \left (1-x \right )^{\frac {5}{2}}}+\frac {2 \sqrt {1+x}}{1155 \left (1-x \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1+x}}{1155 \sqrt {1-x}}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (57) = 114\).
time = 0.27, size = 218, normalized size = 2.69 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{4 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{22 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{132 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{231 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{385 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{1155 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{1155 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.30, size = 101, normalized size = 1.25 \begin {gather*} \frac {152 \, x^{6} - 912 \, x^{5} + 2280 \, x^{4} - 3040 \, x^{3} + 2280 \, x^{2} - {\left (2 \, x^{5} - 12 \, x^{4} + 31 \, x^{3} - 46 \, x^{2} - 243 \, x - 152\right )} \sqrt {x + 1} \sqrt {-x + 1} - 912 \, x + 152}{1155 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 177.97, size = 1751, normalized size = 21.62
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.02, size = 111, normalized size = 1.37 \begin {gather*} \frac {2 \left (\left (\left (\frac 1{105}-\frac {1}{1155} \sqrt {x+1} \sqrt {x+1}\right ) \sqrt {x+1} \sqrt {x+1}-\frac {3}{70}\right ) \sqrt {x+1} \sqrt {x+1}+\frac 1{10}\right ) \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {-x+1}}{\left (-x+1\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.31, size = 94, normalized size = 1.16 \begin {gather*} \frac {\sqrt {1-x}\,\left (\frac {81\,x\,\sqrt {x+1}}{385}+\frac {152\,\sqrt {x+1}}{1155}+\frac {46\,x^2\,\sqrt {x+1}}{1155}-\frac {31\,x^3\,\sqrt {x+1}}{1155}+\frac {4\,x^4\,\sqrt {x+1}}{385}-\frac {2\,x^5\,\sqrt {x+1}}{1155}\right )}{x^6-6\,x^5+15\,x^4-20\,x^3+15\,x^2-6\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________