Optimal. Leaf size=142 \[ -\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (853+1140 x)}{10000}+\frac {13153 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10000 \sqrt {10}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 155, 158,
152, 56, 222} \begin {gather*} \frac {13153 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10000 \sqrt {10}}-\frac {128 \sqrt {1-2 x} (3 x+2)^3}{25 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac {378}{125} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (1140 x+853)}{10000} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 56
Rule 99
Rule 152
Rule 155
Rule 158
Rule 222
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {(3-27 x) \sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (\frac {1029}{2}-1701 x\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {2 \int \frac {(2+3 x) \left (-1953+\frac {17955 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1125}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (853+1140 x)}{10000}+\frac {13153 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{20000}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (853+1140 x)}{10000}+\frac {13153 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{10000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (853+1140 x)}{10000}+\frac {13153 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10000 \sqrt {10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 74, normalized size = 0.52 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \left (31171+129910 x+118395 x^2-83700 x^3-108000 x^4\right )}{(3+5 x)^{3/2}}-39459 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{300000} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 147, normalized size = 1.04
method | result | size |
default | \(\frac {\left (-2160000 x^{4} \sqrt {-10 x^{2}-x +3}+986475 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-1674000 x^{3} \sqrt {-10 x^{2}-x +3}+1183770 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +2367900 x^{2} \sqrt {-10 x^{2}-x +3}+355131 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2598200 x \sqrt {-10 x^{2}-x +3}+623420 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{600000 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.53, size = 211, normalized size = 1.49 \begin {gather*} -\frac {35937}{1000000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {7457}{250000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {9}{625} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {297}{2500} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {6831}{50000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {891}{12500} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1875 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1250 \, {\left (5 \, x + 3\right )}} - \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{9375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {877 \, \sqrt {-10 \, x^{2} - x + 3}}{9375 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.40, size = 101, normalized size = 0.71 \begin {gather*} -\frac {39459 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (108000 \, x^{4} + 83700 \, x^{3} - 118395 \, x^{2} - 129910 \, x - 31171\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{600000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.63, size = 184, normalized size = 1.30 \begin {gather*} -\frac {9}{250000} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 65 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 265 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {1}{750000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2316 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {13153}{100000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {579 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{46875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3}{{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________