Optimal. Leaf size=116 \[ -\frac {938 (5 x+3)^{5/2}}{363 \sqrt {1-2 x}}+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {40787 \sqrt {1-2 x} (5 x+3)^{3/2}}{5808}-\frac {40787}{704} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {40787 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64 \sqrt {10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {89, 78, 50, 54, 216} \[ -\frac {938 (5 x+3)^{5/2}}{363 \sqrt {1-2 x}}+\frac {49 (5 x+3)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {40787 \sqrt {1-2 x} (5 x+3)^{3/2}}{5808}-\frac {40787}{704} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {40787 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64 \sqrt {10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 54
Rule 78
Rule 89
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {(3+5 x)^{3/2} \left (\frac {1579}{2}+297 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{1452}\\ &=-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787}{352} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {40787}{704} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787}{128} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {40787}{704} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{64 \sqrt {5}}\\ &=-\frac {40787}{704} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {40787 \sqrt {1-2 x} (3+5 x)^{3/2}}{5808}+\frac {49 (3+5 x)^{5/2}}{66 (1-2 x)^{3/2}}-\frac {938 (3+5 x)^{5/2}}{363 \sqrt {1-2 x}}+\frac {40787 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64 \sqrt {10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 90, normalized size = 0.78 \[ \frac {10 \sqrt {2 x-1} \sqrt {5 x+3} \left (2160 x^3+12780 x^2-52256 x+18351\right )+122361 \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1920 \sqrt {1-2 x} (2 x-1)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.24, size = 96, normalized size = 0.83 \[ -\frac {122361 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2160 \, x^{3} + 12780 \, x^{2} - 52256 \, x + 18351\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3840 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.08, size = 84, normalized size = 0.72 \[ \frac {40787}{640} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 247 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 81574 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1345971 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{24000 \, {\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 137, normalized size = 1.18 \[ \frac {\left (-43200 \sqrt {-10 x^{2}-x +3}\, x^{3}+489444 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-255600 \sqrt {-10 x^{2}-x +3}\, x^{2}-489444 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1045120 \sqrt {-10 x^{2}-x +3}\, x +122361 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-367020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{3840 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 154, normalized size = 1.33 \[ \frac {40787}{1280} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {297}{64} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {49 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{24 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {21 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (2 \, x - 1\right )}} + \frac {539 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {5873 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________