Optimal. Leaf size=113 \[ -\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (168157+69780 x)}{3520}+\frac {126513 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320 \sqrt {10}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {99, 155, 152,
56, 222} \begin {gather*} \frac {126513 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320 \sqrt {10}}+\frac {\sqrt {5 x+3} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac {233 \sqrt {5 x+3} (3 x+2)^2}{66 \sqrt {1-2 x}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (69780 x+168157)}{3520} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 56
Rule 99
Rule 152
Rule 155
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^2 \left (32+\frac {105 x}{2}\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {5349}{2}-\frac {17445 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (168157+69780 x)}{3520}+\frac {126513}{640} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (168157+69780 x)}{3520}+\frac {126513 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{320 \sqrt {5}}\\ &=-\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (168157+69780 x)}{3520}+\frac {126513 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320 \sqrt {10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 78, normalized size = 0.69 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (625431-1786144 x+431244 x^2+71280 x^3\right )+4174929 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{105600 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 137, normalized size = 1.21
method | result | size |
default | \(\frac {\left (16699716 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-1425600 x^{3} \sqrt {-10 x^{2}-x +3}-16699716 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -8624880 x^{2} \sqrt {-10 x^{2}-x +3}+4174929 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+35722880 x \sqrt {-10 x^{2}-x +3}-12508620 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{211200 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(137\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [A]
time = 0.77, size = 96, normalized size = 0.85 \begin {gather*} -\frac {4174929 \, \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 (71280 \, x^{3} + 431244 \, x^{2} - 1786144 \, x + 625431\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{211200 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{3} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.56, size = 84, normalized size = 0.74 \begin {gather*} \frac {126513}{3200} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (891 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 85 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 2783318 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 45924219 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1320000 \, {\left (2 \, x - 1\right )}^{2}} \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 (3\,x+2\right )}^3\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________