Optimal. Leaf size=84 \[ \frac {7 \sqrt {5 x+3} (3 x+2)^2}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (10380 x+25003)}{8800}-\frac {56421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 147, 54, 216} \[ \frac {7 \sqrt {5 x+3} (3 x+2)^2}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (10380 x+25003)}{8800}-\frac {56421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 54
Rule 98
Rule 147
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(2+3 x) \left (159+\frac {519 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (25003+10380 x)}{8800}-\frac {56421 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1600}\\ &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (25003+10380 x)}{8800}-\frac {56421 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{800 \sqrt {5}}\\ &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (25003+10380 x)}{8800}-\frac {56421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 78, normalized size = 0.93 \[ \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (11880 x^2+51678 x-97409\right )-620631 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{88000 \sqrt {-(1-2 x)^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 81, normalized size = 0.96 \[ \frac {620631 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (11880 \, x^{2} + 51678 \, x - 97409\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{176000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.01, size = 71, normalized size = 0.85 \[ -\frac {56421}{8000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (594 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 63 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 620695 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{220000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 106, normalized size = 1.26 \[ -\frac {\left (-237600 \sqrt {-10 x^{2}-x +3}\, x^{2}+1241262 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1033560 \sqrt {-10 x^{2}-x +3}\, x -620631 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1948180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{176000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.18, size = 65, normalized size = 0.77 \[ -\frac {56421}{16000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {27}{40} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {2619}{800} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{44 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^3}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \]
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]
________________________________________________________________________________________