Optimal. Leaf size=84 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, 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 = {100, 152, 56,
222} \begin {gather*} \frac {2493 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}}-\frac {2 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (300 x+979)}{4400} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 56
Rule 100
Rule 152
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {2}{55} \int \frac {\left (-39-\frac {75 x}{2}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{400 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.13, size = 68, normalized size = 0.81 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (9451+19305 x+5940 x^2\right )-27423 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{44000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 99, normalized size = 1.18
method | result | size |
default | \(\frac {\left (137115 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -118800 x^{2} \sqrt {-10 x^{2}-x +3}+82269 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-386100 x \sqrt {-10 x^{2}-x +3}-189020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{88000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.60, size = 65, normalized size = 0.77 \begin {gather*} \frac {2493}{8000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {27}{100} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1431}{2000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{1375 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.72, size = 81, normalized size = 0.96 \begin {gather*} -\frac {27423 \, \sqrt {10} {\left (5 \, x + 3\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 (5940 \, x^{2} + 19305 \, x + 9451\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{88000 \, {\left (5 \, x + 3\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 {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.55, size = 111, normalized size = 1.32 \begin {gather*} -\frac {27}{10000} \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 41 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {2493}{4000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{13750 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{6875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \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 {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________