Optimal. Leaf size=45 \[ -\frac {206 \sqrt {1-2 x}}{1815 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{165 (5 x+3)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {78, 37} \[ -\frac {206 \sqrt {1-2 x}}{1815 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{165 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{165 (3+5 x)^{3/2}}+\frac {103}{165} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{165 (3+5 x)^{3/2}}-\frac {206 \sqrt {1-2 x}}{1815 \sqrt {3+5 x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 0.60 \[ -\frac {2 \sqrt {1-2 x} (103 x+64)}{363 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.97, size = 33, normalized size = 0.73 \[ -\frac {2 \, {\left (103 \, x + 64\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{363 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.96, size = 121, normalized size = 2.69 \[ -\frac {1}{145200} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {828 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {207 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{9075 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 22, normalized size = 0.49 \[ -\frac {2 \left (103 x +64\right ) \sqrt {-2 x +1}}{363 \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.51, size = 48, normalized size = 1.07 \[ -\frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{165 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {206 \, \sqrt {-10 \, x^{2} - x + 3}}{1815 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.50, size = 51, normalized size = 1.13 \[ \frac {\sqrt {5\,x+3}\,\left (\frac {412\,x^2}{9075}+\frac {2\,x}{363}-\frac {128}{9075}\right )}{\frac {6\,x\,\sqrt {1-2\,x}}{5}+\frac {9\,\sqrt {1-2\,x}}{25}+x^2\,\sqrt {1-2\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 x + 2}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________