Optimal. Leaf size=108 \[ -\frac {95 \sqrt {1-2 x}}{8232 (3 x+2)}-\frac {95 \sqrt {1-2 x}}{3528 (3 x+2)^2}-\frac {19 \sqrt {1-2 x}}{252 (3 x+2)^3}+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac {95 \sqrt {1-2 x}}{8232 (3 x+2)}-\frac {95 \sqrt {1-2 x}}{3528 (3 x+2)^2}-\frac {19 \sqrt {1-2 x}}{252 (3 x+2)^3}+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}+\frac {19}{12} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}+\frac {95}{252} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}+\frac {95 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1176}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}+\frac {95 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{8232}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{8232}\\ &=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 42, normalized size = 0.39 \[ \frac {\sqrt {1-2 x} \left (\frac {343}{(3 x+2)^4}-304 \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{28812} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.94, size = 99, normalized size = 0.92 \[ \frac {95 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2565 \, x^{3} + 7125 \, x^{2} + 7942 \, x + 2790\right )} \sqrt {-2 \, x + 1}}{172872 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.01, size = 100, normalized size = 0.93 \[ \frac {95}{172872} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2565 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 21945 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 67963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 70903 \, \sqrt {-2 \, x + 1}}{65856 \, {\left (3 \, x + 2\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 66, normalized size = 0.61 \[ -\frac {95 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{86436}-\frac {1296 \left (-\frac {95 \left (-2 x +1\right )^{\frac {7}{2}}}{197568}+\frac {1045 \left (-2 x +1\right )^{\frac {5}{2}}}{254016}-\frac {1387 \left (-2 x +1\right )^{\frac {3}{2}}}{108864}+\frac {1447 \sqrt {-2 x +1}}{108864}\right )}{\left (-6 x -4\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.19, size = 110, normalized size = 1.02 \[ \frac {95}{172872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2565 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 21945 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 67963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 70903 \, \sqrt {-2 \, x + 1}}{4116 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.20, size = 90, normalized size = 0.83 \[ -\frac {95\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{86436}-\frac {\frac {1447\,\sqrt {1-2\,x}}{6804}-\frac {1387\,{\left (1-2\,x\right )}^{3/2}}{6804}+\frac {1045\,{\left (1-2\,x\right )}^{5/2}}{15876}-\frac {95\,{\left (1-2\,x\right )}^{7/2}}{12348}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]
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]
________________________________________________________________________________________