Optimal. Leaf size=110 \[ \frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {5 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 43, 44, 65,
212} \begin {gather*} \frac {(1-2 x)^{3/2}}{84 (3 x+2)^4}+\frac {15 \sqrt {1-2 x}}{2744 (3 x+2)}+\frac {5 \sqrt {1-2 x}}{392 (3 x+2)^2}-\frac {5 \sqrt {1-2 x}}{28 (3 x+2)^3}+\frac {5 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 44
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}+\frac {45}{28} \int \frac {\sqrt {1-2 x}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}-\frac {5}{28} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}-\frac {15}{392} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}-\frac {15 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2744}\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {15 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2744}\\ &=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {5 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 65, normalized size = 0.59 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \left (-2062-1726 x+3375 x^2+1215 x^3\right )}{2 (2+3 x)^4}+15 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 66, normalized size = 0.60
method | result | size |
risch | \(-\frac {2430 x^{4}+5535 x^{3}-6827 x^{2}-2398 x +2062}{8232 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {5 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\) | \(56\) |
derivativedivides | \(-\frac {1296 \left (\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{21952}-\frac {55 \left (1-2 x \right )^{\frac {5}{2}}}{28224}+\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {5 \sqrt {1-2 x}}{1728}\right )}{\left (-4-6 x \right )^{4}}+\frac {5 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\) | \(66\) |
default | \(-\frac {1296 \left (\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{21952}-\frac {55 \left (1-2 x \right )^{\frac {5}{2}}}{28224}+\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {5 \sqrt {1-2 x}}{1728}\right )}{\left (-4-6 x \right )^{4}}+\frac {5 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\) | \(66\) |
trager | \(\frac {\left (1215 x^{3}+3375 x^{2}-1726 x -2062\right ) \sqrt {1-2 x}}{8232 \left (2+3 x \right )^{4}}+\frac {5 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{19208}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.85, size = 110, normalized size = 1.00 \begin {gather*} -\frac {5}{19208} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1215 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10395 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 10241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 15435 \, \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.14, size = 106, normalized size = 0.96 \begin {gather*} \frac {15 \, \sqrt {7} \sqrt {3} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 7 \, {\left (1215 \, x^{3} + 3375 \, x^{2} - 1726 \, x - 2062\right )} \sqrt {-2 \, x + 1}}{57624 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [A]
time = 1.05, size = 100, normalized size = 0.91 \begin {gather*} -\frac {5}{19208} \, \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 {1215 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 10395 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 10241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 15435 \, \sqrt {-2 \, x + 1}}{65856 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.08, size = 90, normalized size = 0.82 \begin {gather*} \frac {5\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9604}-\frac {\frac {5\,\sqrt {1-2\,x}}{108}+\frac {209\,{\left (1-2\,x\right )}^{3/2}}{6804}-\frac {55\,{\left (1-2\,x\right )}^{5/2}}{1764}+\frac {5\,{\left (1-2\,x\right )}^{7/2}}{1372}}{\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________