Optimal. Leaf size=73 \[ \frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {10}{189} \sqrt {1-2 x} (214+95 x)-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \]
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Rubi [A]
time = 0.01, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 152, 65,
212} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}-\frac {10}{189} \sqrt {1-2 x} (95 x+214)-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 152
Rule 212
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^2} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {1}{21} \int \frac {(-92-190 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {10}{189} \sqrt {1-2 x} (214+95 x)+\frac {104}{189} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {10}{189} \sqrt {1-2 x} (214+95 x)-\frac {104}{189} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{21 (2+3 x)}-\frac {10}{189} \sqrt {1-2 x} (214+95 x)-\frac {208 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{189 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 58, normalized size = 0.79 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (4199+8050 x+2625 x^2\right )}{2+3 x}-208 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 54, normalized size = 0.74
method | result | size |
risch | \(\frac {5250 x^{3}+13475 x^{2}+348 x -4199}{189 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {208 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(51\) |
derivativedivides | \(\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{54}-\frac {725 \sqrt {1-2 x}}{54}-\frac {2 \sqrt {1-2 x}}{567 \left (-\frac {4}{3}-2 x \right )}-\frac {208 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(54\) |
default | \(\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{54}-\frac {725 \sqrt {1-2 x}}{54}-\frac {2 \sqrt {1-2 x}}{567 \left (-\frac {4}{3}-2 x \right )}-\frac {208 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3969}\) | \(54\) |
trager | \(-\frac {\left (2625 x^{2}+8050 x +4199\right ) \sqrt {1-2 x}}{189 \left (2+3 x \right )}-\frac {104 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{3969}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 71, normalized size = 0.97 \begin {gather*} \frac {125}{54} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {104}{3969} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {725}{54} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{189 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 64, normalized size = 0.88 \begin {gather*} \frac {104 \, \sqrt {21} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2625 \, x^{2} + 8050 \, x + 4199\right )} \sqrt {-2 \, x + 1}}{3969 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A]
time = 1.16, size = 74, normalized size = 1.01 \begin {gather*} \frac {125}{54} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {104}{3969} \, \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 {725}{54} \, \sqrt {-2 \, x + 1} + \frac {\sqrt {-2 \, x + 1}}{189 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 55, normalized size = 0.75 \begin {gather*} \frac {2\,\sqrt {1-2\,x}}{567\,\left (2\,x+\frac {4}{3}\right )}-\frac {725\,\sqrt {1-2\,x}}{54}+\frac {125\,{\left (1-2\,x\right )}^{3/2}}{54}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,208{}\mathrm {i}}{3969} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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