Optimal. Leaf size=80 \[ \frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\sqrt {1-2 x} (15676+24825 x)}{66550 (3+5 x)^2}-\frac {7143 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \]
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Rubi [A]
time = 0.01, antiderivative size = 80, 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, 150, 65,
212} \begin {gather*} \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)^2}-\frac {\sqrt {1-2 x} (24825 x+15676)}{66550 (5 x+3)^2}-\frac {7143 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 150
Rule 212
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {1}{11} \int \frac {(-16-3 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^3} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\sqrt {1-2 x} (15676+24825 x)}{66550 (3+5 x)^2}+\frac {7143 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{66550}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\sqrt {1-2 x} (15676+24825 x)}{66550 (3+5 x)^2}-\frac {7143 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{66550}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)^2}-\frac {\sqrt {1-2 x} (15676+24825 x)}{66550 (3+5 x)^2}-\frac {7143 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{33275 \sqrt {55}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 58, normalized size = 0.72 \begin {gather*} \frac {\frac {55 \left (153724+514727 x+430800 x^2\right )}{\sqrt {1-2 x} (3+5 x)^2}-14286 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3660250} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 57, normalized size = 0.71
method | result | size |
risch | \(\frac {430800 x^{2}+514727 x +153724}{66550 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {7143 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}\) | \(46\) |
derivativedivides | \(\frac {\frac {41 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {207 \sqrt {1-2 x}}{3025}}{\left (-6-10 x \right )^{2}}-\frac {7143 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}+\frac {343}{1331 \sqrt {1-2 x}}\) | \(57\) |
default | \(\frac {\frac {41 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {207 \sqrt {1-2 x}}{3025}}{\left (-6-10 x \right )^{2}}-\frac {7143 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1830125}+\frac {343}{1331 \sqrt {1-2 x}}\) | \(57\) |
trager | \(-\frac {\left (430800 x^{2}+514727 x +153724\right ) \sqrt {1-2 x}}{66550 \left (3+5 x \right )^{2} \left (-1+2 x \right )}+\frac {7143 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{3660250}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 83, normalized size = 1.04 \begin {gather*} \frac {7143}{3660250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (107700 \, {\left (2 \, x - 1\right )}^{2} + 945527 \, x + 46024\right )}}{33275 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 84, normalized size = 1.05 \begin {gather*} \frac {7143 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (430800 \, x^{2} + 514727 \, x + 153724\right )} \sqrt {-2 \, x + 1}}{3660250 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\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 = 2.12, size = 77, normalized size = 0.96 \begin {gather*} \frac {7143}{3660250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {343}{1331 \, \sqrt {-2 \, x + 1}} + \frac {1025 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2277 \, \sqrt {-2 \, x + 1}}{133100 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 62, normalized size = 0.78 \begin {gather*} \frac {\frac {171914\,x}{75625}+\frac {8616\,{\left (2\,x-1\right )}^2}{33275}+\frac {8368}{75625}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}-\frac {7143\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1830125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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