Optimal. Leaf size=68 \[ -\frac {1227 \sqrt {1-2 x}}{1210 (5 x+3)}+\frac {49}{22 \sqrt {1-2 x} (5 x+3)}-\frac {138 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \]
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Rubi [A] time = 0.02, antiderivative size = 68, 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 = {89, 78, 63, 206} \begin {gather*} -\frac {1227 \sqrt {1-2 x}}{1210 (5 x+3)}+\frac {49}{22 \sqrt {1-2 x} (5 x+3)}-\frac {138 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac {49}{22 \sqrt {1-2 x} (3+5 x)}-\frac {1}{22} \int \frac {-186+99 x}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=\frac {49}{22 \sqrt {1-2 x} (3+5 x)}-\frac {1227 \sqrt {1-2 x}}{1210 (3+5 x)}+\frac {69}{605} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {49}{22 \sqrt {1-2 x} (3+5 x)}-\frac {1227 \sqrt {1-2 x}}{1210 (3+5 x)}-\frac {69}{605} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {49}{22 \sqrt {1-2 x} (3+5 x)}-\frac {1227 \sqrt {1-2 x}}{1210 (3+5 x)}-\frac {138 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 53, normalized size = 0.78 \begin {gather*} \frac {1227 x+734}{605 \sqrt {1-2 x} (5 x+3)}-\frac {138 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 61, normalized size = 0.90 \begin {gather*} \frac {1227 (1-2 x)-2695}{605 (5 (1-2 x)-11) \sqrt {1-2 x}}-\frac {138 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{605 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 65, normalized size = 0.96 \begin {gather*} \frac {69 \, \sqrt {55} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (1227 \, x + 734\right )} \sqrt {-2 \, x + 1}}{33275 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.40, size = 68, normalized size = 1.00 \begin {gather*} \frac {69}{33275} \, \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 {2 \, {\left (1227 \, x + 734\right )}}{605 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.66 \begin {gather*} -\frac {138 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{33275}+\frac {49}{121 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{3025 \left (-2 x -\frac {6}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 65, normalized size = 0.96 \begin {gather*} \frac {69}{33275} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (1227 \, x + 734\right )}}{605 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 46, normalized size = 0.68 \begin {gather*} \frac {\frac {2454\,x}{3025}+\frac {1468}{3025}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {138\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{33275} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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