Optimal. Leaf size=80 \[ \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)}+\frac {18 \sqrt {1-2 x} (935 x+559)}{3025 (5 x+3)}-\frac {204 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \]
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Rubi [A] time = 0.02, 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 = {98, 146, 63, 206} \begin {gather*} \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} (5 x+3)}+\frac {18 \sqrt {1-2 x} (935 x+559)}{3025 (5 x+3)}-\frac {204 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 146
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^2} \, dx &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{11} \int \frac {(2+3 x) (54+102 x)}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)}+\frac {18 \sqrt {1-2 x} (559+935 x)}{3025 (3+5 x)}+\frac {102 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3025}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)}+\frac {18 \sqrt {1-2 x} (559+935 x)}{3025 (3+5 x)}-\frac {102 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3025}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} (3+5 x)}+\frac {18 \sqrt {1-2 x} (559+935 x)}{3025 (3+5 x)}-\frac {204 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 79, normalized size = 0.99 \begin {gather*} \frac {55 \sqrt {2 x-1} \left (-16335 x^2+19806 x+17762\right )+204 \sqrt {55} \left (10 x^2+x-3\right ) \tan ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{166375 \sqrt {-(1-2 x)^2} (5 x+3)} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.12, size = 70, normalized size = 0.88 \begin {gather*} \frac {16335 (1-2 x)^2+6942 (1-2 x)-94325}{6050 (5 (1-2 x)-11) \sqrt {1-2 x}}-\frac {204 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3025 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 70, normalized size = 0.88 \begin {gather*} \frac {102 \, \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 (16335 \, x^{2} - 19806 \, x - 17762\right )} \sqrt {-2 \, x + 1}}{166375 \, {\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 = 1.32, size = 77, normalized size = 0.96 \begin {gather*} \frac {102}{166375} \, \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 {27}{50} \, \sqrt {-2 \, x + 1} - \frac {42879 \, x + 25723}{3025 \, {\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 = 54, normalized size = 0.68 \begin {gather*} -\frac {204 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{166375}+\frac {27 \sqrt {-2 x +1}}{50}+\frac {343}{242 \sqrt {-2 x +1}}+\frac {2 \sqrt {-2 x +1}}{15125 \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 = 1.38, size = 74, normalized size = 0.92 \begin {gather*} \frac {102}{166375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {27}{50} \, \sqrt {-2 \, x + 1} - \frac {42879 \, x + 25723}{3025 \, {\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 = 1.22, size = 55, normalized size = 0.69 \begin {gather*} \frac {\frac {42879\,x}{15125}+\frac {25723}{15125}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {204\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{166375}+\frac {27\,\sqrt {1-2\,x}}{50} \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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