Optimal. Leaf size=54 \[ -\frac {217}{242 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \]
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Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {87, 63, 206} \[ -\frac {217}{242 \sqrt {1-2 x}}+\frac {49}{66 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 87
Rule 206
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\int \left (\frac {49}{22 (1-2 x)^{5/2}}-\frac {217}{242 (1-2 x)^{3/2}}+\frac {1}{121 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2}}-\frac {217}{242 \sqrt {1-2 x}}+\frac {1}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {49}{66 (1-2 x)^{3/2}}-\frac {217}{242 \sqrt {1-2 x}}-\frac {1}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {49}{66 (1-2 x)^{3/2}}-\frac {217}{242 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 40, normalized size = 0.74 \[ \frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {5}{11} (1-2 x)\right )+33 (45 x-4)}{825 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 69, normalized size = 1.28 \[ \frac {3 \, \sqrt {55} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 385 \, {\left (93 \, x - 8\right )} \sqrt {-2 \, x + 1}}{19965 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.60, size = 61, normalized size = 1.13 \[ \frac {1}{6655} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {7 \, {\left (93 \, x - 8\right )}}{363 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.70 \[ -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{6655}+\frac {49}{66 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {217}{242 \sqrt {-2 x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 51, normalized size = 0.94 \[ \frac {1}{6655} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {7 \, {\left (93 \, x - 8\right )}}{363 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 32, normalized size = 0.59 \[ \frac {\frac {217\,x}{121}-\frac {56}{363}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{6655} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 42.70, size = 90, normalized size = 1.67 \[ \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right )}{121} - \frac {217}{242 \sqrt {1 - 2 x}} + \frac {49}{66 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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