Optimal. Leaf size=120 \[ -\frac {48 \sqrt {1-2 x} (3 x+2)^3}{25 (5 x+3)}-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac {693}{625} \sqrt {1-2 x} (3 x+2)^2+\frac {63 \sqrt {1-2 x} (125 x+92)}{6250}-\frac {5943 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \]
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Rubi [A] time = 0.04, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 149, 153, 147, 63, 206} \begin {gather*} -\frac {48 \sqrt {1-2 x} (3 x+2)^3}{25 (5 x+3)}-\frac {(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac {693}{625} \sqrt {1-2 x} (3 x+2)^2+\frac {63 \sqrt {1-2 x} (125 x+92)}{6250}-\frac {5943 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 97
Rule 147
Rule 149
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}+\frac {1}{10} \int \frac {(3-27 x) \sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {48 \sqrt {1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac {1}{50} \int \frac {(357-1386 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {693}{625} \sqrt {1-2 x} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {48 \sqrt {1-2 x} (2+3 x)^3}{25 (3+5 x)}-\frac {\int \frac {(2+3 x) (-1218+7875 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{1250}\\ &=\frac {693}{625} \sqrt {1-2 x} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {48 \sqrt {1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac {63 \sqrt {1-2 x} (92+125 x)}{6250}+\frac {5943 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{6250}\\ &=\frac {693}{625} \sqrt {1-2 x} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {48 \sqrt {1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac {63 \sqrt {1-2 x} (92+125 x)}{6250}-\frac {5943 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{6250}\\ &=\frac {693}{625} \sqrt {1-2 x} (2+3 x)^2-\frac {(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {48 \sqrt {1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac {63 \sqrt {1-2 x} (92+125 x)}{6250}-\frac {5943 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 68, normalized size = 0.57 \begin {gather*} \frac {\sqrt {1-2 x} \left (-27000 x^4-14400 x^3+37530 x^2+36295 x+8644\right )}{6250 (5 x+3)^2}-\frac {5943 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 88, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {1-2 x} \left (3375 (1-2 x)^4-17100 (1-2 x)^3+12285 (1-2 x)^2+49525 (1-2 x)-65373\right )}{3125 (5 (1-2 x)-11)^2}-\frac {5943 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125 \sqrt {55}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.74, size = 84, normalized size = 0.70 \begin {gather*} \frac {5943 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (27000 \, x^{4} + 14400 \, x^{3} - 37530 \, x^{2} - 36295 \, x - 8644\right )} \sqrt {-2 \, x + 1}}{343750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 102, normalized size = 0.85 \begin {gather*} -\frac {27}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {18}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {5943}{343750} \, \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 {558}{3125} \, \sqrt {-2 \, x + 1} + \frac {193 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 429 \, \sqrt {-2 \, x + 1}}{2500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.62 \begin {gather*} -\frac {5943 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{171875}-\frac {27 \left (-2 x +1\right )^{\frac {5}{2}}}{625}+\frac {18 \left (-2 x +1\right )^{\frac {3}{2}}}{625}+\frac {558 \sqrt {-2 x +1}}{3125}+\frac {\frac {193 \left (-2 x +1\right )^{\frac {3}{2}}}{625}-\frac {429 \sqrt {-2 x +1}}{625}}{\left (-10 x -6\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 101, normalized size = 0.84 \begin {gather*} -\frac {27}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {18}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {5943}{343750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {558}{3125} \, \sqrt {-2 \, x + 1} + \frac {193 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 429 \, \sqrt {-2 \, x + 1}}{625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 83, normalized size = 0.69 \begin {gather*} \frac {558\,\sqrt {1-2\,x}}{3125}+\frac {18\,{\left (1-2\,x\right )}^{3/2}}{625}-\frac {27\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {\frac {429\,\sqrt {1-2\,x}}{15625}-\frac {193\,{\left (1-2\,x\right )}^{3/2}}{15625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,5943{}\mathrm {i}}{171875} \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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