Optimal. Leaf size=94 \[ -\frac {131 (1-2 x)^{5/2}}{6050 (5 x+3)}-\frac {(1-2 x)^{5/2}}{550 (5 x+3)^2}+\frac {119 (1-2 x)^{3/2}}{3025}+\frac {357 \sqrt {1-2 x}}{1375}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]
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Rubi [A] time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 78, 50, 63, 206} \[ -\frac {131 (1-2 x)^{5/2}}{6050 (5 x+3)}-\frac {(1-2 x)^{5/2}}{550 (5 x+3)^2}+\frac {119 (1-2 x)^{3/2}}{3025}+\frac {357 \sqrt {1-2 x}}{1375}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}+\frac {1}{550} \int \frac {(1-2 x)^{3/2} (725+990 x)}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357 \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{1210}\\ &=\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357}{550} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac {357}{250} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac {357}{250} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {357 \sqrt {1-2 x}}{1375}+\frac {119 (1-2 x)^{3/2}}{3025}-\frac {(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac {131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 63, normalized size = 0.67 \[ \frac {\sqrt {1-2 x} \left (-600 x^3+1320 x^2+2105 x+656\right )}{250 (5 x+3)^2}-\frac {357 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{125 \sqrt {55}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 79, normalized size = 0.84 \[ \frac {357 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (600 \, x^{3} - 1320 \, x^{2} - 2105 \, x - 656\right )} \sqrt {-2 \, x + 1}}{13750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 86, normalized size = 0.91 \[ \frac {6}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {357}{13750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {174}{625} \, \sqrt {-2 \, x + 1} + \frac {635 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1419 \, \sqrt {-2 \, x + 1}}{2500 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.70 \[ -\frac {357 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{6875}+\frac {6 \left (-2 x +1\right )^{\frac {3}{2}}}{125}+\frac {174 \sqrt {-2 x +1}}{625}+\frac {\frac {127 \left (-2 x +1\right )^{\frac {3}{2}}}{125}-\frac {1419 \sqrt {-2 x +1}}{625}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 92, normalized size = 0.98 \[ \frac {6}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {357}{13750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {174}{625} \, \sqrt {-2 \, x + 1} + \frac {635 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1419 \, \sqrt {-2 \, x + 1}}{625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 74, normalized size = 0.79 \[ \frac {174\,\sqrt {1-2\,x}}{625}+\frac {6\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {\frac {1419\,\sqrt {1-2\,x}}{15625}-\frac {127\,{\left (1-2\,x\right )}^{3/2}}{3125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,357{}\mathrm {i}}{6875} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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