Optimal. Leaf size=102 \[ -\frac {(1-2 x)^{7/2}}{275 (5 x+3)}-\frac {9}{175} (1-2 x)^{7/2}+\frac {122 (1-2 x)^{5/2}}{6875}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {1342 \sqrt {1-2 x}}{3125}-\frac {1342 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
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Rubi [A] time = 0.03, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 80, 50, 63, 206} \[ -\frac {(1-2 x)^{7/2}}{275 (5 x+3)}-\frac {9}{175} (1-2 x)^{7/2}+\frac {122 (1-2 x)^{5/2}}{6875}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {1342 \sqrt {1-2 x}}{3125}-\frac {1342 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx &=-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {1}{275} \int \frac {(1-2 x)^{5/2} (358+495 x)}{3+5 x} \, dx\\ &=-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {61}{275} \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {61}{125} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {671}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}+\frac {7381 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {7381 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {1342 \sqrt {1-2 x}}{3125}+\frac {122 (1-2 x)^{3/2}}{1875}+\frac {122 (1-2 x)^{5/2}}{6875}-\frac {9}{175} (1-2 x)^{7/2}-\frac {(1-2 x)^{7/2}}{275 (3+5 x)}-\frac {1342 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 68, normalized size = 0.67 \[ \frac {\frac {5 \sqrt {1-2 x} \left (135000 x^4-96300 x^3-75130 x^2+173795 x+90486\right )}{5 x+3}-28182 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{328125} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 80, normalized size = 0.78 \[ \frac {14091 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (135000 \, x^{4} - 96300 \, x^{3} - 75130 \, x^{2} + 173795 \, x + 90486\right )} \sqrt {-2 \, x + 1}}{328125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 106, normalized size = 1.04 \[ \frac {9}{175} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {12}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {128}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {671}{15625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1364}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.71 \[ -\frac {1342 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{15625}-\frac {9 \left (-2 x +1\right )^{\frac {7}{2}}}{175}+\frac {12 \left (-2 x +1\right )^{\frac {5}{2}}}{625}+\frac {128 \left (-2 x +1\right )^{\frac {3}{2}}}{1875}+\frac {1364 \sqrt {-2 x +1}}{3125}+\frac {242 \sqrt {-2 x +1}}{15625 \left (-2 x -\frac {6}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 89, normalized size = 0.87 \[ -\frac {9}{175} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {12}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {128}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {671}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1364}{3125} \, \sqrt {-2 \, x + 1} - \frac {121 \, \sqrt {-2 \, x + 1}}{3125 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 73, normalized size = 0.72 \[ \frac {1364\,\sqrt {1-2\,x}}{3125}-\frac {242\,\sqrt {1-2\,x}}{15625\,\left (2\,x+\frac {6}{5}\right )}+\frac {128\,{\left (1-2\,x\right )}^{3/2}}{1875}+\frac {12\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {9\,{\left (1-2\,x\right )}^{7/2}}{175}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,1342{}\mathrm {i}}{15625} \]
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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