Optimal. Leaf size=76 \[ \frac {(1-2 x)^{5/2}}{21 (3 x+2)}+\frac {76}{189} (1-2 x)^{3/2}+\frac {76}{27} \sqrt {1-2 x}-\frac {76}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 50, 63, 206} \[ \frac {(1-2 x)^{5/2}}{21 (3 x+2)}+\frac {76}{189} (1-2 x)^{3/2}+\frac {76}{27} \sqrt {1-2 x}-\frac {76}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^2} \, dx &=\frac {(1-2 x)^{5/2}}{21 (2+3 x)}+\frac {38}{21} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {76}{189} (1-2 x)^{3/2}+\frac {(1-2 x)^{5/2}}{21 (2+3 x)}+\frac {38}{9} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {76}{27} \sqrt {1-2 x}+\frac {76}{189} (1-2 x)^{3/2}+\frac {(1-2 x)^{5/2}}{21 (2+3 x)}+\frac {266}{27} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {76}{27} \sqrt {1-2 x}+\frac {76}{189} (1-2 x)^{3/2}+\frac {(1-2 x)^{5/2}}{21 (2+3 x)}-\frac {266}{27} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {76}{27} \sqrt {1-2 x}+\frac {76}{189} (1-2 x)^{3/2}+\frac {(1-2 x)^{5/2}}{21 (2+3 x)}-\frac {76}{27} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.76 \[ \frac {1}{81} \left (\frac {3 \sqrt {1-2 x} \left (-60 x^2+212 x+175\right )}{3 x+2}-76 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 70, normalized size = 0.92 \[ \frac {38 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 3 \, {\left (60 \, x^{2} - 212 \, x - 175\right )} \sqrt {-2 \, x + 1}}{81 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 74, normalized size = 0.97 \[ \frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {38}{81} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {74}{27} \, \sqrt {-2 \, x + 1} + \frac {7 \, \sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.71 \[ -\frac {76 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{81}+\frac {10 \left (-2 x +1\right )^{\frac {3}{2}}}{27}+\frac {74 \sqrt {-2 x +1}}{27}-\frac {14 \sqrt {-2 x +1}}{81 \left (-2 x -\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 71, normalized size = 0.93 \[ \frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {38}{81} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {74}{27} \, \sqrt {-2 \, x + 1} + \frac {7 \, \sqrt {-2 \, x + 1}}{27 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 55, normalized size = 0.72 \[ \frac {14\,\sqrt {1-2\,x}}{81\,\left (2\,x+\frac {4}{3}\right )}+\frac {74\,\sqrt {1-2\,x}}{27}+\frac {10\,{\left (1-2\,x\right )}^{3/2}}{27}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,76{}\mathrm {i}}{81} \]
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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