Optimal. Leaf size=76 \[ \frac {20}{147 (1-2 x)^{3/2}}+\frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {60}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.01, 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 = {79, 53, 65, 212}
\begin {gather*} \frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (3 x+2)}+\frac {20}{147 (1-2 x)^{3/2}}-\frac {60}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 212
Rubi steps
\begin {align*} \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx &=\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {10}{7} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)} \, dx\\ &=\frac {20}{147 (1-2 x)^{3/2}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {30}{49} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac {20}{147 (1-2 x)^{3/2}}+\frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}+\frac {90}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {20}{147 (1-2 x)^{3/2}}+\frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {90}{343} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {20}{147 (1-2 x)^{3/2}}+\frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {60}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 72, normalized size = 0.95 \begin {gather*} \frac {2 \left (-539-420 (1-2 x)+270 (1-2 x)^2\right )}{1029 (-7+3 (1-2 x)) (1-2 x)^{3/2}}-\frac {60}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 54, normalized size = 0.71
method | result | size |
risch | \(\frac {1080 x^{2}-240 x -689}{1029 \sqrt {1-2 x}\, \left (-1+2 x \right ) \left (2+3 x \right )}-\frac {60 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(53\) |
derivativedivides | \(\frac {22}{147 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {62}{343 \sqrt {1-2 x}}-\frac {2 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}-\frac {60 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(54\) |
default | \(\frac {22}{147 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {62}{343 \sqrt {1-2 x}}-\frac {2 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}-\frac {60 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(54\) |
trager | \(-\frac {\left (1080 x^{2}-240 x -689\right ) \sqrt {1-2 x}}{1029 \left (-1+2 x \right )^{2} \left (2+3 x \right )}-\frac {30 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2401}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 74, normalized size = 0.97 \begin {gather*} \frac {30}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (270 \, {\left (2 \, x - 1\right )}^{2} + 840 \, x - 959\right )}}{1029 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.22, size = 90, normalized size = 1.18 \begin {gather*} \frac {90 \, \sqrt {7} \sqrt {3} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (1080 \, x^{2} - 240 \, x - 689\right )} \sqrt {-2 \, x + 1}}{7203 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 211.42, size = 204, normalized size = 2.68 \begin {gather*} \frac {12 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{49} + \frac {186 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right )}{343} + \frac {62}{343 \sqrt {1 - 2 x}} + \frac {22}{147 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.72, size = 77, normalized size = 1.01 \begin {gather*} \frac {30}{2401} \, \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 {4 \, {\left (93 \, x - 85\right )}}{1029 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {3 \, \sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 56, normalized size = 0.74 \begin {gather*} -\frac {\frac {80\,x}{147}+\frac {60\,{\left (2\,x-1\right )}^2}{343}-\frac {274}{441}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}-\frac {60\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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