Optimal. Leaf size=108 \[ -\frac {635 \sqrt {1-2 x}}{8232 (3 x+2)}-\frac {635 \sqrt {1-2 x}}{3528 (3 x+2)^2}+\frac {13 \sqrt {1-2 x}}{252 (3 x+2)^3}-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}-\frac {635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 108, 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, 51, 63, 206} \begin {gather*} -\frac {635 \sqrt {1-2 x}}{8232 (3 x+2)}-\frac {635 \sqrt {1-2 x}}{3528 (3 x+2)^2}+\frac {13 \sqrt {1-2 x}}{252 (3 x+2)^3}-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}-\frac {635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=-\frac {\sqrt {1-2 x}}{252 (2+3 x)^4}+\frac {1}{252} \int \frac {1127+2100 x}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {\sqrt {1-2 x}}{252 (2+3 x)^4}+\frac {13 \sqrt {1-2 x}}{252 (2+3 x)^3}+\frac {635}{252} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {\sqrt {1-2 x}}{252 (2+3 x)^4}+\frac {13 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {635 \sqrt {1-2 x}}{3528 (2+3 x)^2}+\frac {635 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{1176}\\ &=-\frac {\sqrt {1-2 x}}{252 (2+3 x)^4}+\frac {13 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {635 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {635 \sqrt {1-2 x}}{8232 (2+3 x)}+\frac {635 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{8232}\\ &=-\frac {\sqrt {1-2 x}}{252 (2+3 x)^4}+\frac {13 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {635 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {635 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {635 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{8232}\\ &=-\frac {\sqrt {1-2 x}}{252 (2+3 x)^4}+\frac {13 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {635 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {635 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 47, normalized size = 0.44 \begin {gather*} \frac {\sqrt {1-2 x} \left (\frac {343 (39 x+25)}{(3 x+2)^4}-5080 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{86436} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 79, normalized size = 0.73 \begin {gather*} \frac {\left (17145 (1-2 x)^3-146685 (1-2 x)^2+399399 (1-2 x)-351379\right ) \sqrt {1-2 x}}{4116 (3 (1-2 x)-7)^4}-\frac {635 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 99, normalized size = 0.92 \begin {gather*} \frac {635 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (17145 \, x^{3} + 47625 \, x^{2} + 39366 \, x + 10190\right )} \sqrt {-2 \, x + 1}}{172872 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 100, normalized size = 0.93 \begin {gather*} \frac {635}{172872} \, \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 {17145 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 146685 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 399399 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 351379 \, \sqrt {-2 \, x + 1}}{65856 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.61 \begin {gather*} -\frac {635 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{86436}+\frac {\frac {5715 \left (-2 x +1\right )^{\frac {7}{2}}}{1372}-\frac {6985 \left (-2 x +1\right )^{\frac {5}{2}}}{196}+\frac {2717 \left (-2 x +1\right )^{\frac {3}{2}}}{28}-\frac {7171 \sqrt {-2 x +1}}{84}}{\left (-6 x -4\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 110, normalized size = 1.02 \begin {gather*} \frac {635}{172872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17145 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 146685 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 399399 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 351379 \, \sqrt {-2 \, x + 1}}{4116 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 90, normalized size = 0.83 \begin {gather*} -\frac {635\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{86436}-\frac {\frac {7171\,\sqrt {1-2\,x}}{6804}-\frac {2717\,{\left (1-2\,x\right )}^{3/2}}{2268}+\frac {6985\,{\left (1-2\,x\right )}^{5/2}}{15876}-\frac {635\,{\left (1-2\,x\right )}^{7/2}}{12348}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \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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