Optimal. Leaf size=208 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}+\frac{12068887 \sqrt{1-2 x}}{1323 (3 x+2) (5 x+3)}+\frac{924025 \sqrt{1-2 x}}{1512 (3 x+2)^2 (5 x+3)}+\frac{16549 \sqrt{1-2 x}}{270 (3 x+2)^3 (5 x+3)}+\frac{1379 \sqrt{1-2 x}}{180 (3 x+2)^4 (5 x+3)}-\frac{323422735 \sqrt{1-2 x}}{3528 (5 x+3)}-\frac{2231141147 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}+111650 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0930197, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}+\frac{12068887 \sqrt{1-2 x}}{1323 (3 x+2) (5 x+3)}+\frac{924025 \sqrt{1-2 x}}{1512 (3 x+2)^2 (5 x+3)}+\frac{16549 \sqrt{1-2 x}}{270 (3 x+2)^3 (5 x+3)}+\frac{1379 \sqrt{1-2 x}}{180 (3 x+2)^4 (5 x+3)}-\frac{323422735 \sqrt{1-2 x}}{3528 (5 x+3)}-\frac{2231141147 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}+111650 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1}{15} \int \frac{(263-295 x) \sqrt{1-2 x}}{(2+3 x)^5 (3+5 x)^2} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1379 \sqrt{1-2 x}}{180 (2+3 x)^4 (3+5 x)}-\frac{1}{180} \int \frac{-37432+59695 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1379 \sqrt{1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac{16549 \sqrt{1-2 x}}{270 (2+3 x)^3 (3+5 x)}-\frac{\int \frac{-5374285+8109010 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx}{3780}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1379 \sqrt{1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac{16549 \sqrt{1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac{924025 \sqrt{1-2 x}}{1512 (2+3 x)^2 (3+5 x)}-\frac{\int \frac{-587414870+808521875 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx}{52920}\\ &=\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1379 \sqrt{1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac{16549 \sqrt{1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac{924025 \sqrt{1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac{12068887 \sqrt{1-2 x}}{1323 (2+3 x) (3+5 x)}-\frac{\int \frac{-44297056545+50689325400 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{370440}\\ &=-\frac{323422735 \sqrt{1-2 x}}{3528 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1379 \sqrt{1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac{16549 \sqrt{1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac{924025 \sqrt{1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac{12068887 \sqrt{1-2 x}}{1323 (2+3 x) (3+5 x)}+\frac{\int \frac{-1829861506935+1120659776775 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{4074840}\\ &=-\frac{323422735 \sqrt{1-2 x}}{3528 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1379 \sqrt{1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac{16549 \sqrt{1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac{924025 \sqrt{1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac{12068887 \sqrt{1-2 x}}{1323 (2+3 x) (3+5 x)}+\frac{2231141147 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{1176}-3070375 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{323422735 \sqrt{1-2 x}}{3528 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1379 \sqrt{1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac{16549 \sqrt{1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac{924025 \sqrt{1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac{12068887 \sqrt{1-2 x}}{1323 (2+3 x) (3+5 x)}-\frac{2231141147 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1176}+3070375 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{323422735 \sqrt{1-2 x}}{3528 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)}+\frac{1379 \sqrt{1-2 x}}{180 (2+3 x)^4 (3+5 x)}+\frac{16549 \sqrt{1-2 x}}{270 (2+3 x)^3 (3+5 x)}+\frac{924025 \sqrt{1-2 x}}{1512 (2+3 x)^2 (3+5 x)}+\frac{12068887 \sqrt{1-2 x}}{1323 (2+3 x) (3+5 x)}-\frac{2231141147 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}+111650 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.160445, size = 105, normalized size = 0.5 \[ -\frac{\sqrt{1-2 x} \left (130986207675 x^5+432275892930 x^4+570477768855 x^3+376323861626 x^2+124085884254 x+16360698684\right )}{5880 (3 x+2)^5 (5 x+3)}-\frac{2231141147 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}+111650 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 109, normalized size = 0.5 \begin{align*} 972\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{54012347\, \left ( 1-2\,x \right ) ^{9/2}}{7056}}-{\frac{46563587\, \left ( 1-2\,x \right ) ^{7/2}}{648}}+{\frac{307361449\, \left ( 1-2\,x \right ) ^{5/2}}{1215}}-{\frac{2308578797\, \left ( 1-2\,x \right ) ^{3/2}}{5832}}+{\frac{2709545797\,\sqrt{1-2\,x}}{11664}} \right ) }-{\frac{2231141147\,\sqrt{21}}{12348}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+6050\,{\frac{\sqrt{1-2\,x}}{-2\,x-6/5}}+111650\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.8184, size = 246, normalized size = 1.18 \begin{align*} -55825 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2231141147}{24696} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{130986207675 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 1519482824235 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 7049980295610 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 16353496911178 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 18965427342155 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 8796956467915 \, \sqrt{-2 \, x + 1}}{2940 \,{\left (1215 \,{\left (2 \, x - 1\right )}^{6} + 16848 \,{\left (2 \, x - 1\right )}^{5} + 97335 \,{\left (2 \, x - 1\right )}^{4} + 299880 \,{\left (2 \, x - 1\right )}^{3} + 519645 \,{\left (2 \, x - 1\right )}^{2} + 960400 \, x - 295323\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41178, size = 659, normalized size = 3.17 \begin{align*} \frac{6893271000 \, \sqrt{55}{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11155705735 \, \sqrt{21}{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (130986207675 \, x^{5} + 432275892930 \, x^{4} + 570477768855 \, x^{3} + 376323861626 \, x^{2} + 124085884254 \, x + 16360698684\right )} \sqrt{-2 \, x + 1}}{123480 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.49848, size = 231, normalized size = 1.11 \begin{align*} -55825 \, \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{2231141147}{24696} \, \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{15125 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{21875000535 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 205345418670 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 722914128048 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1131203610530 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 663838720265 \, \sqrt{-2 \, x + 1}}{94080 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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