Optimal. Leaf size=181 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}+\frac{288770 \sqrt{1-2 x}}{189 (3 x+2) (5 x+3)}+\frac{22109 \sqrt{1-2 x}}{216 (3 x+2)^2 (5 x+3)}+\frac{287 \sqrt{1-2 x}}{27 (3 x+2)^3 (5 x+3)}-\frac{7738475 \sqrt{1-2 x}}{504 (5 x+3)}-\frac{53384095 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}+18700 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0756392, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}+\frac{288770 \sqrt{1-2 x}}{189 (3 x+2) (5 x+3)}+\frac{22109 \sqrt{1-2 x}}{216 (3 x+2)^2 (5 x+3)}+\frac{287 \sqrt{1-2 x}}{27 (3 x+2)^3 (5 x+3)}-\frac{7738475 \sqrt{1-2 x}}{504 (5 x+3)}-\frac{53384095 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}+18700 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)^5 (3+5 x)^2} \, dx &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac{1}{12} \int \frac{(230-229 x) \sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac{287 \sqrt{1-2 x}}{27 (2+3 x)^3 (3+5 x)}-\frac{1}{108} \int \frac{-25717+38806 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac{287 \sqrt{1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac{22109 \sqrt{1-2 x}}{216 (2+3 x)^2 (3+5 x)}-\frac{\int \frac{-2810990+3869075 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx}{1512}\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac{287 \sqrt{1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac{22109 \sqrt{1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac{288770 \sqrt{1-2 x}}{189 (2+3 x) (3+5 x)}-\frac{\int \frac{-211977465+242566800 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{10584}\\ &=-\frac{7738475 \sqrt{1-2 x}}{504 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac{287 \sqrt{1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac{22109 \sqrt{1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac{288770 \sqrt{1-2 x}}{189 (2+3 x) (3+5 x)}+\frac{\int \frac{-8756550495+5362763175 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{116424}\\ &=-\frac{7738475 \sqrt{1-2 x}}{504 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac{287 \sqrt{1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac{22109 \sqrt{1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac{288770 \sqrt{1-2 x}}{189 (2+3 x) (3+5 x)}+\frac{53384095}{168} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-514250 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{7738475 \sqrt{1-2 x}}{504 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac{287 \sqrt{1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac{22109 \sqrt{1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac{288770 \sqrt{1-2 x}}{189 (2+3 x) (3+5 x)}-\frac{53384095}{168} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+514250 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{7738475 \sqrt{1-2 x}}{504 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)}+\frac{287 \sqrt{1-2 x}}{27 (2+3 x)^3 (3+5 x)}+\frac{22109 \sqrt{1-2 x}}{216 (2+3 x)^2 (3+5 x)}+\frac{288770 \sqrt{1-2 x}}{189 (2+3 x) (3+5 x)}-\frac{53384095 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}+18700 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.142951, size = 100, normalized size = 0.55 \[ -\frac{\sqrt{1-2 x} \left (208938825 x^4+550239720 x^3+543154477 x^2+238179048 x+39145938\right )}{168 (3 x+2)^4 (5 x+3)}-\frac{53384095 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}+18700 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 100, normalized size = 0.6 \begin{align*} 162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{1242775\, \left ( 1-2\,x \right ) ^{7/2}}{504}}-{\frac{11266013\, \left ( 1-2\,x \right ) ^{5/2}}{648}}+{\frac{79444085\, \left ( 1-2\,x \right ) ^{3/2}}{1944}}-{\frac{62254745\,\sqrt{1-2\,x}}{1944}} \right ) }-{\frac{53384095\,\sqrt{21}}{1764}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+1210\,{\frac{\sqrt{1-2\,x}}{-2\,x-6/5}}+18700\,{\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 = 2.40055, size = 221, normalized size = 1.22 \begin{align*} -9350 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{53384095}{3528} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{208938825 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 1936234740 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 6727689178 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 10387861820 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 6013803565 \, \sqrt{-2 \, x + 1}}{84 \,{\left (405 \,{\left (2 \, x - 1\right )}^{5} + 4671 \,{\left (2 \, x - 1\right )}^{4} + 21546 \,{\left (2 \, x - 1\right )}^{3} + 49686 \,{\left (2 \, x - 1\right )}^{2} + 114562 \, x - 30870\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44218, size = 555, normalized size = 3.07 \begin{align*} \frac{32986800 \, \sqrt{55}{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 53384095 \, \sqrt{21}{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (208938825 \, x^{4} + 550239720 \, x^{3} + 543154477 \, x^{2} + 238179048 \, x + 39145938\right )} \sqrt{-2 \, x + 1}}{3528 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\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 = 2.226, size = 209, normalized size = 1.15 \begin{align*} -9350 \, \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{53384095}{3528} \, \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{3025 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{33554925 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 236586273 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 556108595 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 435783215 \, \sqrt{-2 \, x + 1}}{1344 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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