Optimal. Leaf size=127 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^3}{1210 (5 x+3)^2}-\frac{1344 \sqrt{1-2 x} (3 x+2)^2}{33275 (5 x+3)}+\frac{441 \sqrt{1-2 x} (1125 x+3344)}{332750}-\frac{4557 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{166375 \sqrt{55}} \]
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Rubi [A] time = 0.0402577, antiderivative size = 127, normalized size of antiderivative = 1., 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 = {98, 149, 147, 63, 206} \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^3}{1210 (5 x+3)^2}-\frac{1344 \sqrt{1-2 x} (3 x+2)^2}{33275 (5 x+3)}+\frac{441 \sqrt{1-2 x} (1125 x+3344)}{332750}-\frac{4557 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{166375 \sqrt{55}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx &=\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{1}{11} \int \frac{(2+3 x)^3 (110+207 x)}{\sqrt{1-2 x} (3+5 x)^3} \, dx\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^3}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{\int \frac{(2+3 x)^2 (8043+14301 x)}{\sqrt{1-2 x} (3+5 x)^2} \, dx}{1210}\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^3}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{1344 \sqrt{1-2 x} (2+3 x)^2}{33275 (3+5 x)}-\frac{\int \frac{(2+3 x) (293118+496125 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{66550}\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^3}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{1344 \sqrt{1-2 x} (2+3 x)^2}{33275 (3+5 x)}+\frac{441 \sqrt{1-2 x} (3344+1125 x)}{332750}+\frac{4557 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{332750}\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^3}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{1344 \sqrt{1-2 x} (2+3 x)^2}{33275 (3+5 x)}+\frac{441 \sqrt{1-2 x} (3344+1125 x)}{332750}-\frac{4557 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{332750}\\ &=-\frac{71 \sqrt{1-2 x} (2+3 x)^3}{1210 (3+5 x)^2}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} (3+5 x)^2}-\frac{1344 \sqrt{1-2 x} (2+3 x)^2}{33275 (3+5 x)}+\frac{441 \sqrt{1-2 x} (3344+1125 x)}{332750}-\frac{4557 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{166375 \sqrt{55}}\\ \end{align*}
Mathematica [C] time = 0.0982227, size = 96, normalized size = 0.76 \[ \frac{\frac{2415 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{\sqrt{1-2 x}}-\frac{55 \left (490050 x^4+3822390 x^3-68385 x^2-3786773 x-1485319\right )}{\sqrt{1-2 x} (5 x+3)^2}-609 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1663750} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 75, normalized size = 0.6 \begin{align*} -{\frac{81}{500} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1539}{625}\sqrt{1-2\,x}}+{\frac{16807}{5324}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{4}{33275\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{337}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3729}{100}\sqrt{1-2\,x}} \right ) }-{\frac{4557\,\sqrt{55}}{9150625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.98628, size = 136, normalized size = 1.07 \begin{align*} -\frac{81}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4557}{18301250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1539}{625} \, \sqrt{-2 \, x + 1} + \frac{262616115 \,{\left (2 \, x - 1\right )}^{2} + 2310992332 \, x + 115533209}{3327500 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51923, size = 301, normalized size = 2.37 \begin{align*} \frac{4557 \, \sqrt{55}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (5390550 \, x^{4} + 42046290 \, x^{3} - 764310 \, x^{2} - 41668993 \, x - 16342856\right )} \sqrt{-2 \, x + 1}}{18301250 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.43548, size = 128, normalized size = 1.01 \begin{align*} -\frac{81}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4557}{18301250} \, \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{1539}{625} \, \sqrt{-2 \, x + 1} + \frac{16807}{5324 \, \sqrt{-2 \, x + 1}} + \frac{1685 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3729 \, \sqrt{-2 \, x + 1}}{3327500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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