Optimal. Leaf size=164 \[ \frac{(5 x+3)^{3/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{123 (5 x+3)^{3/2} (3 x+2)^3}{22 \sqrt{1-2 x}}-\frac{3315}{352} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (10798680 x+22868329)}{281600}-\frac{1626211523 \sqrt{1-2 x} \sqrt{5 x+3}}{1126400}+\frac{1626211523 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]
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Rubi [A] time = 0.0502907, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 150, 153, 147, 50, 54, 216} \[ \frac{(5 x+3)^{3/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{123 (5 x+3)^{3/2} (3 x+2)^3}{22 \sqrt{1-2 x}}-\frac{3315}{352} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (10798680 x+22868329)}{281600}-\frac{1626211523 \sqrt{1-2 x} \sqrt{5 x+3}}{1126400}+\frac{1626211523 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 153
Rule 147
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^3 \sqrt{3+5 x} \left (51+\frac{165 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-7734-\frac{49725 x}{4}\right ) (2+3 x)^2 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{3315}{352} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{(2+3 x) \sqrt{3+5 x} \left (\frac{3817455}{4}+\frac{12148515 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{1320}\\ &=-\frac{3315}{352} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac{1626211523 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{563200}\\ &=-\frac{1626211523 \sqrt{1-2 x} \sqrt{3+5 x}}{1126400}-\frac{3315}{352} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac{1626211523 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{204800}\\ &=-\frac{1626211523 \sqrt{1-2 x} \sqrt{3+5 x}}{1126400}-\frac{3315}{352} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac{1626211523 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{102400 \sqrt{5}}\\ &=-\frac{1626211523 \sqrt{1-2 x} \sqrt{3+5 x}}{1126400}-\frac{3315}{352} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac{1626211523 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{102400 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0800697, size = 84, normalized size = 0.51 \[ \frac{4878634569 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (15552000 x^5+83548800 x^4+236669040 x^3+633940524 x^2-2034703904 x+739060191\right )}{3072000 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 171, normalized size = 1. \begin{align*}{\frac{1}{6144000\, \left ( 2\,x-1 \right ) ^{2}} \left ( -311040000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-1670976000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+19514538276\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-4733380800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-19514538276\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-12678810480\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4878634569\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +40694078080\,x\sqrt{-10\,{x}^{2}-x+3}-14781203820\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 3.48355, size = 325, normalized size = 1.98 \begin{align*} \frac{81}{64} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1666460963}{2048000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{251559}{12800} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{10161}{1280} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{2079}{32} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{29403}{5120} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{43659}{640} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{34897797}{102400} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2401 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{96 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{1029 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1323 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (2 \, x - 1\right )}} + \frac{26411 \, \sqrt{-10 \, x^{2} - x + 3}}{192 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{491519 \, \sqrt{-10 \, x^{2} - x + 3}}{192 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53369, size = 367, normalized size = 2.24 \begin{align*} -\frac{4878634569 \, \sqrt{10}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (15552000 \, x^{5} + 83548800 \, x^{4} + 236669040 \, x^{3} + 633940524 \, x^{2} - 2034703904 \, x + 739060191\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{6144000 \,{\left (4 \, x^{2} - 4 \, x + 1\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 = 2.46075, size = 149, normalized size = 0.91 \begin{align*} \frac{1626211523}{1024000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9 \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 427 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 42657 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9855815 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 3252423046 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 53664980259 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{38400000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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