Optimal. Leaf size=144 \[ -\frac{32735 \sqrt{5 x+3}}{15092 \sqrt{1-2 x}}+\frac{2865 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)}+\frac{27 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{102345 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
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Rubi [A] time = 0.0488349, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ -\frac{32735 \sqrt{5 x+3}}{15092 \sqrt{1-2 x}}+\frac{2865 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)}+\frac{27 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{102345 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx &=\frac{\sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{1}{21} \int \frac{\frac{69}{2}-90 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{27 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^2}+\frac{1}{294} \int \frac{\frac{4935}{4}-5670 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{27 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^2}+\frac{2865 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)}+\frac{\int \frac{-\frac{85785}{8}-\frac{300825 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{2058}\\ &=-\frac{32735 \sqrt{3+5 x}}{15092 \sqrt{1-2 x}}+\frac{\sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{27 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^2}+\frac{2865 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)}-\frac{\int -\frac{23641695}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{79233}\\ &=-\frac{32735 \sqrt{3+5 x}}{15092 \sqrt{1-2 x}}+\frac{\sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{27 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^2}+\frac{2865 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)}+\frac{102345 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5488}\\ &=-\frac{32735 \sqrt{3+5 x}}{15092 \sqrt{1-2 x}}+\frac{\sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{27 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^2}+\frac{2865 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)}+\frac{102345 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2744}\\ &=-\frac{32735 \sqrt{3+5 x}}{15092 \sqrt{1-2 x}}+\frac{\sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{27 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^2}+\frac{2865 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)}-\frac{102345 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2744 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.059882, size = 90, normalized size = 0.62 \[ \frac{-7 \sqrt{5 x+3} \left (1767690 x^3+1549935 x^2-377658 x-421184\right )-1125795 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{211288 \sqrt{1-2 x} (3 x+2)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 257, normalized size = 1.8 \begin{align*}{\frac{1}{422576\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) } \left ( 60792930\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+91189395\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+20264310\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+24747660\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-22515900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+21699090\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-9006360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -5287212\,x\sqrt{-10\,{x}^{2}-x+3}-5896576\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{4}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81131, size = 359, normalized size = 2.49 \begin{align*} -\frac{1125795 \, \sqrt{7}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \,{\left (1767690 \, x^{3} + 1549935 \, x^{2} - 377658 \, x - 421184\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{422576 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\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 [B] time = 3.16761, size = 464, normalized size = 3.22 \begin{align*} \frac{20469}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{32 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{132055 \,{\left (2 \, x - 1\right )}} + \frac{297 \,{\left (4937 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1785280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 188708800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9604 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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