Optimal. Leaf size=151 \[ \frac{565 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{5 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^2}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^3}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{7435 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
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Rubi [A] time = 0.0497381, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {99, 151, 12, 93, 204} \[ \frac{565 \sqrt{1-2 x} \sqrt{5 x+3}}{2744 (3 x+2)}-\frac{5 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^2}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{7 (3 x+2)^3}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{7435 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 151
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{2}{7} \int \frac{-\frac{53}{2}-45 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}-\frac{2}{147} \int \frac{-\frac{525}{4}-210 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}-\frac{5 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}-\frac{\int \frac{-\frac{5355}{8}-\frac{525 x}{2}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{1029}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}-\frac{5 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{565 \sqrt{1-2 x} \sqrt{3+5 x}}{2744 (2+3 x)}-\frac{\int -\frac{156135}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{7203}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}-\frac{5 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{565 \sqrt{1-2 x} \sqrt{3+5 x}}{2744 (2+3 x)}+\frac{7435 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5488}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}-\frac{5 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{565 \sqrt{1-2 x} \sqrt{3+5 x}}{2744 (2+3 x)}+\frac{7435 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2744}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{7 (2+3 x)^3}-\frac{5 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^2}+\frac{565 \sqrt{1-2 x} \sqrt{3+5 x}}{2744 (2+3 x)}-\frac{7435 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2744 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0560544, size = 90, normalized size = 0.6 \[ \frac{7 \sqrt{5 x+3} \left (-10170 x^3-8055 x^2+3114 x+2512\right )-7435 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{1-2 x} (3 x+2)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 257, normalized size = 1.7 \begin{align*}{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) } \left ( 401490\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+602235\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+133830\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+142380\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-148700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+112770\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-59480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -43596\,x\sqrt{-10\,{x}^{2}-x+3}-35168\,\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 [A] time = 1.96863, size = 285, normalized size = 1.89 \begin{align*} \frac{7435}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2825 \, x}{4116 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1145}{2744 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{63 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{23}{252 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{125}{1176 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79992, size = 342, normalized size = 2.26 \begin{align*} -\frac{7435 \, \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 (10170 \, x^{3} + 8055 \, x^{2} - 3114 \, x - 2512\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{38416 \,{\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 = 3.53292, size = 464, normalized size = 3.07 \begin{align*} \frac{1487}{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{16 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{12005 \,{\left (2 \, x - 1\right )}} - \frac{99 \,{\left (527 \, \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} - 253120 \, \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} - 36299200 \, \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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