Optimal. Leaf size=180 \[ \frac{107245 \sqrt{1-2 x} \sqrt{5 x+3}}{153664 (3 x+2)}+\frac{835 \sqrt{1-2 x} \sqrt{5 x+3}}{10976 (3 x+2)^2}-\frac{13 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)^3}-\frac{27 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^4}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^4}-\frac{1244755 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0634631, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{107245 \sqrt{1-2 x} \sqrt{5 x+3}}{153664 (3 x+2)}+\frac{835 \sqrt{1-2 x} \sqrt{5 x+3}}{10976 (3 x+2)^2}-\frac{13 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)^3}-\frac{27 \sqrt{1-2 x} \sqrt{5 x+3}}{196 (3 x+2)^4}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^4}-\frac{1244755 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]
Antiderivative was successfully verified.
[In]
[Out]
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)^5} \, dx &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^4}-\frac{2}{7} \int \frac{-\frac{71}{2}-60 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^4}-\frac{27 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^4}-\frac{1}{98} \int \frac{-\frac{989}{4}-405 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)^4}-\frac{27 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^4}-\frac{13 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)^3}-\frac{\int \frac{-\frac{13125}{8}-1365 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{2058}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^4}-\frac{27 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^4}-\frac{13 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)^3}+\frac{835 \sqrt{1-2 x} \sqrt{3+5 x}}{10976 (2+3 x)^2}-\frac{\int \frac{-\frac{516915}{16}+\frac{87675 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{28812}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^4}-\frac{27 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^4}-\frac{13 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)^3}+\frac{835 \sqrt{1-2 x} \sqrt{3+5 x}}{10976 (2+3 x)^2}+\frac{107245 \sqrt{1-2 x} \sqrt{3+5 x}}{153664 (2+3 x)}-\frac{\int -\frac{26139855}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{201684}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^4}-\frac{27 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^4}-\frac{13 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)^3}+\frac{835 \sqrt{1-2 x} \sqrt{3+5 x}}{10976 (2+3 x)^2}+\frac{107245 \sqrt{1-2 x} \sqrt{3+5 x}}{153664 (2+3 x)}+\frac{1244755 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{307328}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^4}-\frac{27 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^4}-\frac{13 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)^3}+\frac{835 \sqrt{1-2 x} \sqrt{3+5 x}}{10976 (2+3 x)^2}+\frac{107245 \sqrt{1-2 x} \sqrt{3+5 x}}{153664 (2+3 x)}+\frac{1244755 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{153664}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^4}-\frac{27 \sqrt{1-2 x} \sqrt{3+5 x}}{196 (2+3 x)^4}-\frac{13 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)^3}+\frac{835 \sqrt{1-2 x} \sqrt{3+5 x}}{10976 (2+3 x)^2}+\frac{107245 \sqrt{1-2 x} \sqrt{3+5 x}}{153664 (2+3 x)}-\frac{1244755 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{153664 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0692395, size = 95, normalized size = 0.53 \[ \frac{-7 \sqrt{5 x+3} \left (5791230 x^4+8897265 x^3+2075184 x^2-2239092 x-917264\right )-1244755 \sqrt{7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1075648 \sqrt{1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.014, size = 305, normalized size = 1.7 \begin{align*}{\frac{1}{2151296\, \left ( 2+3\,x \right ) ^{4} \left ( 2\,x-1 \right ) } \left ( 201650310\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+436909005\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+268867080\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+81077220\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-29874120\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+124561710\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-79664320\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+29052576\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-19916080\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -31347288\,x\sqrt{-10\,{x}^{2}-x+3}-12841696\,\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.83827, size = 400, normalized size = 2.22 \begin{align*} \frac{1244755}{2151296} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{536225 \, x}{230496 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{189585}{153664 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{84 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{227}{3528 \,{\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{599}{14112 \,{\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{12725}{65856 \,{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.97011, size = 416, normalized size = 2.31 \begin{align*} -\frac{1244755 \, \sqrt{7}{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\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 (5791230 \, x^{4} + 8897265 \, x^{3} + 2075184 \, x^{2} - 2239092 \, x - 917264\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2151296 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 3.81296, size = 547, normalized size = 3.04 \begin{align*} \frac{248951}{4302592} \, \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}}{84035 \,{\left (2 \, x - 1\right )}} - \frac{33 \,{\left (264101 \, \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 )}^{7} - 272107080 \, \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} - 72200520000 \, \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} - 5707629760000 \, \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 )}}{537824 \,{\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 )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]