Optimal. Leaf size=122 \[ \frac{2 (5 x+3)^{5/2}}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{5 \sqrt{1-2 x} (5 x+3)^{3/2}}{98 (3 x+2)^2}+\frac{165 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}+\frac{1815 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0294422, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{2 (5 x+3)^{5/2}}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{5 \sqrt{1-2 x} (5 x+3)^{3/2}}{98 (3 x+2)^2}+\frac{165 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}+\frac{1815 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx &=\frac{2 (3+5 x)^{5/2}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{5}{7} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{5 \sqrt{1-2 x} (3+5 x)^{3/2}}{98 (2+3 x)^2}+\frac{2 (3+5 x)^{5/2}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{165}{196} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{165 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}+\frac{5 \sqrt{1-2 x} (3+5 x)^{3/2}}{98 (2+3 x)^2}+\frac{2 (3+5 x)^{5/2}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{1815 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=\frac{165 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}+\frac{5 \sqrt{1-2 x} (3+5 x)^{3/2}}{98 (2+3 x)^2}+\frac{2 (3+5 x)^{5/2}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{1815 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=\frac{165 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}+\frac{5 \sqrt{1-2 x} (3+5 x)^{3/2}}{98 (2+3 x)^2}+\frac{2 (3+5 x)^{5/2}}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{1815 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0418913, size = 85, normalized size = 0.7 \[ \frac{7 \sqrt{5 x+3} \left (8110 x^2+11525 x+4068\right )+1815 \sqrt{7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9604 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 209, normalized size = 1.7 \begin{align*} -{\frac{1}{19208\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) } \left ( 32670\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+27225\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-7260\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+113540\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-7260\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +161350\,x\sqrt{-10\,{x}^{2}-x+3}+56952\,\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 [A] time = 3.77205, size = 193, normalized size = 1.58 \begin{align*} -\frac{1815}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{20275 \, x}{6174 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{83665}{37044 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{378 \,{\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}{1764 \,{\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 = 1.74149, size = 298, normalized size = 2.44 \begin{align*} \frac{1815 \, \sqrt{7}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 (8110 \, x^{2} + 11525 \, x + 4068\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{19208 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 = 3.39227, size = 381, normalized size = 3.12 \begin{align*} -\frac{363}{38416} \, \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{242 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1715 \,{\left (2 \, x - 1\right )}} + \frac{121 \,{\left (\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} + 360 \, \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 )}}{98 \,{\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 )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]