Optimal. Leaf size=86 \[ -\frac{515 \sqrt{1-2 x}}{77 \sqrt{5 x+3}}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) \sqrt{5 x+3}}+\frac{321 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0269292, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {103, 152, 12, 93, 204} \[ -\frac{515 \sqrt{1-2 x}}{77 \sqrt{5 x+3}}+\frac{3 \sqrt{1-2 x}}{7 (3 x+2) \sqrt{5 x+3}}+\frac{321 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 103
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx &=\frac{3 \sqrt{1-2 x}}{7 (2+3 x) \sqrt{3+5 x}}+\frac{1}{7} \int \frac{\frac{67}{2}-30 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{515 \sqrt{1-2 x}}{77 \sqrt{3+5 x}}+\frac{3 \sqrt{1-2 x}}{7 (2+3 x) \sqrt{3+5 x}}-\frac{2}{77} \int \frac{3531}{4 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{515 \sqrt{1-2 x}}{77 \sqrt{3+5 x}}+\frac{3 \sqrt{1-2 x}}{7 (2+3 x) \sqrt{3+5 x}}-\frac{321}{14} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{515 \sqrt{1-2 x}}{77 \sqrt{3+5 x}}+\frac{3 \sqrt{1-2 x}}{7 (2+3 x) \sqrt{3+5 x}}-\frac{321}{7} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{515 \sqrt{1-2 x}}{77 \sqrt{3+5 x}}+\frac{3 \sqrt{1-2 x}}{7 (2+3 x) \sqrt{3+5 x}}+\frac{321 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{7 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0388287, size = 69, normalized size = 0.8 \[ \frac{321 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}}-\frac{\sqrt{1-2 x} (1545 x+997)}{77 (3 x+2) \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 154, normalized size = 1.8 \begin{align*} -{\frac{1}{2156+3234\,x} \left ( 52965\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+67089\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+21186\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +21630\,x\sqrt{-10\,{x}^{2}-x+3}+13958\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{2} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.516, size = 258, normalized size = 3. \begin{align*} \frac{3531 \, \sqrt{7}{\left (15 \, x^{2} + 19 \, x + 6\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 (1545 \, x + 997\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1078 \,{\left (15 \, x^{2} + 19 \, x + 6\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 = 2.28657, size = 340, normalized size = 3.95 \begin{align*} -\frac{321}{980} \, \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{5}{22} \, \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 )} - \frac{198 \, \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 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]