Optimal. Leaf size=115 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)}-\frac{185}{126} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{125}{54} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{173 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{189 \sqrt{7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0416468, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {98, 154, 157, 54, 216, 93, 204} \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)}-\frac{185}{126} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{125}{54} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{173 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{189 \sqrt{7}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^2} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}-\frac{1}{21} \int \frac{\left (-\frac{189}{2}-185 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{185}{126} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac{1}{126} \int \frac{1516+\frac{4375 x}{2}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{185}{126} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac{173}{378} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx+\frac{625}{108} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{185}{126} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac{173}{189} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{1}{54} \left (125 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{185}{126} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)}+\frac{125}{54} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{173 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{189 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0954193, size = 94, normalized size = 0.82 \[ \frac{-\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} (525 x+352)}{3 x+2}-6125 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-692 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5292} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 146, normalized size = 1.3 \begin{align*}{\frac{1}{21168+31752\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 18375\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+2076\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+12250\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1384\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -44100\,x\sqrt{-10\,{x}^{2}-x+3}-29568\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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 = 5.50791, size = 101, normalized size = 0.88 \begin{align*} \frac{125}{216} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{173}{2646} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{25}{18} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{\sqrt{-10 \, x^{2} - x + 3}}{63 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.93486, size = 392, normalized size = 3.41 \begin{align*} -\frac{6125 \, \sqrt{5} \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 692 \, \sqrt{7}{\left (3 \, x + 2\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 ) + 84 \,{\left (525 \, x + 352\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{10584 \,{\left (3 \, x + 2\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 = 2.33508, size = 377, normalized size = 3.28 \begin{align*} \frac{173}{26460} \, \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{125}{216} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{5}{18} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{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 )}}{63 \,{\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]