Optimal. Leaf size=150 \[ \frac{1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac{23}{216} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{53}{192} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{15863 \sqrt{1-2 x} \sqrt{5 x+3}}{20736}+\frac{648919 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{62208 \sqrt{10}}+\frac{14}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.064069, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {101, 154, 157, 54, 216, 93, 204} \[ \frac{1}{12} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac{23}{216} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{53}{192} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{15863 \sqrt{1-2 x} \sqrt{5 x+3}}{20736}+\frac{648919 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{62208 \sqrt{10}}+\frac{14}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 101
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{2+3 x} \, dx &=\frac{1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{12} \int \frac{\left (-29-\frac{115 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=\frac{23}{216} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{1}{540} \int \frac{\left (-\frac{425}{2}-\frac{7155 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{53}{192} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{23}{216} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{\int \frac{\sqrt{3+5 x} \left (\frac{94995}{4}+\frac{237945 x}{8}\right )}{\sqrt{1-2 x} (2+3 x)} \, dx}{6480}\\ &=-\frac{15863 \sqrt{1-2 x} \sqrt{3+5 x}}{20736}-\frac{53}{192} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{23}{216} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{\int \frac{-\frac{3181875}{8}-\frac{9733785 x}{16}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{38880}\\ &=-\frac{15863 \sqrt{1-2 x} \sqrt{3+5 x}}{20736}-\frac{53}{192} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{23}{216} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{49}{243} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx+\frac{648919 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{124416}\\ &=-\frac{15863 \sqrt{1-2 x} \sqrt{3+5 x}}{20736}-\frac{53}{192} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{23}{216} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{98}{243} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{648919 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{62208 \sqrt{5}}\\ &=-\frac{15863 \sqrt{1-2 x} \sqrt{3+5 x}}{20736}-\frac{53}{192} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{23}{216} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1}{12} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{648919 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{62208 \sqrt{10}}+\frac{14}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0851233, size = 110, normalized size = 0.73 \[ \frac{30 \sqrt{5 x+3} \left (172800 x^4-75840 x^3-121992 x^2+53578 x+2389\right )-648919 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+35840 \sqrt{7-14 x} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{622080 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 132, normalized size = 0.9 \begin{align*}{\frac{1}{1244160}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -5184000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-316800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+648919\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -35840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3501360\,x\sqrt{-10\,{x}^{2}-x+3}+143340\,\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 = 2.53709, size = 132, normalized size = 0.88 \begin{align*} \frac{5}{12} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{7}{432} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{2675}{1728} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{648919}{1244160} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{7}{243} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3397}{20736} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61113, size = 377, normalized size = 2.51 \begin{align*} -\frac{1}{20736} \,{\left (86400 \, x^{3} + 5280 \, x^{2} - 58356 \, x - 2389\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + \frac{7}{243} \, \sqrt{7} \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 ) - \frac{648919}{1244160} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.18038, size = 269, normalized size = 1.79 \begin{align*} -\frac{7}{2430} \, \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{1}{518400} \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} - 313 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 2385 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 79315 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{648919}{1244160} \, \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]