Optimal. Leaf size=166 \[ \frac{2184369575 \sqrt{1-2 x}}{996072 \sqrt{5 x+3}}-\frac{21891025 \sqrt{1-2 x}}{90552 (5 x+3)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (3 x+2) (5 x+3)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{41307885 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0609974, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ \frac{2184369575 \sqrt{1-2 x}}{996072 \sqrt{5 x+3}}-\frac{21891025 \sqrt{1-2 x}}{90552 (5 x+3)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (3 x+2) (5 x+3)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{41307885 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 103
Rule 151
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{5/2}} \, dx &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{1}{21} \int \frac{\frac{165}{2}-120 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{1}{294} \int \frac{\frac{40335}{4}-14625 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac{\int \frac{\frac{7422495}{8}-1190025 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{2058}\\ &=-\frac{21891025 \sqrt{1-2 x}}{90552 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}-\frac{\int \frac{\frac{837775605}{16}-\frac{197019225 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{33957}\\ &=-\frac{21891025 \sqrt{1-2 x}}{90552 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac{2184369575 \sqrt{1-2 x}}{996072 \sqrt{3+5 x}}+\frac{2 \int \frac{44984286765}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{373527}\\ &=-\frac{21891025 \sqrt{1-2 x}}{90552 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac{2184369575 \sqrt{1-2 x}}{996072 \sqrt{3+5 x}}+\frac{41307885 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5488}\\ &=-\frac{21891025 \sqrt{1-2 x}}{90552 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac{2184369575 \sqrt{1-2 x}}{996072 \sqrt{3+5 x}}+\frac{41307885 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2744}\\ &=-\frac{21891025 \sqrt{1-2 x}}{90552 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{325 \sqrt{1-2 x}}{196 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{79335 \sqrt{1-2 x}}{2744 (2+3 x) (3+5 x)^{3/2}}+\frac{2184369575 \sqrt{1-2 x}}{996072 \sqrt{3+5 x}}-\frac{41307885 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2744 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0736513, size = 84, normalized size = 0.51 \[ \frac{\sqrt{1-2 x} \left (294889892625 x^4+760212086400 x^3+734310313245 x^2+314968389410 x+50617099616\right )}{996072 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{41307885 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.014, size = 298, normalized size = 1.8 \begin{align*}{\frac{1}{13945008\, \left ( 2+3\,x \right ) ^{3}} \left ( 10121464522125\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+32388686470800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+41430528110565\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+4128458496750\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+26480750142330\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+10642969209600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+8457045911820\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+10280344385430\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1079622882360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +4409557451740\,x\sqrt{-10\,{x}^{2}-x+3}+708639394624\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \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{5}{2}}{\left (3 \, x + 2\right )}^{4} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68692, size = 470, normalized size = 2.83 \begin{align*} -\frac{14994762255 \, \sqrt{7}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 (294889892625 \, x^{4} + 760212086400 \, x^{3} + 734310313245 \, x^{2} + 314968389410 \, x + 50617099616\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{13945008 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 [B] time = 3.46463, size = 591, normalized size = 3.56 \begin{align*} -\frac{125}{5808} \, \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} + \frac{8261577}{76832} \, \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{8125}{121} \, \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{1485 \,{\left (13759 \, \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} + 6614720 \, \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} + 818950720 \, \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 )}}{1372 \,{\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 )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]