Optimal. Leaf size=193 \[ -\frac{508 (1-2 x)^{3/2} (3 x+2)^4}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}+\frac{2514}{625} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^3+\frac{23991 (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2}{25000}+\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (118392 x+64435)}{4000000}+\frac{8026963 \sqrt{1-2 x} \sqrt{5 x+3}}{40000000}+\frac{88296593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40000000 \sqrt{10}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.066504, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 150, 153, 147, 50, 54, 216} \[ -\frac{508 (1-2 x)^{3/2} (3 x+2)^4}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}+\frac{2514}{625} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^3+\frac{23991 (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2}{25000}+\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (118392 x+64435)}{4000000}+\frac{8026963 \sqrt{1-2 x} \sqrt{5 x+3}}{40000000}+\frac{88296593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40000000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 150
Rule 153
Rule 147
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt{3+5 x}}+\frac{4}{75} \int \frac{(552-3771 x) \sqrt{1-2 x} (2+3 x)^3}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt{3+5 x}}+\frac{2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{2 \int \frac{\sqrt{1-2 x} (2+3 x)^2 \left (-2406+\frac{71973 x}{2}\right )}{\sqrt{3+5 x}} \, dx}{1875}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt{3+5 x}}+\frac{23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}{25000}+\frac{2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}+\frac{\int \frac{\left (\frac{25095}{2}-\frac{932337 x}{4}\right ) \sqrt{1-2 x} (2+3 x)}{\sqrt{3+5 x}} \, dx}{37500}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt{3+5 x}}+\frac{23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}{25000}+\frac{2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}+\frac{21 (1-2 x)^{3/2} \sqrt{3+5 x} (64435+118392 x)}{4000000}+\frac{8026963 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{8000000}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt{3+5 x}}+\frac{8026963 \sqrt{1-2 x} \sqrt{3+5 x}}{40000000}+\frac{23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}{25000}+\frac{2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}+\frac{21 (1-2 x)^{3/2} \sqrt{3+5 x} (64435+118392 x)}{4000000}+\frac{88296593 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{80000000}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt{3+5 x}}+\frac{8026963 \sqrt{1-2 x} \sqrt{3+5 x}}{40000000}+\frac{23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}{25000}+\frac{2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}+\frac{21 (1-2 x)^{3/2} \sqrt{3+5 x} (64435+118392 x)}{4000000}+\frac{88296593 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{40000000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt{3+5 x}}+\frac{8026963 \sqrt{1-2 x} \sqrt{3+5 x}}{40000000}+\frac{23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}{25000}+\frac{2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}+\frac{21 (1-2 x)^{3/2} \sqrt{3+5 x} (64435+118392 x)}{4000000}+\frac{88296593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{40000000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0671638, size = 98, normalized size = 0.51 \[ \frac{-10 \left (3110400000 x^7+1697760000 x^6-4464936000 x^5-1391171400 x^4+3137091690 x^3+1095371425 x^2-558948208 x-210855251\right )-264889779 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1200000000 \sqrt{1-2 x} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 181, normalized size = 0.9 \begin{align*}{\frac{1}{2400000000} \left ( 31104000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}+32529600000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-28384560000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+6622244475\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-28103994000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7946693370\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+17318919900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2384008011\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +19613174200\,x\sqrt{-10\,{x}^{2}-x+3}+4217105020\,\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 [C] time = 1.87883, size = 478, normalized size = 2.48 \begin{align*} \frac{81}{15625} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{891}{25000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{70759953}{800000000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{27401}{1250000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{8811}{500000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3125 \,{\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3125 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{18 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3125 \,{\left (5 \, x + 3\right )}} + \frac{584793}{2000000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{13450239}{40000000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{3267}{62500} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{11 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{18750 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{99 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6250 \,{\left (5 \, x + 3\right )}} - \frac{121 \, \sqrt{-10 \, x^{2} - x + 3}}{93750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{638 \, \sqrt{-10 \, x^{2} - x + 3}}{9375 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.81742, size = 404, normalized size = 2.09 \begin{align*} -\frac{264889779 \, \sqrt{10}{\left (25 \, x^{2} + 30 \, x + 9\right )} \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 ) - 20 \,{\left (1555200000 \, x^{6} + 1626480000 \, x^{5} - 1419228000 \, x^{4} - 1405199700 \, x^{3} + 865945995 \, x^{2} + 980658710 \, x + 210855251\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2400000000 \,{\left (25 \, x^{2} + 30 \, x + 9\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 = 2.91671, size = 290, normalized size = 1.5 \begin{align*} \frac{1}{1000000000} \,{\left (12 \,{\left (24 \,{\left (12 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} - 613 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 19439 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1264235 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 10674335 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{18750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{88296593}{400000000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{561 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{312500 \, \sqrt{5 \, x + 3}} + \frac{11 \,{\left (\frac{765 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1171875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]