3.2582 \(\int \frac{(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=157 \[ \frac{(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{373 (5 x+3)^{5/2} (3 x+2)^2}{66 \sqrt{1-2 x}}-\frac{9444023 \sqrt{1-2 x} (5 x+3)^{3/2}}{33792}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (40164 x+81191)}{1408}-\frac{9444023 \sqrt{1-2 x} \sqrt{5 x+3}}{4096}+\frac{103884253 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4096 \sqrt{10}} \]

[Out]

(-9444023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096 - (9444023*Sqrt[1 - 2*x]*(3 + 5*x)^(
3/2))/33792 - (373*(2 + 3*x)^2*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^
3*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(81191 +
40164*x))/1408 + (103884253*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096*Sqrt[10])

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Rubi [A]  time = 0.240923, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{373 (5 x+3)^{5/2} (3 x+2)^2}{66 \sqrt{1-2 x}}-\frac{9444023 \sqrt{1-2 x} (5 x+3)^{3/2}}{33792}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (40164 x+81191)}{1408}-\frac{9444023 \sqrt{1-2 x} \sqrt{5 x+3}}{4096}+\frac{103884253 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4096 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^3*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]

[Out]

(-9444023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096 - (9444023*Sqrt[1 - 2*x]*(3 + 5*x)^(
3/2))/33792 - (373*(2 + 3*x)^2*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^
3*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(81191 +
40164*x))/1408 + (103884253*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096*Sqrt[10])

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Rubi in Sympy [A]  time = 29.4856, size = 151, normalized size = 0.96 \[ - \frac{3369 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{224} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}} \left (\frac{185196375 x}{2} + \frac{3137518125}{16}\right )}{504000} - \frac{9444023 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4096} + \frac{103884253 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{40960} - \frac{373 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}}{42 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**3*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

-3369*sqrt(-2*x + 1)*(3*x + 2)**2*(5*x + 3)**(3/2)/224 - sqrt(-2*x + 1)*(5*x + 3
)**(3/2)*(185196375*x/2 + 3137518125/16)/504000 - 9444023*sqrt(-2*x + 1)*sqrt(5*
x + 3)/4096 + 103884253*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/40960 - 373*(3*
x + 2)**3*(5*x + 3)**(3/2)/(42*sqrt(-2*x + 1)) + (3*x + 2)**3*(5*x + 3)**(5/2)/(
3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.168806, size = 84, normalized size = 0.54 \[ \frac{311652759 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (1036800 x^5+5477760 x^4+15301008 x^3+40614996 x^2-129940960 x+47216961\right )}{122880 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^3*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]

[Out]

(-10*Sqrt[3 + 5*x]*(47216961 - 129940960*x + 40614996*x^2 + 15301008*x^3 + 54777
60*x^4 + 1036800*x^5) + 311652759*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*S
qrt[1 - 2*x]])/(122880*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.019, size = 171, normalized size = 1.1 \[{\frac{1}{245760\, \left ( -1+2\,x \right ) ^{2}} \left ( -20736000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-109555200\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-306020160\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1246611036\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-812299920\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-944339220\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^3*(3+5*x)^(5/2)/(1-2*x)^(5/2),x)

[Out]

1/245760*(-20736000*x^5*(-10*x^2-x+3)^(1/2)-109555200*x^4*(-10*x^2-x+3)^(1/2)+12
46611036*10^(1/2)*arcsin(20/11*x+1/11)*x^2-306020160*x^3*(-10*x^2-x+3)^(1/2)-124
6611036*10^(1/2)*arcsin(20/11*x+1/11)*x-812299920*x^2*(-10*x^2-x+3)^(1/2)+311652
759*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-944339220*(-1
0*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.52485, size = 439, normalized size = 2.8 \[ \frac{2606989}{2048} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{395307}{81920} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{495}{256} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{343 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{16 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac{63 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{64 \,{\left (2 \, x - 1\right )}} - \frac{16335}{1024} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{68607}{4096} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{114345}{512} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{18865 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{192 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{24255 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{128 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{3465 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{128 \,{\left (2 \, x - 1\right )}} + \frac{207515 \, \sqrt{-10 \, x^{2} - x + 3}}{384 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{3721795 \, \sqrt{-10 \, x^{2} - x + 3}}{768 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^3/(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

2606989/2048*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 395307/81920*I*sqrt(5)*sqr
t(2)*arcsin(20/11*x - 21/11) + 495/256*(-10*x^2 - x + 3)^(3/2) - 343/16*(-10*x^2
- x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 441/32*(-10*x^2 - x + 3)^
(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 63/16*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x +
1) - 27/64*(-10*x^2 - x + 3)^(5/2)/(2*x - 1) - 16335/1024*sqrt(10*x^2 - 21*x + 8
)*x + 68607/4096*sqrt(10*x^2 - 21*x + 8) - 114345/512*sqrt(-10*x^2 - x + 3) - 18
865/192*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 24255/128*(-10*x^2
- x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 3465/128*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) +
207515/384*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 3721795/768*sqrt(-10*x^2 -
x + 3)/(2*x - 1)

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Fricas [A]  time = 0.231026, size = 134, normalized size = 0.85 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (1036800 \, x^{5} + 5477760 \, x^{4} + 15301008 \, x^{3} + 40614996 \, x^{2} - 129940960 \, x + 47216961\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 311652759 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{245760 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^3/(-2*x + 1)^(5/2),x, algorithm="fricas")

[Out]

-1/245760*sqrt(10)*(2*sqrt(10)*(1036800*x^5 + 5477760*x^4 + 15301008*x^3 + 40614
996*x^2 - 129940960*x + 47216961)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 311652759*(4*x^
2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(4
*x^2 - 4*x + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**3*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242795, size = 149, normalized size = 0.95 \[ \frac{103884253}{40960} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \,{\left (36 \,{\left (8 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 137 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 13627 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9444023 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 1038842530 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 17140901745 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{7680000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^3/(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

103884253/40960*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/7680000*(4*(3*(
36*(8*(12*sqrt(5)*(5*x + 3) + 137*sqrt(5))*(5*x + 3) + 13627*sqrt(5))*(5*x + 3)
+ 9444023*sqrt(5))*(5*x + 3) - 1038842530*sqrt(5))*(5*x + 3) + 17140901745*sqrt(
5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2