3.2524 \(\int \frac{(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=138 \[ \frac{9}{80} \sqrt{1-2 x} (5 x+3)^{7/2}+\frac{49 (5 x+3)^{7/2}}{22 \sqrt{1-2 x}}+\frac{25397 \sqrt{1-2 x} (5 x+3)^{5/2}}{3520}+\frac{25397}{512} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{838101 \sqrt{1-2 x} \sqrt{5 x+3}}{2048}-\frac{9219111 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048 \sqrt{10}} \]

[Out]

(838101*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048 + (25397*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)
)/512 + (25397*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/3520 + (49*(3 + 5*x)^(7/2))/(22*Sq
rt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/80 - (9219111*ArcSin[Sqrt[2/11]
*Sqrt[3 + 5*x]])/(2048*Sqrt[10])

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Rubi [A]  time = 0.169689, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{9}{80} \sqrt{1-2 x} (5 x+3)^{7/2}+\frac{49 (5 x+3)^{7/2}}{22 \sqrt{1-2 x}}+\frac{25397 \sqrt{1-2 x} (5 x+3)^{5/2}}{3520}+\frac{25397}{512} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{838101 \sqrt{1-2 x} \sqrt{5 x+3}}{2048}-\frac{9219111 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(838101*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048 + (25397*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)
)/512 + (25397*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/3520 + (49*(3 + 5*x)^(7/2))/(22*Sq
rt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/80 - (9219111*ArcSin[Sqrt[2/11]
*Sqrt[3 + 5*x]])/(2048*Sqrt[10])

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Rubi in Sympy [A]  time = 14.2799, size = 126, normalized size = 0.91 \[ \frac{9 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{80} + \frac{25397 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{3520} + \frac{25397 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{512} + \frac{838101 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2048} - \frac{9219111 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{20480} + \frac{49 \left (5 x + 3\right )^{\frac{7}{2}}}{22 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/80 + 25397*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/352
0 + 25397*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/512 + 838101*sqrt(-2*x + 1)*sqrt(5*x +
3)/2048 - 9219111*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/20480 + 49*(5*x + 3)
**(7/2)/(22*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.100928, size = 74, normalized size = 0.54 \[ \frac{9219111 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (57600 x^4+243520 x^3+517096 x^2+966014 x-1405233\right )}{20480 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(-1405233 + 966014*x + 517096*x^2 + 243520*x^3 + 57600*x^4) +
9219111*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(20480*Sqrt[1 - 2*x])

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Maple [A]  time = 0.017, size = 140, normalized size = 1. \[ -{\frac{1}{-40960+81920\,x} \left ( -1152000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4870400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-10341920\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-9219111\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -19320280\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40960*(-1152000*x^4*(-10*x^2-x+3)^(1/2)-4870400*x^3*(-10*x^2-x+3)^(1/2)+18438
222*10^(1/2)*arcsin(20/11*x+1/11)*x-10341920*x^2*(-10*x^2-x+3)^(1/2)-9219111*10^
(1/2)*arcsin(20/11*x+1/11)-19320280*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)
^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49685, size = 147, normalized size = 1.07 \[ -\frac{1125 \, x^{5}}{8 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{21725 \, x^{4}}{32 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{414505 \, x^{3}}{256 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3190679 \, x^{2}}{1024 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{9219111}{40960} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{4128123 \, x}{2048 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4215699}{2048 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1125/8*x^5/sqrt(-10*x^2 - x + 3) - 21725/32*x^4/sqrt(-10*x^2 - x + 3) - 414505/
256*x^3/sqrt(-10*x^2 - x + 3) - 3190679/1024*x^2/sqrt(-10*x^2 - x + 3) + 9219111
/40960*sqrt(10)*arcsin(-20/11*x - 1/11) + 4128123/2048*x/sqrt(-10*x^2 - x + 3) +
4215699/2048/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.230255, size = 113, normalized size = 0.82 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (57600 \, x^{4} + 243520 \, x^{3} + 517096 \, x^{2} + 966014 \, x - 1405233\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 9219111 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{40960 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40960*sqrt(10)*(2*sqrt(10)*(57600*x^4 + 243520*x^3 + 517096*x^2 + 966014*x - 1
405233)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 9219111*(2*x - 1)*arctan(1/20*sqrt(10)*(2
0*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(2*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)**2*(3+5*x)**(5/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.238138, size = 131, normalized size = 0.95 \[ -\frac{9219111}{20480} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 329 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 25397 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1396835 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 46095555 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{256000 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9219111/20480*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/256000*(2*(4*(8*
(36*sqrt(5)*(5*x + 3) + 329*sqrt(5))*(5*x + 3) + 25397*sqrt(5))*(5*x + 3) + 1396
835*sqrt(5))*(5*x + 3) - 46095555*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x -
1)