3.2452 \(\int \frac{(2+3 x)^2 (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{3}{40} \sqrt{1-2 x} (3 x+2) (5 x+3)^{5/2}-\frac{251}{800} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{14529 \sqrt{1-2 x} (5 x+3)^{3/2}}{6400}-\frac{479457 \sqrt{1-2 x} \sqrt{5 x+3}}{25600}+\frac{5274027 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600 \sqrt{10}} \]

[Out]

(-479457*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600 - (14529*Sqrt[1 - 2*x]*(3 + 5*x)^(3/
2))/6400 - (251*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*
(3 + 5*x)^(5/2))/40 + (5274027*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600*Sqrt[10]
)

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Rubi [A]  time = 0.140917, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{40} \sqrt{1-2 x} (3 x+2) (5 x+3)^{5/2}-\frac{251}{800} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{14529 \sqrt{1-2 x} (5 x+3)^{3/2}}{6400}-\frac{479457 \sqrt{1-2 x} \sqrt{5 x+3}}{25600}+\frac{5274027 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-479457*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600 - (14529*Sqrt[1 - 2*x]*(3 + 5*x)^(3/
2))/6400 - (251*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*
(3 + 5*x)^(5/2))/40 + (5274027*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600*Sqrt[10]
)

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Rubi in Sympy [A]  time = 11.6318, size = 109, normalized size = 0.9 \[ - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}} \left (9 x + 6\right )}{40} - \frac{251 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{800} - \frac{14529 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{6400} - \frac{479457 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{25600} + \frac{5274027 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{256000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(5*x + 3)**(5/2)*(9*x + 6)/40 - 251*sqrt(-2*x + 1)*(5*x + 3)**(5
/2)/800 - 14529*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/6400 - 479457*sqrt(-2*x + 1)*sqr
t(5*x + 3)/25600 + 5274027*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/256000

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Mathematica [A]  time = 0.0957549, size = 65, normalized size = 0.54 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (144000 x^3+469600 x^2+698580 x+760653\right )-5274027 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{256000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(760653 + 698580*x + 469600*x^2 + 144000*x^3) -
5274027*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/256000

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Maple [A]  time = 0.014, size = 104, normalized size = 0.9 \[{\frac{1}{512000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -2880000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-9392000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+5274027\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -13971600\,x\sqrt{-10\,{x}^{2}-x+3}-15213060\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/512000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-2880000*x^3*(-10*x^2-x+3)^(1/2)-9392000*x
^2*(-10*x^2-x+3)^(1/2)+5274027*10^(1/2)*arcsin(20/11*x+1/11)-13971600*x*(-10*x^2
-x+3)^(1/2)-15213060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.48966, size = 101, normalized size = 0.83 \[ -\frac{45}{8} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{587}{32} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{34929}{1280} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{5274027}{512000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{760653}{25600} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-45/8*sqrt(-10*x^2 - x + 3)*x^3 - 587/32*sqrt(-10*x^2 - x + 3)*x^2 - 34929/1280*
sqrt(-10*x^2 - x + 3)*x - 5274027/512000*sqrt(10)*arcsin(-20/11*x - 1/11) - 7606
53/25600*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.217052, size = 90, normalized size = 0.74 \[ -\frac{1}{512000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (144000 \, x^{3} + 469600 \, x^{2} + 698580 \, x + 760653\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 5274027 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/512000*sqrt(10)*(2*sqrt(10)*(144000*x^3 + 469600*x^2 + 698580*x + 760653)*sqr
t(5*x + 3)*sqrt(-2*x + 1) - 5274027*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x +
3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 57.2444, size = 398, normalized size = 3.29 \[ \frac{2 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} + \frac{12 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (\frac{3 \sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{1936} + \frac{\sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} + \frac{18 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (\frac{7 \sqrt{2} \left (- 20 x - 1\right ) \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{3872} + \frac{2 \sqrt{2} \left (- 10 x + 5\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac{\sqrt{2} \sqrt{- 10 x + 5} \sqrt{5 x + 3}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x +
3)/968 - sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x +
3)/11)/8)/8, (x >= -3/5) & (x < 1/2)))/125 + 12*sqrt(5)*Piecewise((1331*sqrt(2)*
(3*sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/1936 + sqrt(2)*(-10*x + 5)*
*(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 5*asin
(sqrt(22)*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5) & (x < 1/2)))/125 + 18*sqrt(5)*P
iecewise((14641*sqrt(2)*(7*sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/387
2 + 2*sqrt(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 + sqrt(2)*sqrt(-10*x + 5)
*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048
- sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/
128)/32, (x >= -3/5) & (x < 1/2)))/125

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GIAC/XCAS [A]  time = 0.232241, size = 85, normalized size = 0.7 \[ -\frac{1}{256000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (180 \, x + 371\right )}{\left (5 \, x + 3\right )} + 14529\right )}{\left (5 \, x + 3\right )} + 479457\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 5274027 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/256000*sqrt(5)*(2*(4*(8*(180*x + 371)*(5*x + 3) + 14529)*(5*x + 3) + 479457)*
sqrt(5*x + 3)*sqrt(-10*x + 5) - 5274027*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3)))