3.2371 \(\int (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=172 \[ -\frac{3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{3 (5 x+3)^{3/2} (26700 x+33857) (1-2 x)^{7/2}}{280000}-\frac{255169 \sqrt{5 x+3} (1-2 x)^{7/2}}{640000}+\frac{2806859 \sqrt{5 x+3} (1-2 x)^{5/2}}{19200000}+\frac{30875449 \sqrt{5 x+3} (1-2 x)^{3/2}}{76800000}+\frac{339629939 \sqrt{5 x+3} \sqrt{1-2 x}}{256000000}+\frac{3735929329 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000000 \sqrt{10}} \]

[Out]

(339629939*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000000 + (30875449*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/76800000 + (2806859*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/19200000 - (2551
69*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/640000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5
*x)^(3/2))/70 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)*(33857 + 26700*x))/280000 + (
3735929329*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000000*Sqrt[10])

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Rubi [A]  time = 0.210121, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{3 (5 x+3)^{3/2} (26700 x+33857) (1-2 x)^{7/2}}{280000}-\frac{255169 \sqrt{5 x+3} (1-2 x)^{7/2}}{640000}+\frac{2806859 \sqrt{5 x+3} (1-2 x)^{5/2}}{19200000}+\frac{30875449 \sqrt{5 x+3} (1-2 x)^{3/2}}{76800000}+\frac{339629939 \sqrt{5 x+3} \sqrt{1-2 x}}{256000000}+\frac{3735929329 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(339629939*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000000 + (30875449*(1 - 2*x)^(3/2)*Sq
rt[3 + 5*x])/76800000 + (2806859*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/19200000 - (2551
69*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/640000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5
*x)^(3/2))/70 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)*(33857 + 26700*x))/280000 + (
3735929329*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000000*Sqrt[10])

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Rubi in Sympy [A]  time = 18.8809, size = 158, normalized size = 0.92 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{70} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}} \left (60075 x + \frac{304713}{4}\right )}{210000} + \frac{255169 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1600000} + \frac{2806859 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{9600000} + \frac{30875449 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{64000000} - \frac{339629939 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{256000000} + \frac{3735929329 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2560000000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(7/2)*(3*x + 2)**2*(5*x + 3)**(3/2)/70 - (-2*x + 1)**(7/2)*(5*x +
3)**(3/2)*(60075*x + 304713/4)/210000 + 255169*(-2*x + 1)**(5/2)*(5*x + 3)**(3/
2)/1600000 + 2806859*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/9600000 + 30875449*sqrt(
-2*x + 1)*(5*x + 3)**(3/2)/64000000 - 339629939*sqrt(-2*x + 1)*sqrt(5*x + 3)/256
000000 + 3735929329*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/2560000000

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Mathematica [A]  time = 0.140554, size = 80, normalized size = 0.47 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (82944000000 x^6+97459200000 x^5-52468992000 x^4-85095638400 x^3+9906627680 x^2+29819034260 x-679278531\right )-78454515909 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{53760000000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-679278531 + 29819034260*x + 9906627680*x^2 - 8
5095638400*x^3 - 52468992000*x^4 + 97459200000*x^5 + 82944000000*x^6) - 78454515
909*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/53760000000

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Maple [A]  time = 0.014, size = 155, normalized size = 0.9 \[{\frac{1}{107520000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1658880000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-1049379840000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1701912768000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-13585570620\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/107520000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1658880000000*x^6*(-10*x^2-x+3)^(1/2
)[phone]*x^5*(-10*x^2-x+3)^(1/2)-1049379840000*x^4*(-10*x^2-x+3)^(1/2)-17
01912768000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)+7845451
5909*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-1358557062
0*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.51048, size = 163, normalized size = 0.95 \[ -\frac{54}{35} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{1161}{700} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{47529}{70000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{5697497}{5600000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{5531929}{67200000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{30875449}{12800000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{3735929329}{5120000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{30875449}{256000000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-54/35*(-10*x^2 - x + 3)^(3/2)*x^4 - 1161/700*(-10*x^2 - x + 3)^(3/2)*x^3 + 4752
9/70000*(-10*x^2 - x + 3)^(3/2)*x^2 + 5697497/5600000*(-10*x^2 - x + 3)^(3/2)*x
- 5531929/67200000*(-10*x^2 - x + 3)^(3/2) + 30875449/12800000*sqrt(-10*x^2 - x
+ 3)*x - 3735929329/5120000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 30875449/25600
0000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.218289, size = 111, normalized size = 0.65 \[ \frac{1}{107520000000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (82944000000 \, x^{6} + 97459200000 \, x^{5} - 52468992000 \, x^{4} - 85095638400 \, x^{3} + 9906627680 \, x^{2} + 29819034260 \, x - 679278531\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 78454515909 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/107520000000*sqrt(10)*(2*sqrt(10)*(82944000000*x^6 + 97459200000*x^5 - 5246899
2000*x^4 - 85095638400*x^3 + 9906627680*x^2 + 29819034260*x - 679278531)*sqrt(5*
x + 3)*sqrt(-2*x + 1) + 78454515909*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x +
3)*sqrt(-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((1-2*x)**(5/2)*(2+3*x)**3*(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279506, size = 548, normalized size = 3.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/89600000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x
+ 3) - 3968469)*(5*x + 3) + 33617829)*(5*x + 3) - 276044685)*(5*x + 3) + 8735611
5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5
*x + 3))) + 9/640000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)
*(5*x + 3) - 318159)*(5*x + 3) + 3237255)*(5*x + 3) - 2656665)*sqrt(5*x + 3)*sqr
t(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 3/1280000
0*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x +
3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)
*sqrt(5*x + 3))) - 29/960000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x
+ 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)
*sqrt(5*x + 3))) + 1/6000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3
)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/50*sqrt
(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(2
2)*sqrt(5*x + 3)))