3.2273 \(\int \sqrt{1-2 x} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=116 \[ -\frac{1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{5 x+3}+\frac{1331}{512} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{512 \sqrt{10}} \]

[Out]

(1331*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/512 - (605*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/256
- (55*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/96 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/8
+ (14641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(512*Sqrt[10])

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Rubi [A]  time = 0.102331, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac{605}{256} (1-2 x)^{3/2} \sqrt{5 x+3}+\frac{1331}{512} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{512 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(1331*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/512 - (605*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/256
- (55*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/96 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/8
+ (14641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(512*Sqrt[10])

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Rubi in Sympy [A]  time = 9.76436, size = 104, normalized size = 0.9 \[ \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{20} - \frac{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{240} - \frac{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{384} - \frac{1331 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{512} + \frac{14641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{5120} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(5*x + 3)**(7/2)/20 - 11*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/240 - 12
1*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/384 - 1331*sqrt(-2*x + 1)*sqrt(5*x + 3)/512 +
14641*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/5120

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Mathematica [A]  time = 0.0728457, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (9600 x^3+15520 x^2+5836 x-4005\right )-43923 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15360} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-4005 + 5836*x + 15520*x^2 + 9600*x^3) - 43923*
Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/15360

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Maple [A]  time = 0.008, size = 104, normalized size = 0.9 \[{\frac{1}{20} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}\sqrt{1-2\,x}}-{\frac{11}{240} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{121}{384} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{512}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{14641\,\sqrt{10}}{10240}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(3+5*x)^(7/2)*(1-2*x)^(1/2)-11/240*(3+5*x)^(5/2)*(1-2*x)^(1/2)-121/384*(3+5
*x)^(3/2)*(1-2*x)^(1/2)-1331/512*(1-2*x)^(1/2)*(3+5*x)^(1/2)+14641/10240*((1-2*x
)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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Maxima [A]  time = 1.48436, size = 95, normalized size = 0.82 \[ -\frac{5}{8} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{91}{96} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{605}{128} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{14641}{10240} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{121}{512} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/8*(-10*x^2 - x + 3)^(3/2)*x - 91/96*(-10*x^2 - x + 3)^(3/2) + 605/128*sqrt(-1
0*x^2 - x + 3)*x - 14641/10240*sqrt(10)*arcsin(-20/11*x - 1/11) + 121/512*sqrt(-
10*x^2 - x + 3)

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Fricas [A]  time = 0.217327, size = 90, normalized size = 0.78 \[ \frac{1}{30720} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (9600 \, x^{3} + 15520 \, x^{2} + 5836 \, x - 4005\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 43923 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30720*sqrt(10)*(2*sqrt(10)*(9600*x^3 + 15520*x^2 + 5836*x - 4005)*sqrt(5*x + 3
)*sqrt(-2*x + 1) + 43923*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*
x + 1))))

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Sympy [A]  time = 37.5278, size = 272, normalized size = 2.34 \[ \begin{cases} \frac{125 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{10 x - 5}} - \frac{1925 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{24 \sqrt{10 x - 5}} - \frac{605 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{192 \sqrt{10 x - 5}} - \frac{6655 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{768 \sqrt{10 x - 5}} + \frac{14641 i \sqrt{x + \frac{3}{5}}}{512 \sqrt{10 x - 5}} - \frac{14641 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{5120} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{14641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{5120} - \frac{125 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{- 10 x + 5}} + \frac{1925 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{24 \sqrt{- 10 x + 5}} + \frac{605 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{192 \sqrt{- 10 x + 5}} + \frac{6655 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{768 \sqrt{- 10 x + 5}} - \frac{14641 \sqrt{x + \frac{3}{5}}}{512 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((125*I*(x + 3/5)**(9/2)/(2*sqrt(10*x - 5)) - 1925*I*(x + 3/5)**(7/2)/(
24*sqrt(10*x - 5)) - 605*I*(x + 3/5)**(5/2)/(192*sqrt(10*x - 5)) - 6655*I*(x + 3
/5)**(3/2)/(768*sqrt(10*x - 5)) + 14641*I*sqrt(x + 3/5)/(512*sqrt(10*x - 5)) - 1
4641*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/5120, 10*Abs(x + 3/5)/11 > 1),
(14641*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/5120 - 125*(x + 3/5)**(9/2)/(2
*sqrt(-10*x + 5)) + 1925*(x + 3/5)**(7/2)/(24*sqrt(-10*x + 5)) + 605*(x + 3/5)**
(5/2)/(192*sqrt(-10*x + 5)) + 6655*(x + 3/5)**(3/2)/(768*sqrt(-10*x + 5)) - 1464
1*sqrt(x + 3/5)/(512*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.245229, size = 220, normalized size = 1.9 \[ \frac{1}{76800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{800} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/76800*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/
800*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 36
3*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/400*sqrt(5)*(2*(20*x + 1)*sqr
t(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))