3.2374 \(\int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=116 \[ -\frac{1}{8} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{11}{240} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{121}{960} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{1331 \sqrt{5 x+3} \sqrt{1-2 x}}{3200}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200 \sqrt{10}} \]

[Out]

(1331*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3200 + (121*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/96
0 + (11*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/240 - ((1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/8 +
(14641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*Sqrt[10])

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Rubi [A]  time = 0.109075, 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 \[ -\frac{1}{8} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{11}{240} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{121}{960} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{1331 \sqrt{5 x+3} \sqrt{1-2 x}}{3200}+\frac{14641 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(1331*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3200 + (121*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/96
0 + (11*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/240 - ((1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/8 +
(14641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*Sqrt[10])

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Rubi in Sympy [A]  time = 9.77037, size = 104, normalized size = 0.9 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{20} + \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{120} + \frac{121 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{800} - \frac{1331 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3200} + \frac{14641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{32000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/20 + 11*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/12
0 + 121*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/800 - 1331*sqrt(-2*x + 1)*sqrt(5*x + 3)/
3200 + 14641*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/32000

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Mathematica [A]  time = 0.0918614, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (9600 x^3-12640 x^2+3020 x+4443\right )-43923 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{96000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(4443 + 3020*x - 12640*x^2 + 9600*x^3) - 43923*S
qrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/96000

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Maple [A]  time = 0.007, size = 104, normalized size = 0.9 \[{\frac{1}{20} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}}+{\frac{11}{120} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}}+{\frac{121}{800} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{3200}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{14641\,\sqrt{10}}{64000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(1-2*x)^(5/2)*(3+5*x)^(3/2)+11/120*(1-2*x)^(3/2)*(3+5*x)^(3/2)+121/800*(3+5
*x)^(3/2)*(1-2*x)^(1/2)-1331/3200*(1-2*x)^(1/2)*(3+5*x)^(1/2)+14641/64000*((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 = 1.48296, size = 95, normalized size = 0.82 \[ -\frac{1}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{17}{120} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{121}{160} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{14641}{64000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{121}{3200} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(-10*x^2 - x + 3)^(3/2)*x + 17/120*(-10*x^2 - x + 3)^(3/2) + 121/160*sqrt(
-10*x^2 - x + 3)*x - 14641/64000*sqrt(10)*arcsin(-20/11*x - 1/11) + 121/3200*sqr
t(-10*x^2 - x + 3)

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Fricas [A]  time = 0.214622, size = 90, normalized size = 0.78 \[ \frac{1}{192000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (9600 \, x^{3} - 12640 \, x^{2} + 3020 \, x + 4443\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 43923 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192000*sqrt(10)*(2*sqrt(10)*(9600*x^3 - 12640*x^2 + 3020*x + 4443)*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 = 35.7772, size = 269, normalized size = 2.32 \[ \begin{cases} \frac{10 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{\sqrt{10 x - 5}} - \frac{253 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{6 \sqrt{10 x - 5}} + \frac{15367 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{240 \sqrt{10 x - 5}} - \frac{177023 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4800 \sqrt{10 x - 5}} + \frac{14641 i \sqrt{x + \frac{3}{5}}}{3200 \sqrt{10 x - 5}} - \frac{14641 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{32000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{14641 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{32000} - \frac{10 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{\sqrt{- 10 x + 5}} + \frac{253 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{6 \sqrt{- 10 x + 5}} - \frac{15367 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{240 \sqrt{- 10 x + 5}} + \frac{177023 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{4800 \sqrt{- 10 x + 5}} - \frac{14641 \sqrt{x + \frac{3}{5}}}{3200 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((10*I*(x + 3/5)**(9/2)/sqrt(10*x - 5) - 253*I*(x + 3/5)**(7/2)/(6*sqrt
(10*x - 5)) + 15367*I*(x + 3/5)**(5/2)/(240*sqrt(10*x - 5)) - 177023*I*(x + 3/5)
**(3/2)/(4800*sqrt(10*x - 5)) + 14641*I*sqrt(x + 3/5)/(3200*sqrt(10*x - 5)) - 14
641*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/32000, 10*Abs(x + 3/5)/11 > 1),
(14641*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/32000 - 10*(x + 3/5)**(9/2)/sq
rt(-10*x + 5) + 253*(x + 3/5)**(7/2)/(6*sqrt(-10*x + 5)) - 15367*(x + 3/5)**(5/2
)/(240*sqrt(-10*x + 5)) + 177023*(x + 3/5)**(3/2)/(4800*sqrt(-10*x + 5)) - 14641
*sqrt(x + 3/5)/(3200*sqrt(-10*x + 5)), True))

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GIAC/XCAS [A]  time = 0.24523, size = 220, normalized size = 1.9 \[ \frac{1}{480000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{6000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/480000*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))) + 1/400*sqrt(5)*(2*(20*x + 1)*s
qrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))