3.2422 \(\int \frac{(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{2 (1-2 x)^{7/2}}{55 \sqrt{5 x+3}}+\frac{7}{275} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{7}{100} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{231 \sqrt{5 x+3} \sqrt{1-2 x}}{1000}+\frac{2541 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(7/2))/(55*Sqrt[3 + 5*x]) + (231*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1000
+ (7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/100 + (7*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/275
+ (2541*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1000*Sqrt[10])

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Rubi [A]  time = 0.119008, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 (1-2 x)^{7/2}}{55 \sqrt{5 x+3}}+\frac{7}{275} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{7}{100} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{231 \sqrt{5 x+3} \sqrt{1-2 x}}{1000}+\frac{2541 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - 2*x)^(7/2))/(55*Sqrt[3 + 5*x]) + (231*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1000
+ (7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/100 + (7*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/275
+ (2541*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1000*Sqrt[10])

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Rubi in Sympy [A]  time = 11.7375, size = 105, normalized size = 0.91 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{7}{2}}}{55 \sqrt{5 x + 3}} + \frac{7 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{275} + \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{100} + \frac{231 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1000} + \frac{2541 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{10000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-2*x + 1)**(7/2)/(55*sqrt(5*x + 3)) + 7*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/275
+ 7*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/100 + 231*sqrt(-2*x + 1)*sqrt(5*x + 3)/1000
+ 2541*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/10000

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Mathematica [A]  time = 0.140674, size = 65, normalized size = 0.56 \[ \frac{\frac{10 \sqrt{1-2 x} \left (800 x^3-1340 x^2+1125 x+943\right )}{\sqrt{5 x+3}}-2541 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{10000} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*(943 + 1125*x - 1340*x^2 + 800*x^3))/Sqrt[3 + 5*x] - 2541*Sqr
t[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/10000

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Maple [A]  time = 0.016, size = 116, normalized size = 1. \[{\frac{1}{20000} \left ( 16000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+12705\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-26800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+7623\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +22500\,x\sqrt{-10\,{x}^{2}-x+3}+18860\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20000*(16000*x^3*(-10*x^2-x+3)^(1/2)+12705*10^(1/2)*arcsin(20/11*x+1/11)*x-268
00*x^2*(-10*x^2-x+3)^(1/2)+7623*10^(1/2)*arcsin(20/11*x+1/11)+22500*x*(-10*x^2-x
+3)^(1/2)+18860*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(
1/2)

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Maxima [A]  time = 1.50753, size = 124, normalized size = 1.07 \[ -\frac{8 \, x^{4}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{87 \, x^{3}}{25 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{359 \, x^{2}}{100 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2541}{20000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{761 \, x}{1000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{943}{1000 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/5*x^4/sqrt(-10*x^2 - x + 3) + 87/25*x^3/sqrt(-10*x^2 - x + 3) - 359/100*x^2/s
qrt(-10*x^2 - x + 3) - 2541/20000*sqrt(10)*arcsin(-20/11*x - 1/11) - 761/1000*x/
sqrt(-10*x^2 - x + 3) + 943/1000/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.230741, size = 107, normalized size = 0.92 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (800 \, x^{3} - 1340 \, x^{2} + 1125 \, x + 943\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 2541 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{20000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20000*sqrt(10)*(2*sqrt(10)*(800*x^3 - 1340*x^2 + 1125*x + 943)*sqrt(5*x + 3)*s
qrt(-2*x + 1) + 2541*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))))/(5*x + 3)

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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+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.288294, size = 167, normalized size = 1.44 \[ \frac{1}{25000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 139 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 3597 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{2541}{10000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{6250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{3125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25000*(4*(8*sqrt(5)*(5*x + 3) - 139*sqrt(5))*(5*x + 3) + 3597*sqrt(5))*sqrt(5*
x + 3)*sqrt(-10*x + 5) + 2541/10000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))
- 121/6250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 242/31
25*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))