3.2566 \(\int \frac{(2+3 x) \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{7 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{3 \sqrt{5 x+3}}{2 \sqrt{1-2 x}}+\frac{3}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-3*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (7*(3 + 5*x)^(3/2))/(33*(1 - 2*x)^(3/2))
+ (3*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/2

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Rubi [A]  time = 0.074046, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{7 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{3 \sqrt{5 x+3}}{2 \sqrt{1-2 x}}+\frac{3}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (7*(3 + 5*x)^(3/2))/(33*(1 - 2*x)^(3/2))
+ (3*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/2

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Rubi in Sympy [A]  time = 7.88819, size = 65, normalized size = 0.88 \[ \frac{3 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4} - \frac{3 \sqrt{5 x + 3}}{2 \sqrt{- 2 x + 1}} + \frac{7 \left (5 x + 3\right )^{\frac{3}{2}}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/4 - 3*sqrt(5*x + 3)/(2*sqrt(-2*x + 1)
) + 7*(5*x + 3)**(3/2)/(33*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.111145, size = 64, normalized size = 0.86 \[ \frac{2 \sqrt{5 x+3} (268 x-57)+99 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{132 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x]*(-57 + 268*x) + 99*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]
*Sqrt[1 - 2*x]])/(132*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.015, size = 103, normalized size = 1.4 \[{\frac{1}{264\, \left ( -1+2\,x \right ) ^{2}} \left ( 396\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-396\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+99\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1072\,x\sqrt{-10\,{x}^{2}-x+3}-228\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/264*(396*10^(1/2)*arcsin(20/11*x+1/11)*x^2-396*10^(1/2)*arcsin(20/11*x+1/11)*x
+99*10^(1/2)*arcsin(20/11*x+1/11)+1072*x*(-10*x^2-x+3)^(1/2)-228*(-10*x^2-x+3)^(
1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50722, size = 65, normalized size = 0.88 \[ \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{3 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{10 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 10/33*sqrt(-10*x^2 - x + 3)/(2*x -
1)

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Fricas [A]  time = 0.2277, size = 115, normalized size = 1.55 \[ \frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (268 \, x - 57\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 99 \, \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{264 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/264*sqrt(2)*(2*sqrt(2)*(268*x - 57)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 99*sqrt(5)*
(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*
x + 1))))/(4*x^2 - 4*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right ) \sqrt{5 x + 3}}{\left (- 2 x + 1\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)*sqrt(5*x + 3)/(-2*x + 1)**(5/2), x)

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GIAC/XCAS [A]  time = 0.228133, size = 78, normalized size = 1.05 \[ \frac{3}{4} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (268 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1089 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1650 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/1650*(268*sqrt(5)*(5*x + 3)
- 1089*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2