3.2594 \(\int \frac{(2+3 x)^2}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{448 \sqrt{5 x+3}}{363 \sqrt{1-2 x}}+\frac{49 \sqrt{5 x+3}}{66 (1-2 x)^{3/2}}+\frac{9 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2 \sqrt{10}} \]

[Out]

(49*Sqrt[3 + 5*x])/(66*(1 - 2*x)^(3/2)) - (448*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x]
) + (9*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sqrt[10])

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Rubi [A]  time = 0.0983234, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{448 \sqrt{5 x+3}}{363 \sqrt{1-2 x}}+\frac{49 \sqrt{5 x+3}}{66 (1-2 x)^{3/2}}+\frac{9 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(49*Sqrt[3 + 5*x])/(66*(1 - 2*x)^(3/2)) - (448*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x]
) + (9*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sqrt[10])

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Rubi in Sympy [A]  time = 8.31524, size = 65, normalized size = 0.9 \[ \frac{9 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{20} - \frac{448 \sqrt{5 x + 3}}{363 \sqrt{- 2 x + 1}} + \frac{49 \sqrt{5 x + 3}}{66 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/20 - 448*sqrt(5*x + 3)/(363*sqrt(-2*x
+ 1)) + 49*sqrt(5*x + 3)/(66*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.145639, size = 64, normalized size = 0.89 \[ \frac{70 \sqrt{5 x+3} (256 x-51)+3267 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{7260 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(70*Sqrt[3 + 5*x]*(-51 + 256*x) + 3267*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/
11]*Sqrt[1 - 2*x]])/(7260*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.02, size = 103, normalized size = 1.4 \[{\frac{1}{14520\, \left ( -1+2\,x \right ) ^{2}} \left ( 13068\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-13068\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+3267\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +35840\,x\sqrt{-10\,{x}^{2}-x+3}-7140\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14520*(13068*10^(1/2)*arcsin(20/11*x+1/11)*x^2-13068*10^(1/2)*arcsin(20/11*x+1
/11)*x+3267*10^(1/2)*arcsin(20/11*x+1/11)+35840*x*(-10*x^2-x+3)^(1/2)-7140*(-10*
x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.48025, size = 84, normalized size = 1.17 \[ \frac{9}{40} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{66 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{448 \, \sqrt{-10 \, x^{2} - x + 3}}{363 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/40*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 49/66*sqrt(-10*x^2 - x + 3)/(4*x^2
- 4*x + 1) + 448/363*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.223716, size = 107, normalized size = 1.49 \[ \frac{\sqrt{10}{\left (14 \, \sqrt{10}{\left (256 \, x - 51\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3267 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{14520 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14520*sqrt(10)*(14*sqrt(10)*(256*x - 51)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 3267*(
4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(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. \[ \int \frac{\left (3 x + 2\right )^{2}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.252803, size = 78, normalized size = 1.08 \[ \frac{9}{20} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{7 \,{\left (256 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1023 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{18150 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/20*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 7/18150*(256*sqrt(5)*(5*x +
3) - 1023*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2