3.2505 \(\int \frac{(2+3 x) \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ \frac{7 (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}+\frac{103}{44} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{103 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4 \sqrt{10}} \]

[Out]

(103*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/44 + (7*(3 + 5*x)^(3/2))/(11*Sqrt[1 - 2*x]) -
(103*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4*Sqrt[10])

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Rubi [A]  time = 0.07466, antiderivative size = 72, 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 \[ \frac{7 (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}+\frac{103}{44} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{103 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(103*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/44 + (7*(3 + 5*x)^(3/2))/(11*Sqrt[1 - 2*x]) -
(103*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4*Sqrt[10])

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Rubi in Sympy [A]  time = 7.09688, size = 65, normalized size = 0.9 \[ \frac{103 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{44} - \frac{103 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{40} + \frac{7 \left (5 x + 3\right )^{\frac{3}{2}}}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

103*sqrt(-2*x + 1)*sqrt(5*x + 3)/44 - 103*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/1
1)/40 + 7*(5*x + 3)**(3/2)/(11*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0586986, size = 59, normalized size = 0.82 \[ \frac{10 \sqrt{5 x+3} (17-6 x)+103 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{40 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(10*(17 - 6*x)*Sqrt[3 + 5*x] + 103*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(40*Sqrt[1 - 2*x])

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Maple [A]  time = 0.016, size = 89, normalized size = 1.2 \[ -{\frac{1}{-80+160\,x} \left ( 206\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-103\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -120\,x\sqrt{-10\,{x}^{2}-x+3}+340\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/80*(206*10^(1/2)*arcsin(20/11*x+1/11)*x-103*10^(1/2)*arcsin(20/11*x+1/11)-120
*x*(-10*x^2-x+3)^(1/2)+340*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+
2*x)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5068, size = 68, normalized size = 0.94 \[ -\frac{103}{80} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{4} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{7 \, \sqrt{-10 \, x^{2} - x + 3}}{2 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-103/80*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 3/4*sqrt(-10*x^2 - x + 3) - 7/2
*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.23363, size = 93, normalized size = 1.29 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (6 \, x - 17\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 103 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{80 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/80*sqrt(10)*(2*sqrt(10)*(6*x - 17)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 103*(2*x - 1
)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225252, size = 78, normalized size = 1.08 \[ -\frac{103}{40} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6 \, \sqrt{5}{\left (5 \, x + 3\right )} - 103 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{100 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-103/40*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/100*(6*sqrt(5)*(5*x + 3
) - 103*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)