3.2472 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{3}{40} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3-\frac{259}{800} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (77820 x+187559)}{128000}+\frac{10866247 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{128000 \sqrt{10}} \]

[Out]

(-259*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^
3*Sqrt[3 + 5*x])/40 - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(187559 + 77820*x))/128000
+ (10866247*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(128000*Sqrt[10])

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Rubi [A]  time = 0.186069, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{40} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3-\frac{259}{800} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (77820 x+187559)}{128000}+\frac{10866247 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{128000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x)^4/(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]),x]

[Out]

(-259*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^
3*Sqrt[3 + 5*x])/40 - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(187559 + 77820*x))/128000
+ (10866247*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(128000*Sqrt[10])

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Rubi in Sympy [A]  time = 18.6092, size = 105, normalized size = 0.93 \[ - \frac{3 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{40} - \frac{259 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{800} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{2042775 x}{2} + \frac{19693695}{8}\right )}{240000} + \frac{10866247 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1280000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**4/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)

[Out]

-3*sqrt(-2*x + 1)*(3*x + 2)**3*sqrt(5*x + 3)/40 - 259*sqrt(-2*x + 1)*(3*x + 2)**
2*sqrt(5*x + 3)/800 - sqrt(-2*x + 1)*sqrt(5*x + 3)*(2042775*x/2 + 19693695/8)/24
0000 + 10866247*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/1280000

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Mathematica [A]  time = 0.113637, size = 65, normalized size = 0.58 \[ \frac{-30 \sqrt{1-2 x} \sqrt{5 x+3} \left (86400 x^3+297120 x^2+462540 x+518491\right )-10866247 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1280000} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x)^4/(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]),x]

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(518491 + 462540*x + 297120*x^2 + 86400*x^3) -
10866247*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/1280000

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Maple [A]  time = 0.018, size = 104, normalized size = 0.9 \[{\frac{1}{2560000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -5184000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-17827200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -27752400\,x\sqrt{-10\,{x}^{2}-x+3}-31109460\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x)

[Out]

1/2560000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-5184000*x^3*(-10*x^2-x+3)^(1/2)-17827200
*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-27752400*x*(-10*
x^2-x+3)^(1/2)-31109460*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49344, size = 101, normalized size = 0.89 \[ -\frac{81}{40} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{5571}{800} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{69381}{6400} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{10866247}{2560000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{1555473}{128000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/(sqrt(5*x + 3)*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

-81/40*sqrt(-10*x^2 - x + 3)*x^3 - 5571/800*sqrt(-10*x^2 - x + 3)*x^2 - 69381/64
00*sqrt(-10*x^2 - x + 3)*x - 10866247/2560000*sqrt(10)*arcsin(-20/11*x - 1/11) -
1555473/128000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.222002, size = 90, normalized size = 0.8 \[ -\frac{1}{2560000} \, \sqrt{10}{\left (6 \, \sqrt{10}{\left (86400 \, x^{3} + 297120 \, x^{2} + 462540 \, x + 518491\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 10866247 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/(sqrt(5*x + 3)*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

-1/2560000*sqrt(10)*(6*sqrt(10)*(86400*x^3 + 297120*x^2 + 462540*x + 518491)*sqr
t(5*x + 3)*sqrt(-2*x + 1) - 10866247*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x +
3)*sqrt(-2*x + 1))))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{4}}{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**4/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)

[Out]

Integral((3*x + 2)**4/(sqrt(-2*x + 1)*sqrt(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.233821, size = 85, normalized size = 0.75 \[ -\frac{1}{6400000} \, \sqrt{5}{\left (6 \,{\left (12 \,{\left (8 \,{\left (180 \, x + 403\right )}{\left (5 \, x + 3\right )} + 16609\right )}{\left (5 \, x + 3\right )} + 1646339\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 54331235 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4/(sqrt(5*x + 3)*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

-1/6400000*sqrt(5)*(6*(12*(8*(180*x + 403)*(5*x + 3) + 16609)*(5*x + 3) + 164633
9)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 54331235*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*
x + 3)))