3.2563 \(\int \frac{(2+3 x)^4 \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{\sqrt{5 x+3} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{299 \sqrt{5 x+3} (3 x+2)^3}{66 \sqrt{1-2 x}}-\frac{697}{88} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (7306140 x+17606479)}{70400}+\frac{13246251 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

[Out]

(-697*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/88 - (299*(2 + 3*x)^3*Sqrt[3 + 5*
x])/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^4*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) - (Sqrt
[1 - 2*x]*Sqrt[3 + 5*x]*(17606479 + 7306140*x))/70400 + (13246251*ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10])

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Rubi [A]  time = 0.258207, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{5 x+3} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac{299 \sqrt{5 x+3} (3 x+2)^3}{66 \sqrt{1-2 x}}-\frac{697}{88} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (7306140 x+17606479)}{70400}+\frac{13246251 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-697*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/88 - (299*(2 + 3*x)^3*Sqrt[3 + 5*
x])/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^4*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) - (Sqrt
[1 - 2*x]*Sqrt[3 + 5*x]*(17606479 + 7306140*x))/70400 + (13246251*ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10])

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Rubi in Sympy [A]  time = 26.9749, size = 131, normalized size = 0.92 \[ - \frac{697 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{88} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{82194075 x}{4} + \frac{792291555}{16}\right )}{198000} + \frac{13246251 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{64000} - \frac{299 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{66 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{4} \sqrt{5 x + 3}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-697*sqrt(-2*x + 1)*(3*x + 2)**2*sqrt(5*x + 3)/88 - sqrt(-2*x + 1)*sqrt(5*x + 3)
*(82194075*x/4 + 792291555/16)/198000 + 13246251*sqrt(10)*asin(sqrt(22)*sqrt(5*x
+ 3)/11)/64000 - 299*(3*x + 2)**3*sqrt(5*x + 3)/(66*sqrt(-2*x + 1)) + (3*x + 2)
**4*sqrt(5*x + 3)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.176014, size = 79, normalized size = 0.56 \[ \frac{437126283 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (2851200 x^4+15040080 x^3+52700868 x^2-183672928 x+66038637\right )}{2112000 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[3 + 5*x]*(66038637 - 183672928*x + 52700868*x^2 + 15040080*x^3 + 28512
00*x^4) + 437126283*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])
/(2112000*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.022, size = 154, normalized size = 1.1 \[{\frac{1}{4224000\, \left ( -1+2\,x \right ) ^{2}} \left ( -57024000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-300801600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1748505132\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1054017360\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-1320772740\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4224000*(-57024000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+
1/11)*x^2-300801600*x^3*(-10*x^2-x+3)^(1/2)-1748505132*10^(1/2)*arcsin(20/11*x+1
/11)*x-1054017360*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11
)[phone]*x*(-10*x^2-x+3)^(1/2)-1320772740*(-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 [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Fricas [A]  time = 0.228946, size = 127, normalized size = 0.89 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (2851200 \, x^{4} + 15040080 \, x^{3} + 52700868 \, x^{2} - 183672928 \, x + 66038637\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 437126283 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{4224000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4224000*sqrt(10)*(2*sqrt(10)*(2851200*x^4 + 15040080*x^3 + 52700868*x^2 - 183
672928*x + 66038637)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 437126283*(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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.237615, size = 131, normalized size = 0.92 \[ \frac{13246251}{64000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (891 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 115 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 8919 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 291417650 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4808389113 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{26400000 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13246251/64000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/26400000*(4*(891
*(4*(8*sqrt(5)*(5*x + 3) + 115*sqrt(5))*(5*x + 3) + 8919*sqrt(5))*(5*x + 3) - 29
1417650*sqrt(5))*(5*x + 3) + 4808389113*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(
2*x - 1)^2