3.2601 \(\int \frac{(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{1561 (3 x+2)^3}{726 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{7723 \sqrt{1-2 x} (3 x+2)^2}{39930 \sqrt{5 x+3}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (16227780 x+39109961)}{2129600}+\frac{243189 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

[Out]

(7723*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*Sqrt[3 + 5*x]) - (1561*(2 + 3*x)^3)/(726
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x
]) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(39109961 + 16227780*x))/2129600 + (243189*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sqrt[10])

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Rubi [A]  time = 0.272784, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{1561 (3 x+2)^3}{726 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{7723 \sqrt{1-2 x} (3 x+2)^2}{39930 \sqrt{5 x+3}}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (16227780 x+39109961)}{2129600}+\frac{243189 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(7723*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*Sqrt[3 + 5*x]) - (1561*(2 + 3*x)^3)/(726
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x
]) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(39109961 + 16227780*x))/2129600 + (243189*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sqrt[10])

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Rubi in Sympy [A]  time = 26.8836, size = 133, normalized size = 0.94 \[ \frac{7723 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{39930 \sqrt{5 x + 3}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{60854175 x}{4} + \frac{586649415}{16}\right )}{1996500} + \frac{243189 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16000} - \frac{1561 \left (3 x + 2\right )^{3}}{726 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{7 \left (3 x + 2\right )^{4}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7723*sqrt(-2*x + 1)*(3*x + 2)**2/(39930*sqrt(5*x + 3)) - sqrt(-2*x + 1)*sqrt(5*x
+ 3)*(60854175*x/4 + 586649415/16)/1996500 + 243189*sqrt(10)*asin(sqrt(22)*sqrt
(5*x + 3)/11)/16000 - 1561*(3*x + 2)**3/(726*sqrt(-2*x + 1)*sqrt(5*x + 3)) + 7*(
3*x + 2)**4/(33*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.17696, size = 89, normalized size = 0.63 \[ -\frac{10 \sqrt{5 x+3} \left (77623920 x^4+536898780 x^3-1790987404 x^2-525679641 x+435258129\right )-971053677 \sqrt{10-20 x} \left (10 x^2+x-3\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{63888000 (1-2 x)^{3/2} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[3 + 5*x]*(435258129 - 525679641*x - 1790987404*x^2 + 536898780*x^3 + 7
7623920*x^4) - 971053677*Sqrt[10 - 20*x]*(-3 + x + 10*x^2)*ArcSin[Sqrt[5/11]*Sqr
t[1 - 2*x]])/(63888000*(1 - 2*x)^(3/2)*(3 + 5*x))

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Maple [A]  time = 0.021, size = 168, normalized size = 1.2 \[{\frac{1}{127776000\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 19421073540\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-1552478400\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-7768429416\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-10737975600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-6797375739\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+35819748080\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-8705162580\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/127776000*(1-2*x)^(1/2)*(19421073540*10^(1/2)*arcsin(20/11*x+1/11)*x^3-1552478
400*x^4*(-10*x^2-x+3)^(1/2)-7768429416*10^(1/2)*arcsin(20/11*x+1/11)*x^2-1073797
5600*x^3*(-10*x^2-x+3)^(1/2)-6797375739*10^(1/2)*arcsin(20/11*x+1/11)*x+35819748
080*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)-8705162580*(-10*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^
(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49811, size = 151, normalized size = 1.06 \[ \frac{243 \, x^{3}}{40 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{243189}{32000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{7209 \, x^{2}}{160 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{751566017 \, x}{6388800 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{638622829}{6388800 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{16807}{528 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/40*x^3/sqrt(-10*x^2 - x + 3) + 243189/32000*sqrt(5)*sqrt(2)*arcsin(20/11*x +
1/11) + 7209/160*x^2/sqrt(-10*x^2 - x + 3) - 751566017/6388800*x/sqrt(-10*x^2 -
x + 3) - 638622829/6388800/sqrt(-10*x^2 - x + 3) - 16807/528/(2*sqrt(-10*x^2 -
x + 3)*x - sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.227332, size = 140, normalized size = 0.99 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (77623920 \, x^{4} + 536898780 \, x^{3} - 1790987404 \, x^{2} - 525679641 \, x + 435258129\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 971053677 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{127776000 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/127776000*sqrt(10)*(2*sqrt(10)*(77623920*x^4 + 536898780*x^3 - 1790987404*x^2
- 525679641*x + 435258129)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 971053677*(20*x^3 - 8
*x^2 - 7*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))
/(20*x^3 - 8*x^2 - 7*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.261919, size = 194, normalized size = 1.37 \[ \frac{243189}{16000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1663750 \, \sqrt{5 \, x + 3}} - \frac{{\left (4 \,{\left (323433 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 271 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 3237172310 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 53407238379 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{3993000000 \,{\left (2 \, x - 1\right )}^{2}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{831875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243189/16000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/1663750*sqrt(10)*(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/3993000000*(4*(323433*(12*
sqrt(5)*(5*x + 3) + 271*sqrt(5))*(5*x + 3) - 3237172310*sqrt(5))*(5*x + 3) + 534
07238379*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 2/831875*sqrt(10)*
sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))