3.1944 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=121 \[ -\frac{(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}+\frac{55}{81} (1-2 x)^{5/2} (5 x+3)^2+\frac{220}{729} (1-2 x)^{3/2}-\frac{22}{567} (1-2 x)^{5/2} (100 x+69)+\frac{1540}{729} \sqrt{1-2 x}-\frac{1540}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(1540*Sqrt[1 - 2*x])/729 + (220*(1 - 2*x)^(3/2))/729 + (55*(1 - 2*x)^(5/2)*(3 +
5*x)^2)/81 - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(3*(2 + 3*x)) - (22*(1 - 2*x)^(5/2)*(
69 + 100*x))/567 - (1540*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/729

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Rubi [A]  time = 0.198865, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}+\frac{55}{81} (1-2 x)^{5/2} (5 x+3)^2+\frac{220}{729} (1-2 x)^{3/2}-\frac{22}{567} (1-2 x)^{5/2} (100 x+69)+\frac{1540}{729} \sqrt{1-2 x}-\frac{1540}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(1540*Sqrt[1 - 2*x])/729 + (220*(1 - 2*x)^(3/2))/729 + (55*(1 - 2*x)^(5/2)*(3 +
5*x)^2)/81 - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(3*(2 + 3*x)) - (22*(1 - 2*x)^(5/2)*(
69 + 100*x))/567 - (1540*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/729

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Rubi in Sympy [A]  time = 20.0508, size = 104, normalized size = 0.86 \[ \frac{55 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{2}}{81} - \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (1800 x + 1242\right )}{5103} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{3}}{3 \left (3 x + 2\right )} + \frac{220 \left (- 2 x + 1\right )^{\frac{3}{2}}}{729} + \frac{1540 \sqrt{- 2 x + 1}}{729} - \frac{1540 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2187} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

55*(-2*x + 1)**(5/2)*(5*x + 3)**2/81 - 11*(-2*x + 1)**(5/2)*(1800*x + 1242)/5103
- (-2*x + 1)**(5/2)*(5*x + 3)**3/(3*(3*x + 2)) + 220*(-2*x + 1)**(3/2)/729 + 15
40*sqrt(-2*x + 1)/729 - 1540*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2187

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Mathematica [A]  time = 0.113175, size = 73, normalized size = 0.6 \[ \frac{\frac{3 \sqrt{1-2 x} \left (189000 x^5-17100 x^4-159714 x^3+25275 x^2+65558 x+13759\right )}{3 x+2}-10780 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15309} \]

Antiderivative was successfully verified.

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

[Out]

((3*Sqrt[1 - 2*x]*(13759 + 65558*x + 25275*x^2 - 159714*x^3 - 17100*x^4 + 189000
*x^5))/(2 + 3*x) - 10780*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/15309

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Maple [A]  time = 0.017, size = 81, normalized size = 0.7 \[{\frac{125}{162} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{725}{378} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{2}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{214}{729} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1526}{729}\sqrt{1-2\,x}}-{\frac{98}{2187}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{1540\,\sqrt{21}}{2187}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/162*(1-2*x)^(9/2)-725/378*(1-2*x)^(7/2)+2/27*(1-2*x)^(5/2)+214/729*(1-2*x)^(
3/2)+1526/729*(1-2*x)^(1/2)-98/2187*(1-2*x)^(1/2)/(-4/3-2*x)-1540/2187*arctanh(1
/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.48702, size = 132, normalized size = 1.09 \[ \frac{125}{162} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{725}{378} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{214}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{770}{2187} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1526}{729} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{729 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/162*(-2*x + 1)^(9/2) - 725/378*(-2*x + 1)^(7/2) + 2/27*(-2*x + 1)^(5/2) + 21
4/729*(-2*x + 1)^(3/2) + 770/2187*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) + 1526/729*sqrt(-2*x + 1) + 49/729*sqrt(-2*x + 1)/(
3*x + 2)

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Fricas [A]  time = 0.213866, size = 122, normalized size = 1.01 \[ \frac{\sqrt{3}{\left (5390 \, \sqrt{7}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{3}{\left (189000 \, x^{5} - 17100 \, x^{4} - 159714 \, x^{3} + 25275 \, x^{2} + 65558 \, x + 13759\right )} \sqrt{-2 \, x + 1}\right )}}{15309 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15309*sqrt(3)*(5390*sqrt(7)*(3*x + 2)*log((sqrt(3)*(3*x - 5) + 3*sqrt(7)*sqrt(
-2*x + 1))/(3*x + 2)) + sqrt(3)*(189000*x^5 - 17100*x^4 - 159714*x^3 + 25275*x^2
+ 65558*x + 13759)*sqrt(-2*x + 1))/(3*x + 2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215088, size = 165, normalized size = 1.36 \[ \frac{125}{162} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{725}{378} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{214}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{770}{2187} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1526}{729} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{729 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/162*(2*x - 1)^4*sqrt(-2*x + 1) + 725/378*(2*x - 1)^3*sqrt(-2*x + 1) + 2/27*(
2*x - 1)^2*sqrt(-2*x + 1) + 214/729*(-2*x + 1)^(3/2) + 770/2187*sqrt(21)*ln(1/2*
abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1526/729*sq
rt(-2*x + 1) + 49/729*sqrt(-2*x + 1)/(3*x + 2)