3.1947 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=154 \[ -\frac{55 \sqrt{1-2 x} (5 x+3)^3}{24 (3 x+2)^2}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{54 (3 x+2)^3}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}-\frac{2255 \sqrt{1-2 x} (5 x+3)^2}{378 (3 x+2)}+\frac{275 \sqrt{1-2 x} (4595 x+1123)}{13608}+\frac{645865 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6804 \sqrt{21}} \]

[Out]

(-2255*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(378*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3
)/(12*(2 + 3*x)^4) + (55*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(54*(2 + 3*x)^3) - (55*Sqr
t[1 - 2*x]*(3 + 5*x)^3)/(24*(2 + 3*x)^2) + (275*Sqrt[1 - 2*x]*(1123 + 4595*x))/1
3608 + (645865*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(6804*Sqrt[21])

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Rubi [A]  time = 0.296976, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{55 \sqrt{1-2 x} (5 x+3)^3}{24 (3 x+2)^2}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{54 (3 x+2)^3}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}-\frac{2255 \sqrt{1-2 x} (5 x+3)^2}{378 (3 x+2)}+\frac{275 \sqrt{1-2 x} (4595 x+1123)}{13608}+\frac{645865 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6804 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-2255*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(378*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^3
)/(12*(2 + 3*x)^4) + (55*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(54*(2 + 3*x)^3) - (55*Sqr
t[1 - 2*x]*(3 + 5*x)^3)/(24*(2 + 3*x)^2) + (275*Sqrt[1 - 2*x]*(1123 + 4595*x))/1
3608 + (645865*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(6804*Sqrt[21])

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Rubi in Sympy [A]  time = 20.6064, size = 121, normalized size = 0.79 \[ - \frac{55 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (39537 x + 25245\right )}{666792 \left (3 x + 2\right )^{2}} - \frac{55 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{2}}{378 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{3}}{12 \left (3 x + 2\right )^{4}} - \frac{645865 \left (- 2 x + 1\right )^{\frac{3}{2}}}{333396} - \frac{645865 \sqrt{- 2 x + 1}}{47628} + \frac{645865 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{142884} \]

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)**5,x)

[Out]

-55*(-2*x + 1)**(5/2)*(39537*x + 25245)/(666792*(3*x + 2)**2) - 55*(-2*x + 1)**(
5/2)*(5*x + 3)**2/(378*(3*x + 2)**3) - (-2*x + 1)**(5/2)*(5*x + 3)**3/(12*(3*x +
2)**4) - 645865*(-2*x + 1)**(3/2)/333396 - 645865*sqrt(-2*x + 1)/47628 + 645865
*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/142884

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Mathematica [A]  time = 0.138832, size = 73, normalized size = 0.47 \[ \frac{\frac{21 \sqrt{1-2 x} \left (1512000 x^5-8215200 x^4-32946525 x^3-39158517 x^2-19526798 x-3553918\right )}{(3 x+2)^4}+1291730 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{285768} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(-3553918 - 19526798*x - 39158517*x^2 - 32946525*x^3 - 821520
0*x^4 + 1512000*x^5))/(2 + 3*x)^4 + 1291730*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/285768

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Maple [A]  time = 0.017, size = 84, normalized size = 0.6 \[ -{\frac{500}{729} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{7600}{729}\sqrt{1-2\,x}}-{\frac{4}{9\, \left ( -4-6\,x \right ) ^{4}} \left ( -{\frac{159975}{112} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{4220087}{432} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{28870415}{1296} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{21951755}{1296}\sqrt{1-2\,x}} \right ) }+{\frac{645865\,\sqrt{21}}{142884}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-500/729*(1-2*x)^(3/2)-7600/729*(1-2*x)^(1/2)-4/9*(-159975/112*(1-2*x)^(7/2)+422
0087/432*(1-2*x)^(5/2)-28870415/1296*(1-2*x)^(3/2)[phone]/1296*(1-2*x)^(1/2))/
(-4-6*x)^4+645865/142884*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.4979, size = 173, normalized size = 1.12 \[ -\frac{500}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{645865}{285768} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{7600}{729} \, \sqrt{-2 \, x + 1} + \frac{12957975 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 88621827 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 202092905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 153662285 \, \sqrt{-2 \, x + 1}}{20412 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-500/729*(-2*x + 1)^(3/2) - 645865/285768*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 7600/729*sqrt(-2*x + 1) + 1/20412*(129579
75*(-2*x + 1)^(7/2) - 88621827*(-2*x + 1)^(5/2) + 202092905*(-2*x + 1)^(3/2) - 1
53662285*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 +
8232*x - 1715)

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Fricas [A]  time = 0.210509, size = 154, normalized size = 1. \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (1512000 \, x^{5} - 8215200 \, x^{4} - 32946525 \, x^{3} - 39158517 \, x^{2} - 19526798 \, x - 3553918\right )} \sqrt{-2 \, x + 1} + 645865 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{285768 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/285768*sqrt(21)*(sqrt(21)*(1512000*x^5 - 8215200*x^4 - 32946525*x^3 - 39158517
*x^2 - 19526798*x - 3553918)*sqrt(-2*x + 1) + 645865*(81*x^4 + 216*x^3 + 216*x^2
+ 96*x + 16)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16)

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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)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235267, size = 159, normalized size = 1.03 \[ -\frac{500}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{645865}{285768} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{7600}{729} \, \sqrt{-2 \, x + 1} - \frac{12957975 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 88621827 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 202092905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 153662285 \, \sqrt{-2 \, x + 1}}{326592 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-500/729*(-2*x + 1)^(3/2) - 645865/285768*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sq
rt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 7600/729*sqrt(-2*x + 1) - 1/32659
2*(12957975*(2*x - 1)^3*sqrt(-2*x + 1) + 88621827*(2*x - 1)^2*sqrt(-2*x + 1) - 2
02092905*(-2*x + 1)^(3/2) + 153662285*sqrt(-2*x + 1))/(3*x + 2)^4