3.1873 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=125 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{251 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac{5}{567} \sqrt{1-2 x} (7265 x+2323)-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

[Out]

(251*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(
9*(2 + 3*x)^3) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^2 - (5*Sqrt[1 - 2*x]*(2
323 + 7265*x))/567 - (36038*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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Rubi [A]  time = 0.205132, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{251 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac{5}{567} \sqrt{1-2 x} (7265 x+2323)-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(251*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(
9*(2 + 3*x)^3) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^2 - (5*Sqrt[1 - 2*x]*(2
323 + 7265*x))/567 - (36038*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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Rubi in Sympy [A]  time = 20.1066, size = 104, normalized size = 0.83 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (500850 x + 358146\right )}{71442 \left (3 x + 2\right )} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}}{7 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{3}}{9 \left (3 x + 2\right )^{3}} + \frac{36038 \sqrt{- 2 x + 1}}{3969} - \frac{36038 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{11907} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)*(500850*x + 358146)/(71442*(3*x + 2)) - 2*(-2*x + 1)**(3/2)*(5
*x + 3)**2/(7*(3*x + 2)**2) - (-2*x + 1)**(3/2)*(5*x + 3)**3/(9*(3*x + 2)**3) +
36038*sqrt(-2*x + 1)/3969 - 36038*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/1190
7

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Mathematica [A]  time = 0.128696, size = 68, normalized size = 0.54 \[ \frac{\sqrt{1-2 x} \left (-31500 x^4+81900 x^3+259614 x^2+199243 x+47939\right )}{567 (3 x+2)^3}-\frac{36038 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(47939 + 199243*x + 259614*x^2 + 81900*x^3 - 31500*x^4))/(567*(2
+ 3*x)^3) - (36038*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[{\frac{250}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2050}{243}\sqrt{1-2\,x}}+{\frac{2}{9\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{3938}{21} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{23306}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{26824}{27}\sqrt{1-2\,x}} \right ) }-{\frac{36038\,\sqrt{21}}{11907}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

250/243*(1-2*x)^(3/2)+2050/243*(1-2*x)^(1/2)+2/9*(-3938/21*(1-2*x)^(5/2)+23306/2
7*(1-2*x)^(3/2)-26824/27*(1-2*x)^(1/2))/(-4-6*x)^3-36038/11907*arctanh(1/7*21^(1
/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50661, size = 149, normalized size = 1.19 \[ \frac{250}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{18019}{11907} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2050}{243} \, \sqrt{-2 \, x + 1} + \frac{4 \,{\left (17721 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 81571 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 93884 \, \sqrt{-2 \, x + 1}\right )}}{1701 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

250/243*(-2*x + 1)^(3/2) + 18019/11907*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1
))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2050/243*sqrt(-2*x + 1) + 4/1701*(17721*(-2*
x + 1)^(5/2) - 81571*(-2*x + 1)^(3/2) + 93884*sqrt(-2*x + 1))/(27*(2*x - 1)^3 +
189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.212197, size = 134, normalized size = 1.07 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (31500 \, x^{4} - 81900 \, x^{3} - 259614 \, x^{2} - 199243 \, x - 47939\right )} \sqrt{-2 \, x + 1} - 18019 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{11907 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/11907*sqrt(21)*(sqrt(21)*(31500*x^4 - 81900*x^3 - 259614*x^2 - 199243*x - 479
39)*sqrt(-2*x + 1) - 18019*(27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5)
+ 21*sqrt(-2*x + 1))/(3*x + 2)))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)**(3/2)*(3+5*x)**3/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221543, size = 138, normalized size = 1.1 \[ \frac{250}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{18019}{11907} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2050}{243} \, \sqrt{-2 \, x + 1} + \frac{17721 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 81571 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 93884 \, \sqrt{-2 \, x + 1}}{3402 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

250/243*(-2*x + 1)^(3/2) + 18019/11907*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(
-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2050/243*sqrt(-2*x + 1) + 1/3402*(17
721*(2*x - 1)^2*sqrt(-2*x + 1) - 81571*(-2*x + 1)^(3/2) + 93884*sqrt(-2*x + 1))/
(3*x + 2)^3