3.1921 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=101 \[ \frac{(1-2 x)^{7/2}}{63 (3 x+2)^3}-\frac{53 (1-2 x)^{5/2}}{189 (3 x+2)^2}+\frac{265 (1-2 x)^{3/2}}{567 (3 x+2)}+\frac{530}{567} \sqrt{1-2 x}-\frac{530 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

[Out]

(530*Sqrt[1 - 2*x])/567 + (1 - 2*x)^(7/2)/(63*(2 + 3*x)^3) - (53*(1 - 2*x)^(5/2)
)/(189*(2 + 3*x)^2) + (265*(1 - 2*x)^(3/2))/(567*(2 + 3*x)) - (530*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Rubi [A]  time = 0.103009, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{7/2}}{63 (3 x+2)^3}-\frac{53 (1-2 x)^{5/2}}{189 (3 x+2)^2}+\frac{265 (1-2 x)^{3/2}}{567 (3 x+2)}+\frac{530}{567} \sqrt{1-2 x}-\frac{530 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(530*Sqrt[1 - 2*x])/567 + (1 - 2*x)^(7/2)/(63*(2 + 3*x)^3) - (53*(1 - 2*x)^(5/2)
)/(189*(2 + 3*x)^2) + (265*(1 - 2*x)^(3/2))/(567*(2 + 3*x)) - (530*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Rubi in Sympy [A]  time = 10.8474, size = 87, normalized size = 0.86 \[ \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{63 \left (3 x + 2\right )^{3}} - \frac{53 \left (- 2 x + 1\right )^{\frac{5}{2}}}{189 \left (3 x + 2\right )^{2}} + \frac{265 \left (- 2 x + 1\right )^{\frac{3}{2}}}{567 \left (3 x + 2\right )} + \frac{530 \sqrt{- 2 x + 1}}{567} - \frac{530 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1701} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(7/2)/(63*(3*x + 2)**3) - 53*(-2*x + 1)**(5/2)/(189*(3*x + 2)**2) +
265*(-2*x + 1)**(3/2)/(567*(3*x + 2)) + 530*sqrt(-2*x + 1)/567 - 530*sqrt(21)*at
anh(sqrt(21)*sqrt(-2*x + 1)/7)/1701

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Mathematica [A]  time = 0.109669, size = 63, normalized size = 0.62 \[ \frac{\sqrt{1-2 x} \left (1080 x^3+3627 x^2+2983 x+713\right )}{81 (3 x+2)^3}-\frac{530 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(713 + 2983*x + 3627*x^2 + 1080*x^3))/(81*(2 + 3*x)^3) - (530*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Maple [A]  time = 0.017, size = 66, normalized size = 0.7 \[{\frac{40}{81}\sqrt{1-2\,x}}+{\frac{8}{3\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{163}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{1505}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{6125}{108}\sqrt{1-2\,x}} \right ) }-{\frac{530\,\sqrt{21}}{1701}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

40/81*(1-2*x)^(1/2)+8/3*(-163/12*(1-2*x)^(5/2)+1505/27*(1-2*x)^(3/2)-6125/108*(1
-2*x)^(1/2))/(-4-6*x)^3-530/1701*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.51163, size = 136, normalized size = 1.35 \[ \frac{265}{1701} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{40}{81} \, \sqrt{-2 \, x + 1} + \frac{2 \,{\left (1467 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 6020 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 6125 \, \sqrt{-2 \, x + 1}\right )}}{81 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

265/1701*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) + 40/81*sqrt(-2*x + 1) + 2/81*(1467*(-2*x + 1)^(5/2) - 6020*(-2*x + 1)^(3/2)
+ 6125*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.212904, size = 127, normalized size = 1.26 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (1080 \, x^{3} + 3627 \, x^{2} + 2983 \, x + 713\right )} \sqrt{-2 \, x + 1} + 265 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1701 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1701*sqrt(21)*(sqrt(21)*(1080*x^3 + 3627*x^2 + 2983*x + 713)*sqrt(-2*x + 1) +
265*(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)**(5/2)*(3+5*x)/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214154, size = 126, normalized size = 1.25 \[ \frac{265}{1701} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{40}{81} \, \sqrt{-2 \, x + 1} + \frac{1467 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 6020 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 6125 \, \sqrt{-2 \, x + 1}}{324 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

265/1701*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 40/81*sqrt(-2*x + 1) + 1/324*(1467*(2*x - 1)^2*sqrt(-2*x + 1) - 60
20*(-2*x + 1)^(3/2) + 6125*sqrt(-2*x + 1))/(3*x + 2)^3