3.1887 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=93 \[ \frac{65 \sqrt{1-2 x}}{6 (3 x+2)}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2}+\frac{2243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-22 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2) + (65*Sqrt[1 - 2*x])/(6*(2 + 3*x)) + (2243*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]) - 22*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]]

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Rubi [A]  time = 0.183725, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{65 \sqrt{1-2 x}}{6 (3 x+2)}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2}+\frac{2243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-22 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2) + (65*Sqrt[1 - 2*x])/(6*(2 + 3*x)) + (2243*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]) - 22*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]]

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Rubi in Sympy [A]  time = 21.4851, size = 82, normalized size = 0.88 \[ \frac{65 \sqrt{- 2 x + 1}}{6 \left (3 x + 2\right )} + \frac{7 \sqrt{- 2 x + 1}}{6 \left (3 x + 2\right )^{2}} + \frac{2243 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{63} - 22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

65*sqrt(-2*x + 1)/(6*(3*x + 2)) + 7*sqrt(-2*x + 1)/(6*(3*x + 2)**2) + 2243*sqrt(
21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/63 - 22*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x +
1)/11)

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Mathematica [A]  time = 0.155359, size = 78, normalized size = 0.84 \[ \frac{\sqrt{1-2 x} (195 x+137)}{6 (3 x+2)^2}+\frac{2243 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3 \sqrt{21}}-22 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(137 + 195*x))/(6*(2 + 3*x)^2) + (2243*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/(3*Sqrt[21]) - 22*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.017, size = 66, normalized size = 0.7 \[ -18\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{65\, \left ( 1-2\,x \right ) ^{3/2}}{18}}-{\frac{469\,\sqrt{1-2\,x}}{54}} \right ) }+{\frac{2243\,\sqrt{21}}{63}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-22\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-18*(65/18*(1-2*x)^(3/2)-469/54*(1-2*x)^(1/2))/(-4-6*x)^2+2243/63*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)-22*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50793, size = 149, normalized size = 1.6 \[ 11 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2243}{126} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{195 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 469 \, \sqrt{-2 \, x + 1}}{3 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
2243/126*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) - 1/3*(195*(-2*x + 1)^(3/2) - 469*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 0.235456, size = 158, normalized size = 1.7 \[ \frac{\sqrt{21}{\left (66 \, \sqrt{55} \sqrt{21}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (195 \, x + 137\right )} \sqrt{-2 \, x + 1} + 2243 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{126 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/126*sqrt(21)*(66*sqrt(55)*sqrt(21)*(9*x^2 + 12*x + 4)*log((5*x + sqrt(55)*sqrt
(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*(195*x + 137)*sqrt(-2*x + 1) + 2243*(9*x^2
+ 12*x + 4)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(9*x^2 + 1
2*x + 4)

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Sympy [A]  time = 142.934, size = 372, normalized size = 4. \[ \frac{868 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{3} - \frac{392 \left (\begin{cases} \frac{\sqrt{21} \left (\frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )^{2}}\right )}{1029} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{3} - 726 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right ) + 1210 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

868*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sq
rt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*s
qrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/3 - 392*Piecewise((sqrt(21
)*(3*log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1
)/16 + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7
+ 1)**2) + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(-2*x +
1)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3)))/3 - 726*Piecewise((-sqrt(21)*aco
th(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqr
t(-2*x + 1)/7)/21, -2*x + 1 < 7/3)) + 1210*Piecewise((-sqrt(55)*acoth(sqrt(55)*s
qrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)
/11)/55, -2*x + 1 < 11/5))

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GIAC/XCAS [A]  time = 0.214258, size = 144, normalized size = 1.55 \[ 11 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2243}{126} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{195 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 469 \, \sqrt{-2 \, x + 1}}{12 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 2243/126*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21)
+ 3*sqrt(-2*x + 1))) - 1/12*(195*(-2*x + 1)^(3/2) - 469*sqrt(-2*x + 1))/(3*x + 2
)^2