3.1907 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=93 \[ \frac{71 \sqrt{1-2 x}}{10 (5 x+3)}-\frac{11 \sqrt{1-2 x}}{10 (5 x+3)^2}+14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2379 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

[Out]

(-11*Sqrt[1 - 2*x])/(10*(3 + 5*x)^2) + (71*Sqrt[1 - 2*x])/(10*(3 + 5*x)) + 14*Sq
rt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (2379*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]
])/(5*Sqrt[55])

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Rubi [A]  time = 0.183646, 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 \[ \frac{71 \sqrt{1-2 x}}{10 (5 x+3)}-\frac{11 \sqrt{1-2 x}}{10 (5 x+3)^2}+14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2379 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-11*Sqrt[1 - 2*x])/(10*(3 + 5*x)^2) + (71*Sqrt[1 - 2*x])/(10*(3 + 5*x)) + 14*Sq
rt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (2379*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]
])/(5*Sqrt[55])

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Rubi in Sympy [A]  time = 21.6272, size = 82, normalized size = 0.88 \[ \frac{71 \sqrt{- 2 x + 1}}{10 \left (5 x + 3\right )} - \frac{11 \sqrt{- 2 x + 1}}{10 \left (5 x + 3\right )^{2}} + 14 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )} - \frac{2379 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{275} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

71*sqrt(-2*x + 1)/(10*(5*x + 3)) - 11*sqrt(-2*x + 1)/(10*(5*x + 3)**2) + 14*sqrt
(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7) - 2379*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x +
1)/11)/275

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Mathematica [A]  time = 0.156133, size = 78, normalized size = 0.84 \[ \frac{\sqrt{1-2 x} (355 x+202)}{10 (5 x+3)^2}+14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2379 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(202 + 355*x))/(10*(3 + 5*x)^2) + 14*Sqrt[21]*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]] - (2379*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55])

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Maple [A]  time = 0.018, size = 66, normalized size = 0.7 \[ 14\,{\it Artanh} \left ( 1/7\,\sqrt{21}\sqrt{1-2\,x} \right ) \sqrt{21}+50\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{71\, \left ( 1-2\,x \right ) ^{3/2}}{50}}+{\frac{759\,\sqrt{1-2\,x}}{250}} \right ) }-{\frac{2379\,\sqrt{55}}{275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+50*(-71/50*(1-2*x)^(3/2)+759/250
*(1-2*x)^(1/2))/(-6-10*x)^2-2379/275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/
2)

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Maxima [A]  time = 1.52434, size = 149, normalized size = 1.6 \[ \frac{2379}{550} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - 7 \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 759 \, \sqrt{-2 \, x + 1}}{5 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2379/550*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 1/5*(355*(-2*x + 1)^(3/2) - 759*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 1
1)

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Fricas [A]  time = 0.225125, size = 159, normalized size = 1.71 \[ \frac{\sqrt{55}{\left (70 \, \sqrt{55} \sqrt{21}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + \sqrt{55}{\left (355 \, x + 202\right )} \sqrt{-2 \, x + 1} + 2379 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{550 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/550*sqrt(55)*(70*sqrt(55)*sqrt(21)*(25*x^2 + 30*x + 9)*log((3*x - sqrt(21)*sqr
t(-2*x + 1) - 5)/(3*x + 2)) + sqrt(55)*(355*x + 202)*sqrt(-2*x + 1) + 2379*(25*x
^2 + 30*x + 9)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(25*x^2
+ 30*x + 9)

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Sympy [A]  time = 146.733, size = 372, normalized size = 4. \[ \frac{1628 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{5} + \frac{968 \left (\begin{cases} \frac{\sqrt{55} \left (\frac{3 \log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )^{2}}\right )}{6655} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{5} - 294 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right ) + 490 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1628*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*
sqrt(-2*x + 1)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(5
5)*sqrt(-2*x + 1)/11 - 1)))/605, (x <= 1/2) & (x > -3/5)))/5 + 968*Piecewise((sq
rt(55)*(3*log(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(-2*x + 1)
/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(-2*
x + 1)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)) - 1/(16*(sqrt(55)*s
qrt(-2*x + 1)/11 - 1)**2))/6655, (x <= 1/2) & (x > -3/5)))/5 - 294*Piecewise((-s
qrt(21)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(s
qrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3)) + 490*Piecewise((-sqrt(55)*acoth(
sqrt(55)*sqrt(-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 = 0.217633, size = 144, normalized size = 1.55 \[ \frac{2379}{550} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - 7 \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 759 \, \sqrt{-2 \, x + 1}}{20 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2379/550*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 7*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) - 1/20*(355*(-2*x + 1)^(3/2) - 759*sqrt(-2*x + 1))/(5*x + 3)
^2