3.2113 \(\int \frac{2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=88 \[ -\frac{213 \sqrt{1-2 x}}{2662 (5 x+3)}+\frac{71}{605 \sqrt{1-2 x} (5 x+3)}-\frac{1}{110 \sqrt{1-2 x} (5 x+3)^2}-\frac{213 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]

[Out]

-1/(110*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 71/(605*Sqrt[1 - 2*x]*(3 + 5*x)) - (213*Sqr
t[1 - 2*x])/(2662*(3 + 5*x)) - (213*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1331*Sqr
t[55])

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Rubi [A]  time = 0.092413, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{213 \sqrt{1-2 x}}{2662 (5 x+3)}+\frac{71}{605 \sqrt{1-2 x} (5 x+3)}-\frac{1}{110 \sqrt{1-2 x} (5 x+3)^2}-\frac{213 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(110*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 71/(605*Sqrt[1 - 2*x]*(3 + 5*x)) - (213*Sqr
t[1 - 2*x])/(2662*(3 + 5*x)) - (213*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1331*Sqr
t[55])

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Rubi in Sympy [A]  time = 8.53077, size = 71, normalized size = 0.81 \[ - \frac{213 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{73205} + \frac{213}{6655 \sqrt{- 2 x + 1}} - \frac{71}{1210 \sqrt{- 2 x + 1} \left (5 x + 3\right )} - \frac{1}{110 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-213*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/73205 + 213/(6655*sqrt(-2*x + 1)
) - 71/(1210*sqrt(-2*x + 1)*(5*x + 3)) - 1/(110*sqrt(-2*x + 1)*(5*x + 3)**2)

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Mathematica [A]  time = 0.131311, size = 58, normalized size = 0.66 \[ \frac{\frac{55 \left (2130 x^2+1775 x+274\right )}{\sqrt{1-2 x} (5 x+3)^2}-426 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{146410} \]

Antiderivative was successfully verified.

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

[Out]

((55*(274 + 1775*x + 2130*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 426*Sqrt[55]*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/146410

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Maple [A]  time = 0.017, size = 57, normalized size = 0.7 \[{\frac{28}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{100}{1331\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{73}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{33}{4}\sqrt{1-2\,x}} \right ) }-{\frac{213\,\sqrt{55}}{73205}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

28/1331/(1-2*x)^(1/2)+100/1331*(73/20*(1-2*x)^(3/2)-33/4*(1-2*x)^(1/2))/(-6-10*x
)^2-213/73205*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.4936, size = 112, normalized size = 1.27 \[ \frac{213}{146410} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1065 \,{\left (2 \, x - 1\right )}^{2} + 7810 \, x - 517}{1331 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

213/146410*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 1/1331*(1065*(2*x - 1)^2 + 7810*x - 517)/(25*(-2*x + 1)^(5/2) - 110*(-2*
x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]  time = 0.220922, size = 116, normalized size = 1.32 \[ \frac{\sqrt{55}{\left (213 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (2130 \, x^{2} + 1775 \, x + 274\right )}\right )}}{146410 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/146410*sqrt(55)*(213*(25*x^2 + 30*x + 9)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8
) + 55*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(55)*(2130*x^2 + 1775*x + 274))/((25*x^2
+ 30*x + 9)*sqrt(-2*x + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.222317, size = 104, normalized size = 1.18 \[ \frac{213}{146410} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{28}{1331 \, \sqrt{-2 \, x + 1}} + \frac{5 \,{\left (73 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 165 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

213/146410*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) + 28/1331/sqrt(-2*x + 1) + 5/5324*(73*(-2*x + 1)^(3/2) - 165*sqrt
(-2*x + 1))/(5*x + 3)^2