3.2101 \(\int \frac{(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=80 \[ \frac{7 (3 x+2)^2}{11 \sqrt{1-2 x} (5 x+3)}+\frac{18 \sqrt{1-2 x} (935 x+559)}{3025 (5 x+3)}-\frac{204 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*(3 + 5*x)) + (18*Sqrt[1 - 2*x]*(559 + 935*x))/
(3025*(3 + 5*x)) - (204*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3025*Sqrt[55])

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Rubi [A]  time = 0.11436, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{7 (3 x+2)^2}{11 \sqrt{1-2 x} (5 x+3)}+\frac{18 \sqrt{1-2 x} (935 x+559)}{3025 (5 x+3)}-\frac{204 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*(3 + 5*x)) + (18*Sqrt[1 - 2*x]*(559 + 935*x))/
(3025*(3 + 5*x)) - (204*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3025*Sqrt[55])

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Rubi in Sympy [A]  time = 12.1199, size = 66, normalized size = 0.82 \[ \frac{\sqrt{- 2 x + 1} \left (16830 x + 10062\right )}{3025 \left (5 x + 3\right )} - \frac{204 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{166375} + \frac{7 \left (3 x + 2\right )^{2}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(16830*x + 10062)/(3025*(5*x + 3)) - 204*sqrt(55)*atanh(sqrt(55)*
sqrt(-2*x + 1)/11)/166375 + 7*(3*x + 2)**2/(11*sqrt(-2*x + 1)*(5*x + 3))

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Mathematica [A]  time = 0.114532, size = 61, normalized size = 0.76 \[ \frac{\frac{55 \sqrt{1-2 x} \left (16335 x^2-19806 x-17762\right )}{10 x^2+x-3}-204 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{166375} \]

Antiderivative was successfully verified.

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

[Out]

((55*Sqrt[1 - 2*x]*(-17762 - 19806*x + 16335*x^2))/(-3 + x + 10*x^2) - 204*Sqrt[
55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/166375

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Maple [A]  time = 0.02, size = 54, normalized size = 0.7 \[{\frac{27}{50}\sqrt{1-2\,x}}+{\frac{343}{242}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{15125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{204\,\sqrt{55}}{166375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/50*(1-2*x)^(1/2)+343/242/(1-2*x)^(1/2)+2/15125*(1-2*x)^(1/2)/(-6/5-2*x)-204/1
66375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51587, size = 100, normalized size = 1.25 \[ \frac{102}{166375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{27}{50} \, \sqrt{-2 \, x + 1} - \frac{42879 \, x + 25723}{3025 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

102/166375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 27/50*sqrt(-2*x + 1) - 1/3025*(42879*x + 25723)/(5*(-2*x + 1)^(3/2) - 11
*sqrt(-2*x + 1))

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Fricas [A]  time = 0.215287, size = 104, normalized size = 1.3 \[ \frac{\sqrt{55}{\left (102 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (16335 \, x^{2} - 19806 \, x - 17762\right )}\right )}}{166375 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/166375*sqrt(55)*(102*(5*x + 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55*sqr
t(-2*x + 1))/(5*x + 3)) - sqrt(55)*(16335*x^2 - 19806*x - 17762))/((5*x + 3)*sqr
t(-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)**3/(1-2*x)**(3/2)/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.250223, size = 104, normalized size = 1.3 \[ \frac{102}{166375} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{27}{50} \, \sqrt{-2 \, x + 1} - \frac{42879 \, x + 25723}{3025 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

102/166375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq
rt(-2*x + 1))) + 27/50*sqrt(-2*x + 1) - 1/3025*(42879*x + 25723)/(5*(-2*x + 1)^(
3/2) - 11*sqrt(-2*x + 1))