3.2083 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=80 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)}+\frac{2 \sqrt{1-2 x} (2975 x+1978)}{147 (3 x+2)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

[Out]

(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (2*Sqrt[1 - 2*x]*(1978 + 2975*x))
/(147*(2 + 3*x)) - (68*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

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Rubi [A]  time = 0.112864, 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 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)}+\frac{2 \sqrt{1-2 x} (2975 x+1978)}{147 (3 x+2)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (2*Sqrt[1 - 2*x]*(1978 + 2975*x))
/(147*(2 + 3*x)) - (68*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*Sqrt[21])

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Rubi in Sympy [A]  time = 12.3411, size = 66, normalized size = 0.82 \[ \frac{\sqrt{- 2 x + 1} \left (17850 x + 11868\right )}{441 \left (3 x + 2\right )} - \frac{68 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3087} + \frac{11 \left (5 x + 3\right )^{2}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(17850*x + 11868)/(441*(3*x + 2)) - 68*sqrt(21)*atanh(sqrt(21)*sq
rt(-2*x + 1)/7)/3087 + 11*(5*x + 3)**2/(7*sqrt(-2*x + 1)*(3*x + 2))

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Mathematica [A]  time = 0.124862, size = 61, normalized size = 0.76 \[ \frac{\frac{21 \sqrt{1-2 x} \left (6125 x^2-4968 x-6035\right )}{6 x^2+x-2}-68 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3087} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(-6035 - 4968*x + 6125*x^2))/(-2 + x + 6*x^2) - 68*Sqrt[21]*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3087

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Maple [A]  time = 0.02, size = 54, normalized size = 0.7 \[{\frac{125}{18}\sqrt{1-2\,x}}+{\frac{1331}{98}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{2}{1323}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{68\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/18*(1-2*x)^(1/2)+1331/98/(1-2*x)^(1/2)-2/1323*(1-2*x)^(1/2)/(-4/3-2*x)-68/30
87*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49557, size = 100, normalized size = 1.25 \[ \frac{34}{3087} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{125}{18} \, \sqrt{-2 \, x + 1} - \frac{35933 \, x + 23960}{441 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

34/3087*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 125/18*sqrt(-2*x + 1) - 1/441*(35933*x + 23960)/(3*(-2*x + 1)^(3/2) - 7*sqr
t(-2*x + 1))

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Fricas [A]  time = 0.247056, size = 104, normalized size = 1.3 \[ \frac{\sqrt{21}{\left (34 \,{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{21}{\left (6125 \, x^{2} - 4968 \, x - 6035\right )}\right )}}{3087 \,{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3087*sqrt(21)*(34*(3*x + 2)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-
2*x + 1))/(3*x + 2)) - sqrt(21)*(6125*x^2 - 4968*x - 6035))/((3*x + 2)*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((3+5*x)**3/(1-2*x)**(3/2)/(2+3*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.243656, size = 104, normalized size = 1.3 \[ \frac{34}{3087} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{125}{18} \, \sqrt{-2 \, x + 1} - \frac{35933 \, x + 23960}{441 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

34/3087*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 125/18*sqrt(-2*x + 1) - 1/441*(35933*x + 23960)/(3*(-2*x + 1)^(3/2)
- 7*sqrt(-2*x + 1))