3.2044 \(\int \frac{(2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^3} \, dx\)

Optimal. Leaf size=80 \[ -\frac{\sqrt{1-2 x} (3 x+2)^2}{110 (5 x+3)^2}-\frac{9 \sqrt{1-2 x} (715 x+432)}{6050 (5 x+3)}-\frac{1347 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^2)/(110*(3 + 5*x)^2) - (9*Sqrt[1 - 2*x]*(432 + 715*x))
/(6050*(3 + 5*x)) - (1347*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3025*Sqrt[55])

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Rubi [A]  time = 0.111022, 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 \[ -\frac{\sqrt{1-2 x} (3 x+2)^2}{110 (5 x+3)^2}-\frac{9 \sqrt{1-2 x} (715 x+432)}{6050 (5 x+3)}-\frac{1347 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3025 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^2)/(110*(3 + 5*x)^2) - (9*Sqrt[1 - 2*x]*(432 + 715*x))
/(6050*(3 + 5*x)) - (1347*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3025*Sqrt[55])

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Rubi in Sympy [A]  time = 12.6084, size = 68, normalized size = 0.85 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{110 \left (5 x + 3\right )^{2}} - \frac{\sqrt{- 2 x + 1} \left (32175 x + 19440\right )}{30250 \left (5 x + 3\right )} - \frac{1347 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{166375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**2/(110*(5*x + 3)**2) - sqrt(-2*x + 1)*(32175*x + 1944
0)/(30250*(5*x + 3)) - 1347*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/166375

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Mathematica [A]  time = 0.102432, size = 58, normalized size = 0.72 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (32670 x^2+39405 x+11884\right )}{(5 x+3)^2}-2694 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{332750} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(11884 + 39405*x + 32670*x^2))/(3 + 5*x)^2 - 2694*Sqrt[55]*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/332750

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Maple [A]  time = 0.017, size = 57, normalized size = 0.7 \[ -{\frac{27}{125}\sqrt{1-2\,x}}+{\frac{2}{5\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{201}{1210} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{203}{550}\sqrt{1-2\,x}} \right ) }-{\frac{1347\,\sqrt{55}}{166375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/125*(1-2*x)^(1/2)+2/5*(201/1210*(1-2*x)^(3/2)-203/550*(1-2*x)^(1/2))/(-6-10*
x)^2-1347/166375*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50548, size = 112, normalized size = 1.4 \[ \frac{1347}{332750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{27}{125} \, \sqrt{-2 \, x + 1} + \frac{1005 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2233 \, \sqrt{-2 \, x + 1}}{15125 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1347/332750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 27/125*sqrt(-2*x + 1) + 1/15125*(1005*(-2*x + 1)^(3/2) - 2233*sqrt(-2*x
+ 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 0.243604, size = 107, normalized size = 1.34 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (32670 \, x^{2} + 39405 \, x + 11884\right )} \sqrt{-2 \, x + 1} - 1347 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{332750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/332750*sqrt(55)*(sqrt(55)*(32670*x^2 + 39405*x + 11884)*sqrt(-2*x + 1) - 1347
*(25*x^2 + 30*x + 9)*log((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(2
5*x^2 + 30*x + 9)

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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/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.228553, size = 104, normalized size = 1.3 \[ \frac{1347}{332750} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{27}{125} \, \sqrt{-2 \, x + 1} + \frac{1005 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2233 \, \sqrt{-2 \, x + 1}}{60500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1347/332750*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) - 27/125*sqrt(-2*x + 1) + 1/60500*(1005*(-2*x + 1)^(3/2) - 2233*
sqrt(-2*x + 1))/(5*x + 3)^2