3.2112 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=88 \[ -\frac{2589 \sqrt{1-2 x}}{13310 (5 x+3)}-\frac{613 \sqrt{1-2 x}}{605 (5 x+3)^2}+\frac{49}{22 \sqrt{1-2 x} (5 x+3)^2}-\frac{2589 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6655 \sqrt{55}} \]

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (613*Sqrt[1 - 2*x])/(605*(3 + 5*x)^2) - (258
9*Sqrt[1 - 2*x])/(13310*(3 + 5*x)) - (2589*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(6
655*Sqrt[55])

_______________________________________________________________________________________

Rubi [A]  time = 0.113158, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{2589 \sqrt{1-2 x}}{13310 (5 x+3)}-\frac{613 \sqrt{1-2 x}}{605 (5 x+3)^2}+\frac{49}{22 \sqrt{1-2 x} (5 x+3)^2}-\frac{2589 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6655 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (613*Sqrt[1 - 2*x])/(605*(3 + 5*x)^2) - (258
9*Sqrt[1 - 2*x])/(13310*(3 + 5*x)) - (2589*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(6
655*Sqrt[55])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.76342, size = 71, normalized size = 0.81 \[ - \frac{2589 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{366025} + \frac{2589}{33275 \sqrt{- 2 x + 1}} - \frac{137}{6050 \sqrt{- 2 x + 1} \left (5 x + 3\right )} - \frac{1}{550 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2589*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/366025 + 2589/(33275*sqrt(-2*x
+ 1)) - 137/(6050*sqrt(-2*x + 1)*(5*x + 3)) - 1/(550*sqrt(-2*x + 1)*(5*x + 3)**2
)

_______________________________________________________________________________________

Mathematica [A]  time = 0.129668, size = 58, normalized size = 0.66 \[ \frac{\frac{55 \left (25890 x^2+29561 x+8392\right )}{\sqrt{1-2 x} (5 x+3)^2}-5178 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{732050} \]

Antiderivative was successfully verified.

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

[Out]

((55*(8392 + 29561*x + 25890*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 5178*Sqrt[55]*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/732050

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 57, normalized size = 0.7 \[{\frac{98}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{50}{1331\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{139}{50} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1551}{250}\sqrt{1-2\,x}} \right ) }-{\frac{2589\,\sqrt{55}}{366025}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

98/1331/(1-2*x)^(1/2)+50/1331*(139/50*(1-2*x)^(3/2)-1551/250*(1-2*x)^(1/2))/(-6-
10*x)^2-2589/366025*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.49349, size = 112, normalized size = 1.27 \[ \frac{2589}{732050} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{12945 \,{\left (2 \, x - 1\right )}^{2} + 110902 \, x + 3839}{6655 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2589/732050*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) + 1/6655*(12945*(2*x - 1)^2 + 110902*x + 3839)/(25*(-2*x + 1)^(5/2) - 110
*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

_______________________________________________________________________________________

Fricas [A]  time = 0.224875, size = 116, normalized size = 1.32 \[ \frac{\sqrt{55}{\left (2589 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (25890 \, x^{2} + 29561 \, x + 8392\right )}\right )}}{732050 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/732050*sqrt(55)*(2589*(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)*(25890*x^2 + 29561*x + 8392))/((25
*x^2 + 30*x + 9)*sqrt(-2*x + 1))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.244751, size = 104, normalized size = 1.18 \[ \frac{2589}{732050} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{98}{1331 \, \sqrt{-2 \, x + 1}} + \frac{695 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1551 \, \sqrt{-2 \, x + 1}}{26620 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2589/732050*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) + 98/1331/sqrt(-2*x + 1) + 1/26620*(695*(-2*x + 1)^(3/2) - 1551*
sqrt(-2*x + 1))/(5*x + 3)^2