3.2033 \(\int \frac{(2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^2} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\sqrt{1-2 x} (3 x+2)^2}{55 (5 x+3)}-\frac{6}{55} \sqrt{1-2 x} (3 x+11)-\frac{8 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

[Out]

(-6*Sqrt[1 - 2*x]*(11 + 3*x))/55 - (Sqrt[1 - 2*x]*(2 + 3*x)^2)/(55*(3 + 5*x)) -
(8*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(55*Sqrt[55])

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Rubi [A]  time = 0.106922, antiderivative size = 73, 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 \[ -\frac{\sqrt{1-2 x} (3 x+2)^2}{55 (5 x+3)}-\frac{6}{55} \sqrt{1-2 x} (3 x+11)-\frac{8 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[1 - 2*x]*(11 + 3*x))/55 - (Sqrt[1 - 2*x]*(2 + 3*x)^2)/(55*(3 + 5*x)) -
(8*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(55*Sqrt[55])

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Rubi in Sympy [A]  time = 11.814, size = 61, normalized size = 0.84 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{55 \left (5 x + 3\right )} - \frac{\sqrt{- 2 x + 1} \left (1350 x + 4950\right )}{4125} - \frac{8 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{3025} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**2/(55*(5*x + 3)) - sqrt(-2*x + 1)*(1350*x + 4950)/412
5 - 8*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/3025

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Mathematica [A]  time = 0.101784, size = 58, normalized size = 0.79 \[ -\frac{\sqrt{1-2 x} \left (99 x^2+396 x+202\right )}{55 (5 x+3)}-\frac{8 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{55 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(202 + 396*x + 99*x^2))/(55*(3 + 5*x)) - (8*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]])/(55*Sqrt[55])

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Maple [A]  time = 0.016, size = 54, normalized size = 0.7 \[{\frac{9}{50} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{351}{250}\sqrt{1-2\,x}}+{\frac{2}{6875}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{8\,\sqrt{55}}{3025}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/50*(1-2*x)^(3/2)-351/250*(1-2*x)^(1/2)+2/6875*(1-2*x)^(1/2)/(-6/5-2*x)-8/3025*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48865, size = 96, normalized size = 1.32 \[ \frac{9}{50} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4}{3025} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{351}{250} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{1375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/50*(-2*x + 1)^(3/2) + 4/3025*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt
(55) + 5*sqrt(-2*x + 1))) - 351/250*sqrt(-2*x + 1) - 1/1375*sqrt(-2*x + 1)/(5*x
+ 3)

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Fricas [A]  time = 0.251243, size = 93, normalized size = 1.27 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (99 \, x^{2} + 396 \, x + 202\right )} \sqrt{-2 \, x + 1} - 4 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{3025 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3025*sqrt(55)*(sqrt(55)*(99*x^2 + 396*x + 202)*sqrt(-2*x + 1) - 4*(5*x + 3)*l
og((sqrt(55)*(5*x - 8) + 55*sqrt(-2*x + 1))/(5*x + 3)))/(5*x + 3)

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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)**2/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.212993, size = 100, normalized size = 1.37 \[ \frac{9}{50} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4}{3025} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{351}{250} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{1375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/50*(-2*x + 1)^(3/2) + 4/3025*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x +
1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 351/250*sqrt(-2*x + 1) - 1/1375*sqrt(-2*x +
1)/(5*x + 3)