3.2001 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)} \, dx\)

Optimal. Leaf size=54 \[ \frac{25}{18} (1-2 x)^{3/2}-\frac{155}{18} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]

[Out]

(-155*Sqrt[1 - 2*x])/18 + (25*(1 - 2*x)^(3/2))/18 - (2*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(9*Sqrt[21])

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Rubi [A]  time = 0.0698171, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{25}{18} (1-2 x)^{3/2}-\frac{155}{18} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-155*Sqrt[1 - 2*x])/18 + (25*(1 - 2*x)^(3/2))/18 - (2*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(9*Sqrt[21])

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Rubi in Sympy [A]  time = 7.201, size = 48, normalized size = 0.89 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{3}{2}}}{18} - \frac{155 \sqrt{- 2 x + 1}}{18} - \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{189} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(-2*x + 1)**(3/2)/18 - 155*sqrt(-2*x + 1)/18 - 2*sqrt(21)*atanh(sqrt(21)*sqrt
(-2*x + 1)/7)/189

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Mathematica [A]  time = 0.0678124, size = 46, normalized size = 0.85 \[ -\frac{5}{9} \sqrt{1-2 x} (5 x+13)-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*(13 + 5*x))/9 - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[2
1])

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Maple [A]  time = 0.009, size = 38, normalized size = 0.7 \[{\frac{25}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2\,\sqrt{21}}{189}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{155}{18}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/18*(1-2*x)^(3/2)-2/189*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-155/18*(1
-2*x)^(1/2)

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Maxima [A]  time = 1.49978, size = 74, normalized size = 1.37 \[ \frac{25}{18} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{189} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{155}{18} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/18*(-2*x + 1)^(3/2) + 1/189*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt
(21) + 3*sqrt(-2*x + 1))) - 155/18*sqrt(-2*x + 1)

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Fricas [A]  time = 0.220792, size = 72, normalized size = 1.33 \[ -\frac{1}{189} \, \sqrt{21}{\left (5 \, \sqrt{21}{\left (5 \, x + 13\right )} \sqrt{-2 \, x + 1} - \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/189*sqrt(21)*(5*sqrt(21)*(5*x + 13)*sqrt(-2*x + 1) - log((sqrt(21)*(3*x - 5)
+ 21*sqrt(-2*x + 1))/(3*x + 2)))

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Sympy [A]  time = 4.19013, size = 90, normalized size = 1.67 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{3}{2}}}{18} - \frac{155 \sqrt{- 2 x + 1}}{18} + \frac{2 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21}}{3 \sqrt{- 2 x + 1}} \right )}}{21} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{3}{7} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21}}{3 \sqrt{- 2 x + 1}} \right )}}{21} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{3}{7} \end{cases}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(-2*x + 1)**(3/2)/18 - 155*sqrt(-2*x + 1)/18 + 2*Piecewise((-sqrt(21)*acoth(s
qrt(21)/(3*sqrt(-2*x + 1)))/21, 1/(-2*x + 1) > 3/7), (-sqrt(21)*atanh(sqrt(21)/(
3*sqrt(-2*x + 1)))/21, 1/(-2*x + 1) < 3/7))/9

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GIAC/XCAS [A]  time = 0.225691, size = 78, normalized size = 1.44 \[ \frac{25}{18} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{189} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{155}{18} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/18*(-2*x + 1)^(3/2) + 1/189*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1
))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 155/18*sqrt(-2*x + 1)