3.2012 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{25}{12} (1-2 x)^{5/2}+\frac{400}{27} (1-2 x)^{3/2}-\frac{5135}{108} \sqrt{1-2 x}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

[Out]

(-5135*Sqrt[1 - 2*x])/108 + (400*(1 - 2*x)^(3/2))/27 - (25*(1 - 2*x)^(5/2))/12 +
(2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Rubi [A]  time = 0.0852208, antiderivative size = 67, 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 \[ -\frac{25}{12} (1-2 x)^{5/2}+\frac{400}{27} (1-2 x)^{3/2}-\frac{5135}{108} \sqrt{1-2 x}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-5135*Sqrt[1 - 2*x])/108 + (400*(1 - 2*x)^(3/2))/27 - (25*(1 - 2*x)^(5/2))/12 +
(2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Rubi in Sympy [A]  time = 8.71811, size = 60, normalized size = 0.9 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{5}{2}}}{12} + \frac{400 \left (- 2 x + 1\right )^{\frac{3}{2}}}{27} - \frac{5135 \sqrt{- 2 x + 1}}{108} + \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{567} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-2*x + 1)**(5/2)/12 + 400*(-2*x + 1)**(3/2)/27 - 5135*sqrt(-2*x + 1)/108 +
2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/567

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Mathematica [A]  time = 0.0903213, size = 51, normalized size = 0.76 \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}}-\frac{5}{27} \sqrt{1-2 x} \left (45 x^2+115 x+188\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*(188 + 115*x + 45*x^2))/27 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(27*Sqrt[21])

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Maple [A]  time = 0.009, size = 47, normalized size = 0.7 \[{\frac{400}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{25}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{2\,\sqrt{21}}{567}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{5135}{108}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

400/27*(1-2*x)^(3/2)-25/12*(1-2*x)^(5/2)+2/567*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2
))*21^(1/2)-5135/108*(1-2*x)^(1/2)

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Maxima [A]  time = 1.52728, size = 86, normalized size = 1.28 \[ -\frac{25}{12} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{400}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{567} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5135}{108} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/12*(-2*x + 1)^(5/2) + 400/27*(-2*x + 1)^(3/2) - 1/567*sqrt(21)*log(-(sqrt(21
) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5135/108*sqrt(-2*x + 1)

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Fricas [A]  time = 0.244363, size = 78, normalized size = 1.16 \[ -\frac{1}{567} \, \sqrt{21}{\left (5 \, \sqrt{21}{\left (45 \, x^{2} + 115 \, x + 188\right )} \sqrt{-2 \, x + 1} - \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/567*sqrt(21)*(5*sqrt(21)*(45*x^2 + 115*x + 188)*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 = 5.05802, size = 102, normalized size = 1.52 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{5}{2}}}{12} + \frac{400 \left (- 2 x + 1\right )^{\frac{3}{2}}}{27} - \frac{5135 \sqrt{- 2 x + 1}}{108} - \frac{2 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21}}{3 \sqrt{- 2 x + 1}} \right )}}{21} & \text{for}\: \frac{1}{- 2 x + 1} > \frac{3}{7} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21}}{3 \sqrt{- 2 x + 1}} \right )}}{21} & \text{for}\: \frac{1}{- 2 x + 1} < \frac{3}{7} \end{cases}\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(-2*x + 1)**(5/2)/12 + 400*(-2*x + 1)**(3/2)/27 - 5135*sqrt(-2*x + 1)/108 -
2*Piecewise((-sqrt(21)*acoth(sqrt(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))/27

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GIAC/XCAS [A]  time = 0.222391, size = 100, normalized size = 1.49 \[ -\frac{25}{12} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{400}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{567} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{5135}{108} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/12*(2*x - 1)^2*sqrt(-2*x + 1) + 400/27*(-2*x + 1)^(3/2) - 1/567*sqrt(21)*ln(
1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5135/10
8*sqrt(-2*x + 1)