3.1794 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=88 \[ \frac{23 (1-2 x)^{3/2}}{294 (3 x+2)^2}-\frac{(1-2 x)^{3/2}}{189 (3 x+2)^3}-\frac{2381 \sqrt{1-2 x}}{2646 (3 x+2)}+\frac{2381 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1323 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(3/2)/(189*(2 + 3*x)^3) + (23*(1 - 2*x)^(3/2))/(294*(2 + 3*x)^2) - (2
381*Sqrt[1 - 2*x])/(2646*(2 + 3*x)) + (2381*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1
323*Sqrt[21])

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Rubi [A]  time = 0.100315, 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 \[ \frac{23 (1-2 x)^{3/2}}{294 (3 x+2)^2}-\frac{(1-2 x)^{3/2}}{189 (3 x+2)^3}-\frac{2381 \sqrt{1-2 x}}{2646 (3 x+2)}+\frac{2381 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1323 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

-(1 - 2*x)^(3/2)/(189*(2 + 3*x)^3) + (23*(1 - 2*x)^(3/2))/(294*(2 + 3*x)^2) - (2
381*Sqrt[1 - 2*x])/(2646*(2 + 3*x)) + (2381*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1
323*Sqrt[21])

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Rubi in Sympy [A]  time = 10.4553, size = 75, normalized size = 0.85 \[ \frac{23 \left (- 2 x + 1\right )^{\frac{3}{2}}}{294 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{189 \left (3 x + 2\right )^{3}} - \frac{2381 \sqrt{- 2 x + 1}}{2646 \left (3 x + 2\right )} + \frac{2381 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{27783} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**2*(1-2*x)**(1/2)/(2+3*x)**4,x)

[Out]

23*(-2*x + 1)**(3/2)/(294*(3*x + 2)**2) - (-2*x + 1)**(3/2)/(189*(3*x + 2)**3) -
2381*sqrt(-2*x + 1)/(2646*(3*x + 2)) + 2381*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x +
1)/7)/27783

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Mathematica [A]  time = 0.0959549, size = 58, normalized size = 0.66 \[ \frac{4762 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{21 \sqrt{1-2 x} \left (22671 x^2+28751 x+9124\right )}{(3 x+2)^3}}{55566} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

((-21*Sqrt[1 - 2*x]*(9124 + 28751*x + 22671*x^2))/(2 + 3*x)^3 + 4762*Sqrt[21]*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/55566

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Maple [A]  time = 0.017, size = 57, normalized size = 0.7 \[ -108\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{2519\, \left ( 1-2\,x \right ) ^{5/2}}{15876}}+{\frac{3673\, \left ( 1-2\,x \right ) ^{3/2}}{5103}}-{\frac{2381\,\sqrt{1-2\,x}}{2916}} \right ) }+{\frac{2381\,\sqrt{21}}{27783}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^4,x)

[Out]

-108*(-2519/15876*(1-2*x)^(5/2)+3673/5103*(1-2*x)^(3/2)-2381/2916*(1-2*x)^(1/2))
/(-4-6*x)^3+2381/27783*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49279, size = 124, normalized size = 1.41 \[ -\frac{2381}{55566} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{22671 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 102844 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 116669 \, \sqrt{-2 \, x + 1}}{1323 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="maxima")

[Out]

-2381/55566*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/1323*(22671*(-2*x + 1)^(5/2) - 102844*(-2*x + 1)^(3/2) + 116669*sqrt(
-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.223918, size = 120, normalized size = 1.36 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (22671 \, x^{2} + 28751 \, x + 9124\right )} \sqrt{-2 \, x + 1} - 2381 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{55566 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="fricas")

[Out]

-1/55566*sqrt(21)*(sqrt(21)*(22671*x^2 + 28751*x + 9124)*sqrt(-2*x + 1) - 2381*(
27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x +
2)))/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+5*x)**2*(1-2*x)**(1/2)/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215485, size = 113, normalized size = 1.28 \[ -\frac{2381}{55566} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{22671 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 102844 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 116669 \, \sqrt{-2 \, x + 1}}{10584 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*sqrt(-2*x + 1)/(3*x + 2)^4,x, algorithm="giac")

[Out]

-2381/55566*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) - 1/10584*(22671*(2*x - 1)^2*sqrt(-2*x + 1) - 102844*(-2*x + 1)^(
3/2) + 116669*sqrt(-2*x + 1))/(3*x + 2)^3