3.1806 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=107 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)^3}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{189 (3 x+2)^2}+\frac{2 \sqrt{1-2 x} (26075 x+18016)}{3969 (3 x+2)}-\frac{92996 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}} \]

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(189*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/
(9*(2 + 3*x)^3) + (2*Sqrt[1 - 2*x]*(18016 + 26075*x))/(3969*(2 + 3*x)) - (92996*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3969*Sqrt[21])

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Rubi [A]  time = 0.158251, antiderivative size = 107, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)^3}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{189 (3 x+2)^2}+\frac{2 \sqrt{1-2 x} (26075 x+18016)}{3969 (3 x+2)}-\frac{92996 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{3969 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(189*(2 + 3*x)^2) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/
(9*(2 + 3*x)^3) + (2*Sqrt[1 - 2*x]*(18016 + 26075*x))/(3969*(2 + 3*x)) - (92996*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3969*Sqrt[21])

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Rubi in Sympy [A]  time = 18.8735, size = 92, normalized size = 0.86 \[ \frac{\sqrt{- 2 x + 1} \left (312900 x + 216192\right )}{23814 \left (3 x + 2\right )} - \frac{53 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{189 \left (3 x + 2\right )^{2}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{3}}{9 \left (3 x + 2\right )^{3}} - \frac{92996 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{83349} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(312900*x + 216192)/(23814*(3*x + 2)) - 53*sqrt(-2*x + 1)*(5*x +
3)**2/(189*(3*x + 2)**2) - sqrt(-2*x + 1)*(5*x + 3)**3/(9*(3*x + 2)**3) - 92996*
sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/83349

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Mathematica [A]  time = 0.105961, size = 63, normalized size = 0.59 \[ \frac{\frac{21 \sqrt{1-2 x} \left (330750 x^3+695043 x^2+484618 x+112187\right )}{(3 x+2)^3}-92996 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{83349} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(112187 + 484618*x + 695043*x^2 + 330750*x^3))/(2 + 3*x)^3 -
92996*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/83349

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Maple [A]  time = 0.017, size = 66, normalized size = 0.6 \[{\frac{250}{81}\sqrt{1-2\,x}}+{\frac{2}{3\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{3727}{147} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{22046}{189} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3623}{27}\sqrt{1-2\,x}} \right ) }-{\frac{92996\,\sqrt{21}}{83349}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

250/81*(1-2*x)^(1/2)+2/3*(-3727/147*(1-2*x)^(5/2)+22046/189*(1-2*x)^(3/2)-3623/2
7*(1-2*x)^(1/2))/(-4-6*x)^3-92996/83349*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(
1/2)

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Maxima [A]  time = 1.51276, size = 136, normalized size = 1.27 \[ \frac{46498}{83349} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{250}{81} \, \sqrt{-2 \, x + 1} + \frac{2 \,{\left (33543 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 154322 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 177527 \, \sqrt{-2 \, x + 1}\right )}}{3969 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

46498/83349*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 250/81*sqrt(-2*x + 1) + 2/3969*(33543*(-2*x + 1)^(5/2) - 154322*(-2*x +
1)^(3/2) + 177527*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 9
8)

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Fricas [A]  time = 0.21254, size = 127, normalized size = 1.19 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (330750 \, x^{3} + 695043 \, x^{2} + 484618 \, x + 112187\right )} \sqrt{-2 \, x + 1} + 46498 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{83349 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/83349*sqrt(21)*(sqrt(21)*(330750*x^3 + 695043*x^2 + 484618*x + 112187)*sqrt(-2
*x + 1) + 46498*(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)**3*(1-2*x)**(1/2)/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217749, size = 126, normalized size = 1.18 \[ \frac{46498}{83349} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{250}{81} \, \sqrt{-2 \, x + 1} + \frac{33543 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 154322 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 177527 \, \sqrt{-2 \, x + 1}}{15876 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

46498/83349*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) + 250/81*sqrt(-2*x + 1) + 1/15876*(33543*(2*x - 1)^2*sqrt(-2*x +
1) - 154322*(-2*x + 1)^(3/2) + 177527*sqrt(-2*x + 1))/(3*x + 2)^3