3.1803 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{2+3 x} \, dx\)

Optimal. Leaf size=82 \[ -\frac{125}{84} (1-2 x)^{7/2}+\frac{80}{9} (1-2 x)^{5/2}-\frac{5135}{324} (1-2 x)^{3/2}-\frac{2}{81} \sqrt{1-2 x}+\frac{2}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x])/81 - (5135*(1 - 2*x)^(3/2))/324 + (80*(1 - 2*x)^(5/2))/9 - (1
25*(1 - 2*x)^(7/2))/84 + (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi [A]  time = 0.0904358, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{125}{84} (1-2 x)^{7/2}+\frac{80}{9} (1-2 x)^{5/2}-\frac{5135}{324} (1-2 x)^{3/2}-\frac{2}{81} \sqrt{1-2 x}+\frac{2}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/81 - (5135*(1 - 2*x)^(3/2))/324 + (80*(1 - 2*x)^(5/2))/9 - (1
25*(1 - 2*x)^(7/2))/84 + (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi in Sympy [A]  time = 9.52275, size = 71, normalized size = 0.87 \[ - \frac{125 \left (- 2 x + 1\right )^{\frac{7}{2}}}{84} + \frac{80 \left (- 2 x + 1\right )^{\frac{5}{2}}}{9} - \frac{5135 \left (- 2 x + 1\right )^{\frac{3}{2}}}{324} - \frac{2 \sqrt{- 2 x + 1}}{81} + \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]

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),x)

[Out]

-125*(-2*x + 1)**(7/2)/84 + 80*(-2*x + 1)**(5/2)/9 - 5135*(-2*x + 1)**(3/2)/324
- 2*sqrt(-2*x + 1)/81 + 2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/243

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Mathematica [A]  time = 0.0755893, size = 56, normalized size = 0.68 \[ \frac{3 \sqrt{1-2 x} \left (6750 x^3+10035 x^2+2875 x-4804\right )+14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1701} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(-4804 + 2875*x + 10035*x^2 + 6750*x^3) + 14*Sqrt[21]*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/1701

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Maple [A]  time = 0.01, size = 56, normalized size = 0.7 \[ -{\frac{5135}{324} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{80}{9} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{125}{84} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{2\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{2}{81}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5135/324*(1-2*x)^(3/2)+80/9*(1-2*x)^(5/2)-125/84*(1-2*x)^(7/2)+2/243*arctanh(1/
7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2/81*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49839, size = 99, normalized size = 1.21 \[ -\frac{125}{84} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{80}{9} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{5135}{324} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{243} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2}{81} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/84*(-2*x + 1)^(7/2) + 80/9*(-2*x + 1)^(5/2) - 5135/324*(-2*x + 1)^(3/2) - 1
/243*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)))
- 2/81*sqrt(-2*x + 1)

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Fricas [A]  time = 0.216355, size = 92, normalized size = 1.12 \[ \frac{1}{1701} \, \sqrt{3}{\left (\sqrt{3}{\left (6750 \, x^{3} + 10035 \, x^{2} + 2875 \, x - 4804\right )} \sqrt{-2 \, x + 1} + 7 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1701*sqrt(3)*(sqrt(3)*(6750*x^3 + 10035*x^2 + 2875*x - 4804)*sqrt(-2*x + 1) +
7*sqrt(7)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)))

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Sympy [A]  time = 6.91899, size = 110, normalized size = 1.34 \[ - \frac{125 \left (- 2 x + 1\right )^{\frac{7}{2}}}{84} + \frac{80 \left (- 2 x + 1\right )^{\frac{5}{2}}}{9} - \frac{5135 \left (- 2 x + 1\right )^{\frac{3}{2}}}{324} - \frac{2 \sqrt{- 2 x + 1}}{81} - \frac{14 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*(-2*x + 1)**(7/2)/84 + 80*(-2*x + 1)**(5/2)/9 - 5135*(-2*x + 1)**(3/2)/324
- 2*sqrt(-2*x + 1)/81 - 14*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)
/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 <
7/3))/81

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GIAC/XCAS [A]  time = 0.241305, size = 122, normalized size = 1.49 \[ \frac{125}{84} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{80}{9} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{5135}{324} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{243} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2}{81} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/84*(2*x - 1)^3*sqrt(-2*x + 1) + 80/9*(2*x - 1)^2*sqrt(-2*x + 1) - 5135/324*(
-2*x + 1)^(3/2) - 1/243*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) - 2/81*sqrt(-2*x + 1)