3.1805 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{6 (3 x+2)^2}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}+\frac{5 \sqrt{1-2 x} (2815 x+323)}{1134}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(6*
(2 + 3*x)^2) + (5*Sqrt[1 - 2*x]*(323 + 2815*x))/1134 + (7559*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(567*Sqrt[21])

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Rubi [A]  time = 0.156679, antiderivative size = 100, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{6 (3 x+2)^2}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}+\frac{5 \sqrt{1-2 x} (2815 x+323)}{1134}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(6*
(2 + 3*x)^2) + (5*Sqrt[1 - 2*x]*(323 + 2815*x))/1134 + (7559*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(567*Sqrt[21])

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Rubi in Sympy [A]  time = 19.0319, size = 85, normalized size = 0.85 \[ \frac{\sqrt{- 2 x + 1} \left (42225 x + 4845\right )}{3402} - \frac{53 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{63 \left (3 x + 2\right )} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{3}}{6 \left (3 x + 2\right )^{2}} + \frac{7559 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{11907} \]

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)**3,x)

[Out]

sqrt(-2*x + 1)*(42225*x + 4845)/3402 - 53*sqrt(-2*x + 1)*(5*x + 3)**2/(63*(3*x +
2)) - sqrt(-2*x + 1)*(5*x + 3)**3/(6*(3*x + 2)**2) + 7559*sqrt(21)*atanh(sqrt(2
1)*sqrt(-2*x + 1)/7)/11907

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Mathematica [A]  time = 0.112363, size = 63, normalized size = 0.63 \[ \frac{\sqrt{1-2 x} \left (31500 x^3+7350 x^2-32833 x-15815\right )}{1134 (3 x+2)^2}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-15815 - 32833*x + 7350*x^2 + 31500*x^3))/(1134*(2 + 3*x)^2) + (
7559*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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Maple [A]  time = 0.017, size = 66, normalized size = 0.7 \[ -{\frac{125}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{50}{27}\sqrt{1-2\,x}}-{\frac{2}{3\, \left ( -4-6\,x \right ) ^{2}} \left ( -{\frac{211}{126} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{209}{54}\sqrt{1-2\,x}} \right ) }+{\frac{7559\,\sqrt{21}}{11907}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/81*(1-2*x)^(3/2)-50/27*(1-2*x)^(1/2)-2/3*(-211/126*(1-2*x)^(3/2)+209/54*(1-
2*x)^(1/2))/(-4-6*x)^2+7559/11907*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49204, size = 124, normalized size = 1.24 \[ -\frac{125}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{7559}{23814} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{50}{27} \, \sqrt{-2 \, x + 1} + \frac{633 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1463 \, \sqrt{-2 \, x + 1}}{567 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1)
)/(sqrt(21) + 3*sqrt(-2*x + 1))) - 50/27*sqrt(-2*x + 1) + 1/567*(633*(-2*x + 1)^
(3/2) - 1463*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 0.212737, size = 113, normalized size = 1.13 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (31500 \, x^{3} + 7350 \, x^{2} - 32833 \, x - 15815\right )} \sqrt{-2 \, x + 1} + 7559 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{23814 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/23814*sqrt(21)*(sqrt(21)*(31500*x^3 + 7350*x^2 - 32833*x - 15815)*sqrt(-2*x +
1) + 7559*(9*x^2 + 12*x + 4)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x +
2)))/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 152.759, size = 337, normalized size = 3.37 \[ - \frac{125 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} - \frac{50 \sqrt{- 2 x + 1}}{27} - \frac{428 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{81} - \frac{56 \left (\begin{cases} \frac{\sqrt{21} \left (\frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} - 1\right )^{2}}\right )}{1029} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{81} - \frac{370 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*(-2*x + 1)**(3/2)/81 - 50*sqrt(-2*x + 1)/27 - 428*Piecewise((sqrt(21)*(-log
(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/4 - 1/(4*
(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)))/147, (
x <= 1/2) & (x > -2/3)))/81 - 56*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(-2*x +
1)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(-
2*x + 1)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)**2) + 3/(16*(sqrt(21)*s
qrt(-2*x + 1)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)**2))/1029, (x <= 1
/2) & (x > -2/3)))/81 - 370*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))/27

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GIAC/XCAS [A]  time = 0.21487, size = 116, normalized size = 1.16 \[ -\frac{125}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{7559}{23814} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{50}{27} \, \sqrt{-2 \, x + 1} + \frac{633 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1463 \, \sqrt{-2 \, x + 1}}{2268 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-
2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 50/27*sqrt(-2*x + 1) + 1/2268*(633*(-
2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(3*x + 2)^2