3.1861 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=101 \[ \frac{209 (1-2 x)^{5/2}}{2646 (3 x+2)^2}-\frac{(1-2 x)^{5/2}}{189 (3 x+2)^3}-\frac{7559 (1-2 x)^{3/2}}{7938 (3 x+2)}-\frac{7559 \sqrt{1-2 x}}{3969}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

[Out]

(-7559*Sqrt[1 - 2*x])/3969 - (1 - 2*x)^(5/2)/(189*(2 + 3*x)^3) + (209*(1 - 2*x)^
(5/2))/(2646*(2 + 3*x)^2) - (7559*(1 - 2*x)^(3/2))/(7938*(2 + 3*x)) + (7559*ArcT
anh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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Rubi [A]  time = 0.110694, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{209 (1-2 x)^{5/2}}{2646 (3 x+2)^2}-\frac{(1-2 x)^{5/2}}{189 (3 x+2)^3}-\frac{7559 (1-2 x)^{3/2}}{7938 (3 x+2)}-\frac{7559 \sqrt{1-2 x}}{3969}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-7559*Sqrt[1 - 2*x])/3969 - (1 - 2*x)^(5/2)/(189*(2 + 3*x)^3) + (209*(1 - 2*x)^
(5/2))/(2646*(2 + 3*x)^2) - (7559*(1 - 2*x)^(3/2))/(7938*(2 + 3*x)) + (7559*ArcT
anh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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Rubi in Sympy [A]  time = 12.5695, size = 87, normalized size = 0.86 \[ \frac{209 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2646 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{189 \left (3 x + 2\right )^{3}} - \frac{7559 \left (- 2 x + 1\right )^{\frac{3}{2}}}{7938 \left (3 x + 2\right )} - \frac{7559 \sqrt{- 2 x + 1}}{3969} + \frac{7559 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{11907} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

209*(-2*x + 1)**(5/2)/(2646*(3*x + 2)**2) - (-2*x + 1)**(5/2)/(189*(3*x + 2)**3)
- 7559*(-2*x + 1)**(3/2)/(7938*(3*x + 2)) - 7559*sqrt(-2*x + 1)/3969 + 7559*sqr
t(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/11907

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Mathematica [A]  time = 0.113901, size = 63, normalized size = 0.62 \[ \frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}}-\frac{\sqrt{1-2 x} \left (37800 x^3+100809 x^2+82493 x+21424\right )}{1134 (3 x+2)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(21424 + 82493*x + 100809*x^2 + 37800*x^3))/(1134*(2 + 3*x)^3) +
(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 \[ -{\frac{100}{81}\sqrt{1-2\,x}}-{\frac{4}{3\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{2801}{84} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{4093}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{18613}{108}\sqrt{1-2\,x}} \right ) }+{\frac{7559\,\sqrt{21}}{11907}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-100/81*(1-2*x)^(1/2)-4/3*(-2801/84*(1-2*x)^(5/2)+4093/27*(1-2*x)^(3/2)-18613/10
8*(1-2*x)^(1/2))/(-4-6*x)^3+7559/11907*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1
/2)

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Maxima [A]  time = 1.51251, size = 136, normalized size = 1.35 \[ -\frac{7559}{23814} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{100}{81} \, \sqrt{-2 \, x + 1} - \frac{25209 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 114604 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 130291 \, \sqrt{-2 \, x + 1}}{567 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7559/23814*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 100/81*sqrt(-2*x + 1) - 1/567*(25209*(-2*x + 1)^(5/2) - 114604*(-2*x +
1)^(3/2) + 130291*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98
)

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Fricas [A]  time = 0.226029, size = 127, normalized size = 1.26 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (37800 \, x^{3} + 100809 \, x^{2} + 82493 \, x + 21424\right )} \sqrt{-2 \, x + 1} - 7559 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{23814 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/23814*sqrt(21)*(sqrt(21)*(37800*x^3 + 100809*x^2 + 82493*x + 21424)*sqrt(-2*x
+ 1) - 7559*(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((1-2*x)**(3/2)*(3+5*x)**2/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227526, size = 126, normalized size = 1.25 \[ -\frac{7559}{23814} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{100}{81} \, \sqrt{-2 \, x + 1} - \frac{25209 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 114604 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 130291 \, \sqrt{-2 \, x + 1}}{4536 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7559/23814*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) - 100/81*sqrt(-2*x + 1) - 1/4536*(25209*(2*x - 1)^2*sqrt(-2*x + 1
) - 114604*(-2*x + 1)^(3/2) + 130291*sqrt(-2*x + 1))/(3*x + 2)^3