3.1899 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=127 \[ -\frac{335 \sqrt{1-2 x}}{2 (5 x+3)}+\frac{50 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)}-2311 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+204 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-335*Sqrt[1 - 2*x])/(2*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x))
+ (50*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)) - 2311*Sqrt[3/7]*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]] + 204*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.264281, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{335 \sqrt{1-2 x}}{2 (5 x+3)}+\frac{50 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)}-2311 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+204 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-335*Sqrt[1 - 2*x])/(2*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x))
+ (50*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)) - 2311*Sqrt[3/7]*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]] + 204*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 28.3416, size = 109, normalized size = 0.86 \[ - \frac{335 \sqrt{- 2 x + 1}}{2 \left (5 x + 3\right )} + \frac{50 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right ) \left (5 x + 3\right )} + \frac{7 \sqrt{- 2 x + 1}}{6 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} - \frac{2311 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7} + 204 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-335*sqrt(-2*x + 1)/(2*(5*x + 3)) + 50*sqrt(-2*x + 1)/(3*(3*x + 2)*(5*x + 3)) +
7*sqrt(-2*x + 1)/(6*(3*x + 2)**2*(5*x + 3)) - 2311*sqrt(21)*atanh(sqrt(21)*sqrt(
-2*x + 1)/7)/7 + 204*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.165706, size = 90, normalized size = 0.71 \[ -\frac{\sqrt{1-2 x} \left (3015 x^2+3920 x+1271\right )}{2 (3 x+2)^2 (5 x+3)}-2311 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+204 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(1271 + 3920*x + 3015*x^2))/(2*(2 + 3*x)^2*(3 + 5*x)) - 2311*Sqr
t[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 204*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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Maple [A]  time = 0.02, size = 82, normalized size = 0.7 \[ 18\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{45\, \left ( 1-2\,x \right ) ^{3/2}}{2}}-{\frac{959\,\sqrt{1-2\,x}}{18}} \right ) }-{\frac{2311\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+22\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+204\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

18*(45/2*(1-2*x)^(3/2)-959/18*(1-2*x)^(1/2))/(-4-6*x)^2-2311/7*arctanh(1/7*21^(1
/2)*(1-2*x)^(1/2))*21^(1/2)+22*(1-2*x)^(1/2)/(-6/5-2*x)+204*arctanh(1/11*55^(1/2
)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50186, size = 173, normalized size = 1.36 \[ -102 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2311}{14} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3015 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 13870 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 15939 \, \sqrt{-2 \, x + 1}}{45 \,{\left (2 \, x - 1\right )}^{3} + 309 \,{\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 2311/14*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - (3015*(-2*x + 1)^(5/2) - 13870*(-2*x + 1)^(3/2) + 15939*sqrt(-2*x + 1))/(
45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168)

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Fricas [A]  time = 0.220106, size = 196, normalized size = 1.54 \[ \frac{\sqrt{7}{\left (204 \, \sqrt{55} \sqrt{7}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 2311 \, \sqrt{3}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{7}{\left (3015 \, x^{2} + 3920 \, x + 1271\right )} \sqrt{-2 \, x + 1}\right )}}{14 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*sqrt(7)*(204*sqrt(55)*sqrt(7)*(45*x^3 + 87*x^2 + 56*x + 12)*log((5*x - sqrt
(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 2311*sqrt(3)*(45*x^3 + 87*x^2 + 56*x + 12)
*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(7)*(3015*x
^2 + 3920*x + 1271)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)

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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)/(2+3*x)**3/(3+5*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215014, size = 166, normalized size = 1.31 \[ -102 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2311}{14} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} + \frac{405 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 959 \, \sqrt{-2 \, x + 1}}{4 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-102*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) + 2311/14*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21)
+ 3*sqrt(-2*x + 1))) - 55*sqrt(-2*x + 1)/(5*x + 3) + 1/4*(405*(-2*x + 1)^(3/2)
- 959*sqrt(-2*x + 1))/(3*x + 2)^2