3.1979 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=97 \[ -\frac{11 (1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac{803 \sqrt{1-2 x}}{50 (5 x+3)}+98 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2523}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-11*(1 - 2*x)^(3/2))/(10*(3 + 5*x)^2) + (803*Sqrt[1 - 2*x])/(50*(3 + 5*x)) + 98
*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (2523*Sqrt[11/5]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/25

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Rubi [A]  time = 0.192331, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{11 (1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac{803 \sqrt{1-2 x}}{50 (5 x+3)}+98 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2523}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-11*(1 - 2*x)^(3/2))/(10*(3 + 5*x)^2) + (803*Sqrt[1 - 2*x])/(50*(3 + 5*x)) + 98
*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (2523*Sqrt[11/5]*ArcTanh[Sqrt[5/11
]*Sqrt[1 - 2*x]])/25

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Rubi in Sympy [A]  time = 20.7293, size = 83, normalized size = 0.86 \[ - \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}}}{10 \left (5 x + 3\right )^{2}} + \frac{803 \sqrt{- 2 x + 1}}{50 \left (5 x + 3\right )} + \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3} - \frac{2523 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*(-2*x + 1)**(3/2)/(10*(5*x + 3)**2) + 803*sqrt(-2*x + 1)/(50*(5*x + 3)) + 98
*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/3 - 2523*sqrt(55)*atanh(sqrt(55)*sqrt
(-2*x + 1)/11)/125

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Mathematica [A]  time = 0.215576, size = 81, normalized size = 0.84 \[ 98 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1}{250} \left (\frac{55 \sqrt{1-2 x} (375 x+214)}{(5 x+3)^2}-5046 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + ((55*Sqrt[1 - 2*x]*(214 + 375*x)
)/(3 + 5*x)^2 - 5046*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/250

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Maple [A]  time = 0.017, size = 66, normalized size = 0.7 \[{\frac{98\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+550\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -3/10\, \left ( 1-2\,x \right ) ^{3/2}+{\frac{803\,\sqrt{1-2\,x}}{1250}} \right ) }-{\frac{2523\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

98/3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+550*(-3/10*(1-2*x)^(3/2)+803/1
250*(1-2*x)^(1/2))/(-6-10*x)^2-2523/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^
(1/2)

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Maxima [A]  time = 1.49604, size = 149, normalized size = 1.54 \[ \frac{2523}{250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{49}{3} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{11 \,{\left (375 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 803 \, \sqrt{-2 \, x + 1}\right )}}{25 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2523/250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 49/3*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 11/25*(375*(-2*x + 1)^(3/2) - 803*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*
x + 11)

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Fricas [A]  time = 0.242496, size = 188, normalized size = 1.94 \[ \frac{\sqrt{5} \sqrt{3}{\left (2523 \, \sqrt{11} \sqrt{3}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 2450 \, \sqrt{7} \sqrt{5}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 11 \, \sqrt{5} \sqrt{3}{\left (375 \, x + 214\right )} \sqrt{-2 \, x + 1}\right )}}{750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/750*sqrt(5)*sqrt(3)*(2523*sqrt(11)*sqrt(3)*(25*x^2 + 30*x + 9)*log((sqrt(5)*(5
*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 2450*sqrt(7)*sqrt(5)*(25*x^2 +
30*x + 9)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)) + 11*sq
rt(5)*sqrt(3)*(375*x + 214)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215822, size = 144, normalized size = 1.48 \[ \frac{2523}{250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{49}{3} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{11 \,{\left (375 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 803 \, \sqrt{-2 \, x + 1}\right )}}{100 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2523/250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 49/3*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) - 11/100*(375*(-2*x + 1)^(3/2) - 803*sqrt(-2*x + 1))/(5*x
+ 3)^2