3.1959 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3}+\frac{1073 \sqrt{1-2 x}}{9 (3 x+2)}+\frac{112 \sqrt{1-2 x}}{9 (3 x+2)^2}+\frac{74020 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}}-242 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3) + (112*Sqrt[1 - 2*x])/(9*(2 + 3*x)^2) + (107
3*Sqrt[1 - 2*x])/(9*(2 + 3*x)) + (74020*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqr
t[21]) - 242*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.252359, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3}+\frac{1073 \sqrt{1-2 x}}{9 (3 x+2)}+\frac{112 \sqrt{1-2 x}}{9 (3 x+2)^2}+\frac{74020 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}}-242 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3) + (112*Sqrt[1 - 2*x])/(9*(2 + 3*x)^2) + (107
3*Sqrt[1 - 2*x])/(9*(2 + 3*x)) + (74020*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqr
t[21]) - 242*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 27.6992, size = 100, normalized size = 0.88 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{3}} + \frac{1073 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )} + \frac{112 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{2}} + \frac{74020 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{189} - 242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(9*(3*x + 2)**3) + 1073*sqrt(-2*x + 1)/(9*(3*x + 2)) + 112*s
qrt(-2*x + 1)/(9*(3*x + 2)**2) + 74020*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)
/189 - 242*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.168051, size = 83, normalized size = 0.73 \[ \frac{\sqrt{1-2 x} \left (9657 x^2+13198 x+4523\right )}{9 (3 x+2)^3}+\frac{74020 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9 \sqrt{21}}-242 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(4523 + 13198*x + 9657*x^2))/(9*(2 + 3*x)^3) + (74020*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21]) - 242*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]]

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Maple [A]  time = 0.017, size = 75, normalized size = 0.7 \[ -54\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{1073\, \left ( 1-2\,x \right ) ^{5/2}}{27}}-{\frac{45710\, \left ( 1-2\,x \right ) ^{3/2}}{243}}+{\frac{54145\,\sqrt{1-2\,x}}{243}} \right ) }+{\frac{74020\,\sqrt{21}}{189}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-242\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-54*(1073/27*(1-2*x)^(5/2)-45710/243*(1-2*x)^(3/2)+54145/243*(1-2*x)^(1/2))/(-4-
6*x)^3+74020/189*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-242*arctanh(1/11*5
5^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50311, size = 173, normalized size = 1.53 \[ 121 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{37010}{189} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (9657 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 45710 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 54145 \, \sqrt{-2 \, x + 1}\right )}}{9 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
37010/189*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 2/9*(9657*(-2*x + 1)^(5/2) - 45710*(-2*x + 1)^(3/2) + 54145*sqrt(-2*x +
1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.219085, size = 185, normalized size = 1.64 \[ \frac{\sqrt{21}{\left (1089 \, \sqrt{55} \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (9657 \, x^{2} + 13198 \, x + 4523\right )} \sqrt{-2 \, x + 1} + 37010 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{189 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/189*sqrt(21)*(1089*sqrt(55)*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + s
qrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*(9657*x^2 + 13198*x + 4523)*sq
rt(-2*x + 1) + 37010*(27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5) - 21*s
qrt(-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)**(5/2)/(2+3*x)**4/(3+5*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215576, size = 166, normalized size = 1.47 \[ 121 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{37010}{189} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{9657 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 45710 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 54145 \, \sqrt{-2 \, x + 1}}{36 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 37010/189*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) + 1/36*(9657*(2*x - 1)^2*sqrt(-2*x + 1) - 45710*(-2*x + 1
)^(3/2) + 54145*sqrt(-2*x + 1))/(3*x + 2)^3