3.2095 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=92 \[ -\frac{58}{539 \sqrt{1-2 x}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2)}+\frac{228}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-58/(539*Sqrt[1 - 2*x]) + 3/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (228*Sqrt[3/7]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/11

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Rubi [A]  time = 0.209692, antiderivative size = 92, 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 \[ -\frac{58}{539 \sqrt{1-2 x}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2)}+\frac{228}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-58/(539*Sqrt[1 - 2*x]) + 3/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (228*Sqrt[3/7]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/11

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Rubi in Sympy [A]  time = 21.3741, size = 78, normalized size = 0.85 \[ \frac{228 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{343} - \frac{50 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} - \frac{58}{539 \sqrt{- 2 x + 1}} + \frac{3}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

228*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/343 - 50*sqrt(55)*atanh(sqrt(55)*s
qrt(-2*x + 1)/11)/121 - 58/(539*sqrt(-2*x + 1)) + 3/(7*sqrt(-2*x + 1)*(3*x + 2))

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Mathematica [A]  time = 0.184179, size = 87, normalized size = 0.95 \[ \frac{\sqrt{1-2 x} (174 x-115)}{539 \left (6 x^2+x-2\right )}+\frac{228}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-115 + 174*x))/(539*(-2 + x + 6*x^2)) + (228*Sqrt[3/7]*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/
11

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Maple [A]  time = 0.019, size = 63, normalized size = 0.7 \[{\frac{8}{539}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{6}{49}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{228\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{50\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/539/(1-2*x)^(1/2)-6/49*(1-2*x)^(1/2)/(-4/3-2*x)+228/343*arctanh(1/7*21^(1/2)*(
1-2*x)^(1/2))*21^(1/2)-50/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48222, size = 136, normalized size = 1.48 \[ \frac{25}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{114}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (174 \, x - 115\right )}}{539 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) - 114/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 2/539*(174*x - 115)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))

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Fricas [A]  time = 0.241068, size = 186, normalized size = 2.02 \[ \frac{\sqrt{11} \sqrt{7}{\left (1225 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 1254 \, \sqrt{11} \sqrt{3}{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (174 \, x - 115\right )}\right )}}{41503 \,{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/41503*sqrt(11)*sqrt(7)*(1225*sqrt(7)*sqrt(5)*(3*x + 2)*sqrt(-2*x + 1)*log((sqr
t(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 1254*sqrt(11)*sqrt(3)*
(3*x + 2)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x
+ 2)) - sqrt(11)*sqrt(7)*(174*x - 115))/((3*x + 2)*sqrt(-2*x + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.226798, size = 144, normalized size = 1.57 \[ \frac{25}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{114}{343} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (174 \, x - 115\right )}}{539 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) - 114/343*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2
1) + 3*sqrt(-2*x + 1))) + 2/539*(174*x - 115)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x
+ 1))