3.2096 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2525}{3773 \sqrt{1-2 x}}+\frac{225}{98 \sqrt{1-2 x} (3 x+2)}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{8025}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-2525/(3773*Sqrt[1 - 2*x]) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 225/(98*Sqrt[1 -
2*x]*(2 + 3*x)) + (8025*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*
Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.272724, antiderivative size = 112, 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 \[ -\frac{2525}{3773 \sqrt{1-2 x}}+\frac{225}{98 \sqrt{1-2 x} (3 x+2)}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{8025}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-2525/(3773*Sqrt[1 - 2*x]) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 225/(98*Sqrt[1 -
2*x]*(2 + 3*x)) + (8025*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*
Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi in Sympy [A]  time = 27.7415, size = 97, normalized size = 0.87 \[ \frac{8025 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2401} - \frac{250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} - \frac{2525}{3773 \sqrt{- 2 x + 1}} + \frac{225}{98 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{3}{14 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8025*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2401 - 250*sqrt(55)*atanh(sqrt(55
)*sqrt(-2*x + 1)/11)/121 - 2525/(3773*sqrt(-2*x + 1)) + 225/(98*sqrt(-2*x + 1)*(
3*x + 2)) + 3/(14*sqrt(-2*x + 1)*(3*x + 2)**2)

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Mathematica [A]  time = 0.199676, size = 89, normalized size = 0.79 \[ \frac{-45450 x^2-8625 x+16067}{7546 \sqrt{1-2 x} (3 x+2)^2}+\frac{8025}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(16067 - 8625*x - 45450*x^2)/(7546*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (8025*Sqrt[3/7]*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[
1 - 2*x]])/11

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Maple [A]  time = 0.022, size = 75, normalized size = 0.7 \[{\frac{16}{3773}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{486}{343\, \left ( -4-6\,x \right ) ^{2}} \left ({\frac{77}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{553}{54}\sqrt{1-2\,x}} \right ) }+{\frac{8025\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{250\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/3773/(1-2*x)^(1/2)-486/343*(77/18*(1-2*x)^(3/2)-553/54*(1-2*x)^(1/2))/(-4-6*x
)^2+8025/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250/121*arctanh(1/11*
55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49997, size = 161, normalized size = 1.44 \[ \frac{125}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{8025}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{22725 \,{\left (2 \, x - 1\right )}^{2} + 108150 \, x - 54859}{3773 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 8025/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) - 1/3773*(22725*(2*x - 1)^2 + 108150*x - 54859)/(9*(-2*x + 1)^(5/2) -
42*(-2*x + 1)^(3/2) + 49*sqrt(-2*x + 1))

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Fricas [A]  time = 0.23973, size = 213, normalized size = 1.9 \[ \frac{\sqrt{11} \sqrt{7}{\left (85750 \, \sqrt{7} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 88275 \, \sqrt{11} \sqrt{3}{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (45450 \, x^{2} + 8625 \, x - 16067\right )}\right )}}{581042 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/581042*sqrt(11)*sqrt(7)*(85750*sqrt(7)*sqrt(5)*(9*x^2 + 12*x + 4)*sqrt(-2*x +
1)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 88275*sqrt(
11)*sqrt(3)*(9*x^2 + 12*x + 4)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)
*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(45450*x^2 + 8625*x - 16067))/((9
*x^2 + 12*x + 4)*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)**3/(3+5*x),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.250107, size = 157, normalized size = 1.4 \[ \frac{125}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{8025}{4802} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16}{3773 \, \sqrt{-2 \, x + 1}} - \frac{9 \,{\left (33 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 79 \, \sqrt{-2 \, x + 1}\right )}}{196 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) - 8025/4802*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) + 16/3773/sqrt(-2*x + 1) - 9/196*(33*(-2*x + 1)^(3/2)
- 79*sqrt(-2*x + 1))/(3*x + 2)^2