3.2094 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=72 \[ \frac{4}{77 \sqrt{1-2 x}}+\frac{6}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{10}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

4/(77*Sqrt[1 - 2*x]) + (6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (10*Sq
rt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.135894, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{4}{77 \sqrt{1-2 x}}+\frac{6}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{10}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

4/(77*Sqrt[1 - 2*x]) + (6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (10*Sq
rt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi in Sympy [A]  time = 13.7817, size = 61, normalized size = 0.85 \[ \frac{6 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} - \frac{10 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} + \frac{4}{77 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/49 - 10*sqrt(55)*atanh(sqrt(55)*sqrt
(-2*x + 1)/11)/121 + 4/(77*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.21356, size = 71, normalized size = 0.99 \[ \frac{6}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1}{847} \left (\frac{44}{\sqrt{1-2 x}}-70 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + (44/Sqrt[1 - 2*x] - 70*Sqrt[5
5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/847

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Maple [A]  time = 0.016, size = 47, normalized size = 0.7 \[{\frac{6\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{10\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{4}{77}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-10/121*arctanh(1/11*55^(1/2)*(
1-2*x)^(1/2))*55^(1/2)+4/77/(1-2*x)^(1/2)

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Maxima [A]  time = 1.50403, size = 111, normalized size = 1.54 \[ \frac{5}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4}{77 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 3/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) + 4/77/sqrt(-2*x + 1)

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Fricas [A]  time = 0.239555, size = 157, normalized size = 2.18 \[ \frac{\sqrt{11} \sqrt{7}{\left (35 \, \sqrt{7} \sqrt{5} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 33 \, \sqrt{11} \sqrt{3} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 4 \, \sqrt{11} \sqrt{7}\right )}}{5929 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5929*sqrt(11)*sqrt(7)*(35*sqrt(7)*sqrt(5)*sqrt(-2*x + 1)*log((sqrt(11)*(5*x -
8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 33*sqrt(11)*sqrt(3)*sqrt(-2*x + 1)*
log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + 4*sqrt(11)*sqrt(
7))/sqrt(-2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((-2*x + 1)**(3/2)*(3*x + 2)*(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.219971, size = 119, normalized size = 1.65 \[ \frac{5}{121} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3}{49} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4}{77 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/121*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) - 3/49*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 4/77/sqrt(-2*x + 1)