3.1908 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=118 \[ \frac{207 \sqrt{1-2 x}}{2 (5 x+3)}-\frac{103 \sqrt{1-2 x}}{6 (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)^2}+204 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6933 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

[Out]

(-103*Sqrt[1 - 2*x])/(6*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)^
2) + (207*Sqrt[1 - 2*x])/(2*(3 + 5*x)) + 204*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]] - (6933*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt[55]

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Rubi [A]  time = 0.257334, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{207 \sqrt{1-2 x}}{2 (5 x+3)}-\frac{103 \sqrt{1-2 x}}{6 (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)^2}+204 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6933 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-103*Sqrt[1 - 2*x])/(6*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)^
2) + (207*Sqrt[1 - 2*x])/(2*(3 + 5*x)) + 204*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]] - (6933*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/Sqrt[55]

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Rubi in Sympy [A]  time = 28.7544, size = 105, normalized size = 0.89 \[ \frac{207 \sqrt{- 2 x + 1}}{2 \left (5 x + 3\right )} - \frac{103 \sqrt{- 2 x + 1}}{6 \left (5 x + 3\right )^{2}} + \frac{7 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + 204 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )} - \frac{6933 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

207*sqrt(-2*x + 1)/(2*(5*x + 3)) - 103*sqrt(-2*x + 1)/(6*(5*x + 3)**2) + 7*sqrt(
-2*x + 1)/(3*(3*x + 2)*(5*x + 3)**2) + 204*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1
)/7) - 6933*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55

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Mathematica [A]  time = 0.186808, size = 88, normalized size = 0.75 \[ \frac{\sqrt{1-2 x} \left (3105 x^2+3830 x+1178\right )}{2 (3 x+2) (5 x+3)^2}+204 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6933 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(1178 + 3830*x + 3105*x^2))/(2*(2 + 3*x)*(3 + 5*x)^2) + 204*Sqrt[
21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (6933*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/
Sqrt[55]

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Maple [A]  time = 0.02, size = 82, normalized size = 0.7 \[ -14\,{\frac{\sqrt{1-2\,x}}{-4/3-2\,x}}+204\,{\it Artanh} \left ( 1/7\,\sqrt{21}\sqrt{1-2\,x} \right ) \sqrt{21}+50\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{137\, \left ( 1-2\,x \right ) ^{3/2}}{10}}+{\frac{297\,\sqrt{1-2\,x}}{10}} \right ) }-{\frac{6933\,\sqrt{55}}{55}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14*(1-2*x)^(1/2)/(-4/3-2*x)+204*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+50
*(-137/10*(1-2*x)^(3/2)+297/10*(1-2*x)^(1/2))/(-6-10*x)^2-6933/55*arctanh(1/11*5
5^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50273, size = 171, normalized size = 1.45 \[ \frac{6933}{110} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - 102 \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{3105 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 13870 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 15477 \, \sqrt{-2 \, x + 1}}{75 \,{\left (2 \, x - 1\right )}^{3} + 505 \,{\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6933/110*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 102*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + (3105*(-2*x + 1)^(5/2) - 13870*(-2*x + 1)^(3/2) + 15477*sqrt(-2*x + 1))/(
75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266*x - 286)

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Fricas [A]  time = 0.222674, size = 186, normalized size = 1.58 \[ \frac{\sqrt{55}{\left (204 \, \sqrt{55} \sqrt{21}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + \sqrt{55}{\left (3105 \, x^{2} + 3830 \, x + 1178\right )} \sqrt{-2 \, x + 1} + 6933 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{110 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/110*sqrt(55)*(204*sqrt(55)*sqrt(21)*(75*x^3 + 140*x^2 + 87*x + 18)*log((3*x -
sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + sqrt(55)*(3105*x^2 + 3830*x + 1178)*sq
rt(-2*x + 1) + 6933*(75*x^3 + 140*x^2 + 87*x + 18)*log((sqrt(55)*(5*x - 8) + 55*
sqrt(-2*x + 1))/(5*x + 3)))/(75*x^3 + 140*x^2 + 87*x + 18)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215549, size = 166, normalized size = 1.41 \[ \frac{6933}{110} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - 102 \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2} - \frac{5 \,{\left (137 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 297 \, \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6933/110*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 102*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21)
+ 3*sqrt(-2*x + 1))) + 21*sqrt(-2*x + 1)/(3*x + 2) - 5/4*(137*(-2*x + 1)^(3/2)
- 297*sqrt(-2*x + 1))/(5*x + 3)^2