3.1900 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=154 \[ -\frac{46555 \sqrt{1-2 x}}{42 (5 x+3)}+\frac{6949 \sqrt{1-2 x}}{63 (3 x+2) (5 x+3)}+\frac{133 \sqrt{1-2 x}}{18 (3 x+2)^2 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)}-\frac{321161 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}}+1350 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-46555*Sqrt[1 - 2*x])/(42*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*
x)) + (133*Sqrt[1 - 2*x])/(18*(2 + 3*x)^2*(3 + 5*x)) + (6949*Sqrt[1 - 2*x])/(63*
(2 + 3*x)*(3 + 5*x)) - (321161*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21]) +
1350*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.331451, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{46555 \sqrt{1-2 x}}{42 (5 x+3)}+\frac{6949 \sqrt{1-2 x}}{63 (3 x+2) (5 x+3)}+\frac{133 \sqrt{1-2 x}}{18 (3 x+2)^2 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)}-\frac{321161 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}}+1350 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-46555*Sqrt[1 - 2*x])/(42*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*
x)) + (133*Sqrt[1 - 2*x])/(18*(2 + 3*x)^2*(3 + 5*x)) + (6949*Sqrt[1 - 2*x])/(63*
(2 + 3*x)*(3 + 5*x)) - (321161*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21]) +
1350*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 35.2434, size = 133, normalized size = 0.86 \[ - \frac{9311 \sqrt{- 2 x + 1}}{14 \left (3 x + 2\right )} - \frac{2005 \sqrt{- 2 x + 1}}{18 \left (3 x + 2\right ) \left (5 x + 3\right )} + \frac{133 \sqrt{- 2 x + 1}}{18 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} + \frac{7 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{3} \left (5 x + 3\right )} - \frac{321161 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{147} + 1350 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9311*sqrt(-2*x + 1)/(14*(3*x + 2)) - 2005*sqrt(-2*x + 1)/(18*(3*x + 2)*(5*x + 3
)) + 133*sqrt(-2*x + 1)/(18*(3*x + 2)**2*(5*x + 3)) + 7*sqrt(-2*x + 1)/(9*(3*x +
2)**3*(5*x + 3)) - 321161*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/147 + 1350*
sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.15733, size = 95, normalized size = 0.62 \[ -\frac{\sqrt{1-2 x} \left (418995 x^3+824092 x^2+539819 x+117752\right )}{14 (3 x+2)^3 (5 x+3)}-\frac{321161 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}}+1350 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(117752 + 539819*x + 824092*x^2 + 418995*x^3))/(14*(2 + 3*x)^3*(
3 + 5*x)) - (321161*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21]) + 1350*Sqrt[5
5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.02, size = 91, normalized size = 0.6 \[ 108\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{7001\, \left ( 1-2\,x \right ) ^{5/2}}{84}}-{\frac{10603\, \left ( 1-2\,x \right ) ^{3/2}}{27}}+{\frac{49973\,\sqrt{1-2\,x}}{108}} \right ) }-{\frac{321161\,\sqrt{21}}{147}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+110\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+1350\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

108*(7001/84*(1-2*x)^(5/2)-10603/27*(1-2*x)^(3/2)+49973/108*(1-2*x)^(1/2))/(-4-6
*x)^3-321161/147*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+110*(1-2*x)^(1/2)/
(-6/5-2*x)+1350*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50663, size = 197, normalized size = 1.28 \[ -675 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{321161}{294} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{418995 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 2905169 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 6712629 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 5168471 \, \sqrt{-2 \, x + 1}}{7 \,{\left (135 \,{\left (2 \, x - 1\right )}^{4} + 1242 \,{\left (2 \, x - 1\right )}^{3} + 4284 \,{\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-675*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 321161/294*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 1/7*(418995*(-2*x + 1)^(7/2) - 2905169*(-2*x + 1)^(5/2) + 6712629*(-2*
x + 1)^(3/2) - 5168471*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1)^3 + 428
4*(2*x - 1)^2 + 13132*x - 2793)

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Fricas [A]  time = 0.219039, size = 215, normalized size = 1.4 \[ \frac{\sqrt{21}{\left (9450 \, \sqrt{55} \sqrt{21}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - \sqrt{21}{\left (418995 \, x^{3} + 824092 \, x^{2} + 539819 \, x + 117752\right )} \sqrt{-2 \, x + 1} + 321161 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{294 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/294*sqrt(21)*(9450*sqrt(55)*sqrt(21)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24
)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - sqrt(21)*(418995*x^3 + 82
4092*x^2 + 539819*x + 117752)*sqrt(-2*x + 1) + 321161*(135*x^4 + 351*x^3 + 342*x
^2 + 148*x + 24)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(135*x
^4 + 351*x^3 + 342*x^2 + 148*x + 24)

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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)**4/(3+5*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213834, size = 188, normalized size = 1.22 \[ -675 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{321161}{294} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{275 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{63009 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 296884 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 349811 \, \sqrt{-2 \, x + 1}}{56 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-675*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) + 321161/294*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) - 275*sqrt(-2*x + 1)/(5*x + 3) - 1/56*(63009*(2*x - 1)^
2*sqrt(-2*x + 1) - 296884*(-2*x + 1)^(3/2) + 349811*sqrt(-2*x + 1))/(3*x + 2)^3