3.1982 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=172 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}+\frac{113875 \sqrt{1-2 x}}{6 (5 x+3)}+\frac{1256 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)^2}+\frac{581 \sqrt{1-2 x}}{27 (3 x+2)^2 (5 x+3)^2}-\frac{169975 \sqrt{1-2 x}}{54 (5 x+3)^2}+\frac{785570 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}}-23115 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-169975*Sqrt[1 - 2*x])/(54*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3
+ 5*x)^2) + (581*Sqrt[1 - 2*x])/(27*(2 + 3*x)^2*(3 + 5*x)^2) + (1256*Sqrt[1 - 2
*x])/(3*(2 + 3*x)*(3 + 5*x)^2) + (113875*Sqrt[1 - 2*x])/(6*(3 + 5*x)) + (785570*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 23115*Sqrt[55]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]]

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Rubi [A]  time = 0.387671, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}+\frac{113875 \sqrt{1-2 x}}{6 (5 x+3)}+\frac{1256 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)^2}+\frac{581 \sqrt{1-2 x}}{27 (3 x+2)^2 (5 x+3)^2}-\frac{169975 \sqrt{1-2 x}}{54 (5 x+3)^2}+\frac{785570 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}}-23115 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-169975*Sqrt[1 - 2*x])/(54*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3
+ 5*x)^2) + (581*Sqrt[1 - 2*x])/(27*(2 + 3*x)^2*(3 + 5*x)^2) + (1256*Sqrt[1 - 2
*x])/(3*(2 + 3*x)*(3 + 5*x)^2) + (113875*Sqrt[1 - 2*x])/(6*(3 + 5*x)) + (785570*
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 23115*Sqrt[55]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 41.0449, size = 156, normalized size = 0.91 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{9 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{2}} + \frac{113875 \sqrt{- 2 x + 1}}{6 \left (5 x + 3\right )} - \frac{169975 \sqrt{- 2 x + 1}}{54 \left (5 x + 3\right )^{2}} + \frac{1256 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{581 \sqrt{- 2 x + 1}}{27 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} + \frac{785570 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} - 23115 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(9*(3*x + 2)**3*(5*x + 3)**2) + 113875*sqrt(-2*x + 1)/(6*(5*
x + 3)) - 169975*sqrt(-2*x + 1)/(54*(5*x + 3)**2) + 1256*sqrt(-2*x + 1)/(3*(3*x
+ 2)*(5*x + 3)**2) + 581*sqrt(-2*x + 1)/(27*(3*x + 2)**2*(5*x + 3)**2) + 785570*
sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21 - 23115*sqrt(55)*atanh(sqrt(55)*sqr
t(-2*x + 1)/11)

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Mathematica [A]  time = 0.176157, size = 98, normalized size = 0.57 \[ \frac{\sqrt{1-2 x} \left (5124375 x^4+13153400 x^3+12649336 x^2+5401374 x+864074\right )}{2 (3 x+2)^3 (5 x+3)^2}+\frac{785570 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}}-23115 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(864074 + 5401374*x + 12649336*x^2 + 13153400*x^3 + 5124375*x^4))
/(2*(2 + 3*x)^3*(3 + 5*x)^2) + (785570*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21
] - 23115*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.021, size = 103, normalized size = 0.6 \[ -108\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{6883\, \left ( 1-2\,x \right ) ^{5/2}}{6}}-{\frac{145600\, \left ( 1-2\,x \right ) ^{3/2}}{27}}+{\frac{342265\,\sqrt{1-2\,x}}{54}} \right ) }+{\frac{785570\,\sqrt{21}}{21}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+5500\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{273\, \left ( 1-2\,x \right ) ^{3/2}}{20}}+{\frac{2981\,\sqrt{1-2\,x}}{100}} \right ) }-23115\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-108*(6883/6*(1-2*x)^(5/2)-145600/27*(1-2*x)^(3/2)+342265/54*(1-2*x)^(1/2))/(-4-
6*x)^3+785570/21*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+5500*(-273/20*(1-2
*x)^(3/2)+2981/100*(1-2*x)^(1/2))/(-6-10*x)^2-23115*arctanh(1/11*55^(1/2)*(1-2*x
)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50856, size = 220, normalized size = 1.28 \[ \frac{23115}{2} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{392785}{21} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{5124375 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 46804300 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 160263994 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 243823580 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 139064695 \, \sqrt{-2 \, x + 1}}{675 \,{\left (2 \, x - 1\right )}^{5} + 7695 \,{\left (2 \, x - 1\right )}^{4} + 35082 \,{\left (2 \, x - 1\right )}^{3} + 79954 \,{\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

23115/2*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 392785/21*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + (5124375*(-2*x + 1)^(9/2) - 46804300*(-2*x + 1)^(7/2) + 160263994*(-
2*x + 1)^(5/2) - 243823580*(-2*x + 1)^(3/2) + 139064695*sqrt(-2*x + 1))/(675*(2*
x - 1)^5 + 7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*x - 1)^2 + 182182*x -
49588)

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Fricas [A]  time = 0.248025, size = 239, normalized size = 1.39 \[ \frac{\sqrt{21}{\left (23115 \, \sqrt{55} \sqrt{21}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (5124375 \, x^{4} + 13153400 \, x^{3} + 12649336 \, x^{2} + 5401374 \, x + 864074\right )} \sqrt{-2 \, x + 1} + 785570 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{42 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*sqrt(21)*(23115*sqrt(55)*sqrt(21)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2
+ 564*x + 72)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*(51
24375*x^4 + 13153400*x^3 + 12649336*x^2 + 5401374*x + 864074)*sqrt(-2*x + 1) + 7
85570*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((sqrt(21)*(3*x
- 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2
+ 564*x + 72)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216529, size = 204, normalized size = 1.19 \[ \frac{23115}{2} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{392785}{21} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{55 \,{\left (1365 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2981 \, \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (5 \, x + 3\right )}^{2}} + \frac{61947 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 291200 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 342265 \, \sqrt{-2 \, x + 1}}{4 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

23115/2*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) - 392785/21*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) - 55/4*(1365*(-2*x + 1)^(3/2) - 2981*sqrt(-2*x + 1))/
(5*x + 3)^2 + 1/4*(61947*(2*x - 1)^2*sqrt(-2*x + 1) - 291200*(-2*x + 1)^(3/2) +
342265*sqrt(-2*x + 1))/(3*x + 2)^3