3.1889 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=133 \[ \frac{318643 \sqrt{1-2 x}}{1176 (3 x+2)}+\frac{13723 \sqrt{1-2 x}}{504 (3 x+2)^2}+\frac{131 \sqrt{1-2 x}}{36 (3 x+2)^3}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4}+\frac{10990843 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}-550 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4) + (131*Sqrt[1 - 2*x])/(36*(2 + 3*x)^3) + (137
23*Sqrt[1 - 2*x])/(504*(2 + 3*x)^2) + (318643*Sqrt[1 - 2*x])/(1176*(2 + 3*x)) +
(10990843*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) - 550*Sqrt[55]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.302879, antiderivative size = 133, 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 \[ \frac{318643 \sqrt{1-2 x}}{1176 (3 x+2)}+\frac{13723 \sqrt{1-2 x}}{504 (3 x+2)^2}+\frac{131 \sqrt{1-2 x}}{36 (3 x+2)^3}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4}+\frac{10990843 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}-550 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4) + (131*Sqrt[1 - 2*x])/(36*(2 + 3*x)^3) + (137
23*Sqrt[1 - 2*x])/(504*(2 + 3*x)^2) + (318643*Sqrt[1 - 2*x])/(1176*(2 + 3*x)) +
(10990843*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) - 550*Sqrt[55]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 34.7981, size = 119, normalized size = 0.89 \[ \frac{318643 \sqrt{- 2 x + 1}}{1176 \left (3 x + 2\right )} + \frac{13723 \sqrt{- 2 x + 1}}{504 \left (3 x + 2\right )^{2}} + \frac{131 \sqrt{- 2 x + 1}}{36 \left (3 x + 2\right )^{3}} + \frac{7 \sqrt{- 2 x + 1}}{12 \left (3 x + 2\right )^{4}} + \frac{10990843 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{12348} - 550 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

318643*sqrt(-2*x + 1)/(1176*(3*x + 2)) + 13723*sqrt(-2*x + 1)/(504*(3*x + 2)**2)
+ 131*sqrt(-2*x + 1)/(36*(3*x + 2)**3) + 7*sqrt(-2*x + 1)/(12*(3*x + 2)**4) + 1
0990843*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/12348 - 550*sqrt(55)*atanh(sqr
t(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.16575, size = 88, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (8603361 x^3+17494905 x^2+11868230 x+2686470\right )}{1176 (3 x+2)^4}+\frac{10990843 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}-550 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(2686470 + 11868230*x + 17494905*x^2 + 8603361*x^3))/(1176*(2 + 3
*x)^4) + (10990843*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) - 550*Sqrt[5
5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.019, size = 84, normalized size = 0.6 \[ -162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{318643\, \left ( 1-2\,x \right ) ^{7/2}}{3528}}-{\frac{2895233\, \left ( 1-2\,x \right ) ^{5/2}}{4536}}+{\frac{2923727\, \left ( 1-2\,x \right ) ^{3/2}}{1944}}-{\frac{2297099\,\sqrt{1-2\,x}}{1944}} \right ) }+{\frac{10990843\,\sqrt{21}}{12348}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-550\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-162*(318643/3528*(1-2*x)^(7/2)-2895233/4536*(1-2*x)^(5/2)+2923727/1944*(1-2*x)^
(3/2)-2297099/1944*(1-2*x)^(1/2))/(-4-6*x)^4+10990843/12348*arctanh(1/7*21^(1/2)
*(1-2*x)^(1/2))*21^(1/2)-550*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.52141, size = 197, normalized size = 1.48 \[ 275 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{10990843}{24696} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{8603361 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 60799893 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 143262623 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 112557851 \, \sqrt{-2 \, x + 1}}{588 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

275*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
10990843/24696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) - 1/588*(8603361*(-2*x + 1)^(7/2) - 60799893*(-2*x + 1)^(5/2) + 14326
2623*(-2*x + 1)^(3/2) - 112557851*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1
)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 0.23387, size = 212, normalized size = 1.59 \[ \frac{\sqrt{21}{\left (323400 \, \sqrt{55} \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (8603361 \, x^{3} + 17494905 \, x^{2} + 11868230 \, x + 2686470\right )} \sqrt{-2 \, x + 1} + 10990843 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{24696 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24696*sqrt(21)*(323400*sqrt(55)*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x +
16)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*(8603361*x^3 +
17494905*x^2 + 11868230*x + 2686470)*sqrt(-2*x + 1) + 10990843*(81*x^4 + 216*x^
3 + 216*x^2 + 96*x + 16)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2))
)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217053, size = 188, normalized size = 1.41 \[ 275 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{10990843}{24696} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{8603361 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 60799893 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 143262623 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 112557851 \, \sqrt{-2 \, x + 1}}{9408 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

275*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 10990843/24696*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) + 1/9408*(8603361*(2*x - 1)^3*sqrt(-2*x + 1) + 60799
893*(2*x - 1)^2*sqrt(-2*x + 1) - 143262623*(-2*x + 1)^(3/2) + 112557851*sqrt(-2*
x + 1))/(3*x + 2)^4