3.1983 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx\)

Optimal. Leaf size=201 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}+\frac{23680975 \sqrt{1-2 x}}{168 (5 x+3)}+\frac{522385 \sqrt{1-2 x}}{168 (3 x+2) (5 x+3)^2}+\frac{11243 \sqrt{1-2 x}}{72 (3 x+2)^2 (5 x+3)^2}+\frac{1393 \sqrt{1-2 x}}{108 (3 x+2)^3 (5 x+3)^2}-\frac{8836825 \sqrt{1-2 x}}{378 (5 x+3)^2}+\frac{163363895 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28 \sqrt{21}}-171675 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-8836825*Sqrt[1 - 2*x])/(378*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4
*(3 + 5*x)^2) + (1393*Sqrt[1 - 2*x])/(108*(2 + 3*x)^3*(3 + 5*x)^2) + (11243*Sqrt
[1 - 2*x])/(72*(2 + 3*x)^2*(3 + 5*x)^2) + (522385*Sqrt[1 - 2*x])/(168*(2 + 3*x)*
(3 + 5*x)^2) + (23680975*Sqrt[1 - 2*x])/(168*(3 + 5*x)) + (163363895*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/(28*Sqrt[21]) - 171675*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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Rubi [A]  time = 0.45996, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}+\frac{23680975 \sqrt{1-2 x}}{168 (5 x+3)}+\frac{522385 \sqrt{1-2 x}}{168 (3 x+2) (5 x+3)^2}+\frac{11243 \sqrt{1-2 x}}{72 (3 x+2)^2 (5 x+3)^2}+\frac{1393 \sqrt{1-2 x}}{108 (3 x+2)^3 (5 x+3)^2}-\frac{8836825 \sqrt{1-2 x}}{378 (5 x+3)^2}+\frac{163363895 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28 \sqrt{21}}-171675 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-8836825*Sqrt[1 - 2*x])/(378*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4
*(3 + 5*x)^2) + (1393*Sqrt[1 - 2*x])/(108*(2 + 3*x)^3*(3 + 5*x)^2) + (11243*Sqrt
[1 - 2*x])/(72*(2 + 3*x)^2*(3 + 5*x)^2) + (522385*Sqrt[1 - 2*x])/(168*(2 + 3*x)*
(3 + 5*x)^2) + (23680975*Sqrt[1 - 2*x])/(168*(3 + 5*x)) + (163363895*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/(28*Sqrt[21]) - 171675*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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Rubi in Sympy [A]  time = 48.5419, size = 182, normalized size = 0.91 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{12 \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{2}} + \frac{23680975 \sqrt{- 2 x + 1}}{168 \left (5 x + 3\right )} - \frac{8836825 \sqrt{- 2 x + 1}}{378 \left (5 x + 3\right )^{2}} + \frac{522385 \sqrt{- 2 x + 1}}{168 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{11243 \sqrt{- 2 x + 1}}{72 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} + \frac{1393 \sqrt{- 2 x + 1}}{108 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{2}} + \frac{163363895 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{588} - 171675 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(12*(3*x + 2)**4*(5*x + 3)**2) + 23680975*sqrt(-2*x + 1)/(16
8*(5*x + 3)) - 8836825*sqrt(-2*x + 1)/(378*(5*x + 3)**2) + 522385*sqrt(-2*x + 1)
/(168*(3*x + 2)*(5*x + 3)**2) + 11243*sqrt(-2*x + 1)/(72*(3*x + 2)**2*(5*x + 3)*
*2) + 1393*sqrt(-2*x + 1)/(108*(3*x + 2)**3*(5*x + 3)**2) + 163363895*sqrt(21)*a
tanh(sqrt(21)*sqrt(-2*x + 1)/7)/588 - 171675*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x +
1)/11)

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Mathematica [A]  time = 0.215936, size = 105, normalized size = 0.52 \[ \frac{\sqrt{1-2 x} \left (3196931625 x^5+10337268075 x^4+13362164665 x^3+8630749831 x^2+2785562634 x+359378534\right )}{56 (3 x+2)^4 (5 x+3)^2}+\frac{163363895 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28 \sqrt{21}}-171675 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(359378534 + 2785562634*x + 8630749831*x^2 + 13362164665*x^3 + 10
337268075*x^4 + 3196931625*x^5))/(56*(2 + 3*x)^4*(3 + 5*x)^2) + (163363895*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(28*Sqrt[21]) - 171675*Sqrt[55]*ArcTanh[Sqrt[5/11]*
Sqrt[1 - 2*x]]

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Maple [A]  time = 0.023, size = 112, normalized size = 0.6 \[ -162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{3170015\, \left ( 1-2\,x \right ) ^{7/2}}{168}}-{\frac{28695733\, \left ( 1-2\,x \right ) ^{5/2}}{216}}+{\frac{202051885\, \left ( 1-2\,x \right ) ^{3/2}}{648}}-{\frac{52696315\,\sqrt{1-2\,x}}{216}} \right ) }+{\frac{163363895\,\sqrt{21}}{588}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+13750\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{339\, \left ( 1-2\,x \right ) ^{3/2}}{10}}+{\frac{3707\,\sqrt{1-2\,x}}{50}} \right ) }-171675\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-162*(3170015/168*(1-2*x)^(7/2)-28695733/216*(1-2*x)^(5/2)[phone]/648*(1-2*x)
^(3/2)-52696315/216*(1-2*x)^(1/2))/(-4-6*x)^4+163363895/588*arctanh(1/7*21^(1/2)
*(1-2*x)^(1/2))*21^(1/2)+13750*(-339/10*(1-2*x)^(3/2)+3707/50*(1-2*x)^(1/2))/(-6
-10*x)^2-171675*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.52273, size = 246, normalized size = 1.22 \[ \frac{171675}{2} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{163363895}{1176} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3196931625 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 36659194275 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 168116119510 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 385408507778 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441689778145 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 202435240315 \, \sqrt{-2 \, x + 1}}{28 \,{\left (2025 \,{\left (2 \, x - 1\right )}^{6} + 27810 \,{\left (2 \, x - 1\right )}^{5} + 159111 \,{\left (2 \, x - 1\right )}^{4} + 485436 \,{\left (2 \, x - 1\right )}^{3} + 832951 \,{\left (2 \, x - 1\right )}^{2} + 1524292 \, x - 471625\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

171675/2*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 163363895/1176*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) - 1/28*(3196931625*(-2*x + 1)^(11/2) - 36659194275*(-2*x + 1)^(9
/2) + 168116119510*(-2*x + 1)^(7/2) - 385408507778*(-2*x + 1)^(5/2) + 4416897781
45*(-2*x + 1)^(3/2) - 202435240315*sqrt(-2*x + 1))/(2025*(2*x - 1)^6 + 27810*(2*
x - 1)^5 + 159111*(2*x - 1)^4 + 485436*(2*x - 1)^3 + 832951*(2*x - 1)^2 + 152429
2*x - 471625)

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Fricas [A]  time = 0.243986, size = 266, normalized size = 1.32 \[ \frac{\sqrt{21}{\left (4806900 \, \sqrt{55} \sqrt{21}{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (3196931625 \, x^{5} + 10337268075 \, x^{4} + 13362164665 \, x^{3} + 8630749831 \, x^{2} + 2785562634 \, x + 359378534\right )} \sqrt{-2 \, x + 1} + 163363895 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1176 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1176*sqrt(21)*(4806900*sqrt(55)*sqrt(21)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10
824*x^3 + 5224*x^2 + 1344*x + 144)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x
+ 3)) + sqrt(21)*(3196931625*x^5 + 10337268075*x^4 + 13362164665*x^3 + 863074983
1*x^2 + 2785562634*x + 359378534)*sqrt(-2*x + 1) + 163363895*(2025*x^6 + 7830*x^
5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log((sqrt(21)*(3*x - 5) - 2
1*sqrt(-2*x + 1))/(3*x + 2)))/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 522
4*x^2 + 1344*x + 144)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220327, size = 225, normalized size = 1.12 \[ \frac{171675}{2} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{163363895}{1176} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{275 \,{\left (1695 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3707 \, \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (5 \, x + 3\right )}^{2}} + \frac{85590405 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 602610393 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1414363195 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1106622615 \, \sqrt{-2 \, x + 1}}{448 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

171675/2*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 163363895/1176*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1)
)/(sqrt(21) + 3*sqrt(-2*x + 1))) - 275/4*(1695*(-2*x + 1)^(3/2) - 3707*sqrt(-2*x
+ 1))/(5*x + 3)^2 + 1/448*(85590405*(2*x - 1)^3*sqrt(-2*x + 1) + 602610393*(2*x
- 1)^2*sqrt(-2*x + 1) - 1414363195*(-2*x + 1)^(3/2) + 1106622615*sqrt(-2*x + 1)
)/(3*x + 2)^4