3.1972 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^2} \, dx\)

Optimal. Leaf size=181 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}+\frac{288770 \sqrt{1-2 x}}{189 (3 x+2) (5 x+3)}+\frac{22109 \sqrt{1-2 x}}{216 (3 x+2)^2 (5 x+3)}+\frac{287 \sqrt{1-2 x}}{27 (3 x+2)^3 (5 x+3)}-\frac{7738475 \sqrt{1-2 x}}{504 (5 x+3)}-\frac{53384095 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}+18700 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-7738475*Sqrt[1 - 2*x])/(504*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(
3 + 5*x)) + (287*Sqrt[1 - 2*x])/(27*(2 + 3*x)^3*(3 + 5*x)) + (22109*Sqrt[1 - 2*x
])/(216*(2 + 3*x)^2*(3 + 5*x)) + (288770*Sqrt[1 - 2*x])/(189*(2 + 3*x)*(3 + 5*x)
) - (53384095*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) + 18700*Sqrt[55]*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.38267, antiderivative size = 181, 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 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)}+\frac{288770 \sqrt{1-2 x}}{189 (3 x+2) (5 x+3)}+\frac{22109 \sqrt{1-2 x}}{216 (3 x+2)^2 (5 x+3)}+\frac{287 \sqrt{1-2 x}}{27 (3 x+2)^3 (5 x+3)}-\frac{7738475 \sqrt{1-2 x}}{504 (5 x+3)}-\frac{53384095 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}+18700 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-7738475*Sqrt[1 - 2*x])/(504*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(
3 + 5*x)) + (287*Sqrt[1 - 2*x])/(27*(2 + 3*x)^3*(3 + 5*x)) + (22109*Sqrt[1 - 2*x
])/(216*(2 + 3*x)^2*(3 + 5*x)) + (288770*Sqrt[1 - 2*x])/(189*(2 + 3*x)*(3 + 5*x)
) - (53384095*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) + 18700*Sqrt[55]*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 41.8225, size = 156, normalized size = 0.86 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{12 \left (3 x + 2\right )^{4} \left (5 x + 3\right )} - \frac{7738475 \sqrt{- 2 x + 1}}{504 \left (5 x + 3\right )} + \frac{288770 \sqrt{- 2 x + 1}}{189 \left (3 x + 2\right ) \left (5 x + 3\right )} + \frac{22109 \sqrt{- 2 x + 1}}{216 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} + \frac{287 \sqrt{- 2 x + 1}}{27 \left (3 x + 2\right )^{3} \left (5 x + 3\right )} - \frac{53384095 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1764} + 18700 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

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)**2,x)

[Out]

7*(-2*x + 1)**(3/2)/(12*(3*x + 2)**4*(5*x + 3)) - 7738475*sqrt(-2*x + 1)/(504*(5
*x + 3)) + 288770*sqrt(-2*x + 1)/(189*(3*x + 2)*(5*x + 3)) + 22109*sqrt(-2*x + 1
)/(216*(3*x + 2)**2*(5*x + 3)) + 287*sqrt(-2*x + 1)/(27*(3*x + 2)**3*(5*x + 3))
- 53384095*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/1764 + 18700*sqrt(55)*atanh
(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.186144, size = 100, normalized size = 0.55 \[ -\frac{\sqrt{1-2 x} \left (208938825 x^4+550239720 x^3+543154477 x^2+238179048 x+39145938\right )}{168 (3 x+2)^4 (5 x+3)}-\frac{53384095 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}+18700 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(39145938 + 238179048*x + 543154477*x^2 + 550239720*x^3 + 208938
825*x^4))/(168*(2 + 3*x)^4*(3 + 5*x)) - (53384095*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(84*Sqrt[21]) + 18700*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.021, size = 100, normalized size = 0.6 \[ 162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{1242775\, \left ( 1-2\,x \right ) ^{7/2}}{504}}-{\frac{11266013\, \left ( 1-2\,x \right ) ^{5/2}}{648}}+{\frac{79444085\, \left ( 1-2\,x \right ) ^{3/2}}{1944}}-{\frac{62254745\,\sqrt{1-2\,x}}{1944}} \right ) }-{\frac{53384095\,\sqrt{21}}{1764}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+1210\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+18700\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

162*(1242775/504*(1-2*x)^(7/2)-11266013/648*(1-2*x)^(5/2)[phone]/1944*(1-2*x)^
(3/2)-62254745/1944*(1-2*x)^(1/2))/(-4-6*x)^4-53384095/1764*arctanh(1/7*21^(1/2)
*(1-2*x)^(1/2))*21^(1/2)+1210*(1-2*x)^(1/2)/(-6/5-2*x)+18700*arctanh(1/11*55^(1/
2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.56327, size = 221, normalized size = 1.22 \[ -9350 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{53384095}{3528} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{208938825 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 1936234740 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 6727689178 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 10387861820 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 6013803565 \, \sqrt{-2 \, x + 1}}{84 \,{\left (405 \,{\left (2 \, x - 1\right )}^{5} + 4671 \,{\left (2 \, x - 1\right )}^{4} + 21546 \,{\left (2 \, x - 1\right )}^{3} + 49686 \,{\left (2 \, x - 1\right )}^{2} + 114562 \, x - 30870\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9350*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 53384095/3528*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 1/84*(208938825*(-2*x + 1)^(9/2) - 1936234740*(-2*x + 1)^(7/2) + 6
727689178*(-2*x + 1)^(5/2) - 10387861820*(-2*x + 1)^(3/2) + 6013803565*sqrt(-2*x
+ 1))/(405*(2*x - 1)^5 + 4671*(2*x - 1)^4 + 21546*(2*x - 1)^3 + 49686*(2*x - 1)
^2 + 114562*x - 30870)

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Fricas [A]  time = 0.219992, size = 242, normalized size = 1.34 \[ \frac{\sqrt{21}{\left (1570800 \, \sqrt{55} \sqrt{21}{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - \sqrt{21}{\left (208938825 \, x^{4} + 550239720 \, x^{3} + 543154477 \, x^{2} + 238179048 \, x + 39145938\right )} \sqrt{-2 \, x + 1} + 53384095 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{3528 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3528*sqrt(21)*(1570800*sqrt(55)*sqrt(21)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128
*x^2 + 368*x + 48)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - sqrt(21)
*(208938825*x^4 + 550239720*x^3 + 543154477*x^2 + 238179048*x + 39145938)*sqrt(-
2*x + 1) + 53384095*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log(
(sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(405*x^5 + 1323*x^4 + 1728*
x^3 + 1128*x^2 + 368*x + 48)

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22027, size = 209, normalized size = 1.15 \[ -9350 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{53384095}{3528} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{3025 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{33554925 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 236586273 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 556108595 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 435783215 \, \sqrt{-2 \, x + 1}}{1344 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9350*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 53384095/3528*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) - 3025*sqrt(-2*x + 1)/(5*x + 3) - 1/1344*(33554925*
(2*x - 1)^3*sqrt(-2*x + 1) + 236586273*(2*x - 1)^2*sqrt(-2*x + 1) - 556108595*(-
2*x + 1)^(3/2) + 435783215*sqrt(-2*x + 1))/(3*x + 2)^4